GCSE Maths Number Basics (Higher): worked examples and questions with answers

Key words. Place value means the value of a digit because of the column it is in. Units (ones) are the whole numbers; tenths and hundredths are the parts after the decimal point.

Method. The value of a digit depends on its column. Find the column, then multiply the digit by what that column is worth.

1. Write the number in a place value table. Each column is worth ten times the column on its right. The digits sit in columns, and the decimal point separates the whole numbers from the parts.

   thousands	hundreds	tens	units
   5	3	2	4

2. Find the column of the digit 3. Reading the headings, the 3 is in the hundreds column. One in this column is worth 100.
   The hundreds column is worth 100.
3. Multiply the digit by the column value. There are 3 lots of 100.
   3 × 100 = 300

Check. In words, the digit 3 stands for 3 hundreds, which is 300. ✓

Watch out. A digit is worth its column value, not just the digit itself. The 7 in 4 700 is worth 700, not 7.

Key words. Place value means the value of a digit because of the column it is in. Units (ones) are the whole numbers; tenths and hundredths are the parts after the decimal point.

Method. The value of a digit depends on its column. Find the column, then multiply the digit by what that column is worth.

1. Write the number in a place value table. Each column is worth ten times the column on its right. The digits sit in columns, and the decimal point separates the whole numbers from the parts.

   hundred thousands	ten thousands	thousands	hundreds	tens	units
   7	0	6	1	3	5

2. Find the column of the digit 7. Reading the headings, the 7 is in the hundred thousands column. One in this column is worth 100 000.
   The hundred thousands column is worth 100 000.
3. Multiply the digit by the column value. There are 7 lots of 100 000.
   7 × 100 000 = 700 000

Check. In words, the digit 7 stands for 7 hundred thousands, which is 700 000. ✓

Watch out. A digit is worth its column value, not just the digit itself. The 7 in 4 700 is worth 700, not 7.

Key words. Place value means the value of a digit because of the column it is in. Units (ones) are the whole numbers; tenths and hundredths are the parts after the decimal point.

Method. The value of a digit depends on its column. Find the column, then multiply the digit by what that column is worth.

1. Write the number in a place value table. Each column is worth ten times the column on its right. The digits sit in columns, and the decimal point separates the whole numbers from the parts.

   units		tenths	hundredths
   1	.	8	6

2. Find the column of the digit 8. Reading the headings, the 8 is in the tenths column. One in this column is worth 0.1.
   The tenths column is worth 0.1.
3. Multiply the digit by the column value. There are 8 lots of 0.1.
   8 × 0.1 = 0.8

Check. In words, the digit 8 stands for 8 tenths, which is 0.8. ✓

Watch out. A digit is worth its column value, not just the digit itself. The 7 in 4 700 is worth 700, not 7.

Key words. Place value means the value of a digit because of the column it is in. Units (ones) are the whole numbers; tenths and hundredths are the parts after the decimal point.

Method. The value of a digit depends on its column. Find the column, then multiply the digit by what that column is worth.

1. Write the number in a place value table. Each column is worth ten times the column on its right. The digits sit in columns, and the decimal point separates the whole numbers from the parts.

   tens	units		tenths	hundredths	thousandths
   1	2	.	0	4	7

2. Find the column of the digit 7. Reading the headings, the 7 is in the thousandths column. One in this column is worth 0.001.
   The thousandths column is worth 0.001.
3. Multiply the digit by the column value. There are 7 lots of 0.001.
   7 × 0.001 = 0.007

Check. In words, the digit 7 stands for 7 thousandths, which is 0.007. ✓

Watch out. A digit is worth its column value, not just the digit itself. The 7 in 4 700 is worth 700, not 7.

Key words. Place value means the value of a digit because of the column it is in. Units (ones) are the whole numbers; tenths and hundredths are the parts after the decimal point.

Method. The value of a digit depends on its column. Find the column, then multiply the digit by what that column is worth.

1. Write the number in a place value table. Each column is worth ten times the column on its right. The digits sit in columns, and the decimal point separates the whole numbers from the parts.

   millions	hundred thousands	ten thousands	thousands	hundreds	tens	units
   9	0	5	2	3	1	0

2. Find the column of the digit 2. Reading the headings, the 2 is in the thousands column. One in this column is worth 1000.
   The thousands column is worth 1000.
3. Multiply the digit by the column value. There are 2 lots of 1000.
   2 × 1000 = 2000

Check. In words, the digit 2 stands for 2 thousands, which is 2000. ✓

Watch out. A digit is worth its column value, not just the digit itself. The 7 in 4 700 is worth 700, not 7.

Key words. Place value means the value of a digit because of the column it is in. Units (ones) are the whole numbers; tenths and hundredths are the parts after the decimal point.

Method. The value of a digit depends on its column. Find the column, then multiply the digit by what that column is worth.

1. Write the number in a place value table. Each column is worth ten times the column on its right. The digits sit in columns, and the decimal point separates the whole numbers from the parts.

   units		tenths	hundredths
   0	.	0	3

2. Find the column of the digit 0. Reading the headings, the 0 is in the tenths column. One in this column is worth 0.1.
   The tenths column is worth 0.1.
3. Multiply the digit by the column value. There are 0 lots of 0.1.
   0 × 0.1 = 0

Check. In words, the digit 0 stands for 0 tenths, which is 0. ✓

Watch out. A digit is worth its column value, not just the digit itself. The 7 in 4 700 is worth 700, not 7.

Method. Read the number in groups of three digits (millions, thousands, then the rest) and say each group in words.

1. Split the whole number into groups of three digits from the right. Groups are read as millions, thousands and then the last three digits. The gaps in the number show the groups.
   6 thousand | 204
2. Say each group in words. Each group is a number from 0 to 999 followed by its name (million or thousand). Use "and" before the tens and units, e.g. one hundred and five.
   6 thousand = six thousand
   204 = two hundred and four
3. Join the groups. Put the groups together in order, with "and" before the last part if it is less than 100.
   six thousand two hundred and four

Check. Say it out loud: does "six thousand two hundred and four" sound like 6204? ✓

Watch out. Do not write "and" between every part. We say "four thousand and fifty-six", but "four thousand two hundred and six".

Method. Read the number in groups of three digits (millions, thousands, then the rest) and say each group in words.

1. Split the whole number into groups of three digits from the right. Groups are read as millions, thousands and then the last three digits. The gaps in the number show the groups.
   58 thousand | 30
2. Say each group in words. Each group is a number from 0 to 999 followed by its name (million or thousand). Use "and" before the tens and units, e.g. one hundred and five.
   58 thousand = fifty-eight thousand
   30 = thirty
3. Join the groups. Put the groups together in order, with "and" before the last part if it is less than 100.
   fifty-eight thousand and thirty

Check. Say it out loud: does "fifty-eight thousand and thirty" sound like 58 030? ✓

Watch out. Do not write "and" between every part. We say "four thousand and fifty-six", but "four thousand two hundred and six".

Method. Read the number in groups of three digits (millions, thousands, then the rest) and say each group in words.

1. Split the whole number into groups of three digits from the right. Groups are read as millions, thousands and then the last three digits. The gaps in the number show the groups.
   2 million | 100 thousand | 500
2. Say each group in words. Each group is a number from 0 to 999 followed by its name (million or thousand). Use "and" before the tens and units, e.g. one hundred and five.
   2 million = two million
   100 thousand = one hundred thousand
   500 = five hundred
3. Join the groups. Put the groups together in order, with "and" before the last part if it is less than 100.
   two million one hundred thousand five hundred

Check. Say it out loud: does "two million one hundred thousand five hundred" sound like 2 100 500? ✓

Watch out. Do not write "and" between every part. We say "four thousand and fifty-six", but "four thousand two hundred and six".

Method. Read the number in groups of three digits (millions, thousands, then the rest) and say each group in words.

1. Split the whole number into groups of three digits from the right. Groups are read as millions, thousands and then the last three digits. The gaps in the number show the groups.
   4
2. Say each group in words. Each group is a number from 0 to 999 followed by its name (million or thousand). Use "and" before the tens and units, e.g. one hundred and five.
   4 = four
3. Join the groups. Put the groups together in order, with "and" before the last part if it is less than 100. After the decimal point, read each digit on its own.
   four point zero seven

Check. Say it out loud: does "four point zero seven" sound like 4.07? ✓

Watch out. Do not write "and" between every part. We say "four thousand and fifty-six", but "four thousand two hundred and six".

Method. Write each group of the words as digits, in groups of three, then join them.

1. Pick out the groups in the words. Millions, thousands and units are each said as a number from 1 to 999, followed by the group name.
   nine thousand → 9 000
   fifteen → 15
2. Write the three digits for the last group, using 0 as a place holder. Every group after the first must have three digits, so a missing tens or hundreds is written as 0. For example "five" in the last group is 005.
   9   015
3. Write the number. Put the groups side by side.
   9015

Check. Read it back: 9015 is "nine thousand and fifteen". ✓

Watch out. Use a 0 in an empty column. "Forty thousand and six" is 40 006, not 406.

Method. Write each group of the words as digits, in groups of three, then join them.

1. Pick out the groups in the words. Millions, thousands and units are each said as a number from 1 to 999, followed by the group name.
   forty thousand → 40 000
   six → 6
2. Write the three digits for the last group, using 0 as a place holder. Every group after the first must have three digits, so a missing tens or hundreds is written as 0. For example "five" in the last group is 005.
   40   006
3. Write the number. Put the groups side by side.
   40 006

Check. Read it back: 40 006 is "forty thousand and six". ✓

Watch out. Use a 0 in an empty column. "Forty thousand and six" is 40 006, not 406.

Method. Write each group of the words as digits, in groups of three, then join them.

1. Pick out the groups in the words. Millions, thousands and units are each said as a number from 1 to 999, followed by the group name.
   three hundred and eighty thousand → 380 000
   two hundred and fifty → 250
2. Write the three digits for the last group, using 0 as a place holder. Every group after the first must have three digits, so a missing tens or hundreds is written as 0. For example "five" in the last group is 005.
   380   250
3. Write the number. Put the groups side by side.
   380 250

Check. Read it back: 380 250 is "three hundred and eighty thousand two hundred and fifty". ✓

Watch out. Use a 0 in an empty column. "Forty thousand and six" is 40 006, not 406.

Method. Write each group of the words as digits, in groups of three, then join them.

1. Pick out the groups in the words. Millions, thousands and units are each said as a number from 1 to 999, followed by the group name.
   two million → 2 000 000
   seven thousand → 7 000
2. Write the three digits for the last group, using 0 as a place holder. Every group after the first must have three digits, so a missing tens or hundreds is written as 0. For example "five" in the last group is 005.
   2   007   000
3. Write the number. Put the groups side by side.
   2 007 000

Check. Read it back: 2 007 000 is "two million seven thousand". ✓

Watch out. Use a 0 in an empty column. "Forty thousand and six" is 40 006, not 406.

Method. Compare numbers column by column starting from the biggest column on the left. Make decimals the same length first.

1. Give both numbers the same number of decimal places by adding zeros on the end. Adding a zero after the last decimal digit does not change the value, but it makes the columns match so we can compare digit by digit.
   3.07 = 3.07
   3.7 = 3.70
2. Compare the digits from the left, one column at a time. The first column where the digits are different decides which number is bigger. Compare whole numbers first, then tenths, then hundredths.
   3.07 and 3.70
3. Find the first column where the digits are different. In that column one number has 0 and the other has 7.
   0 < 7, so 3.07 < 3.7

Check. Subtract: 3.7 − 3.07 = 0.63. A positive result confirms which is bigger. ✓

Watch out. Do not compare decimals by their length. 0.5 is bigger than 0.45, even though 45 is bigger than 5.

Method. Compare numbers column by column starting from the biggest column on the left. Make decimals the same length first.

1. Give both numbers the same number of decimal places by adding zeros on the end. Adding a zero after the last decimal digit does not change the value, but it makes the columns match so we can compare digit by digit.
   0.8 = 0.8
   0.8 = 0.8
2. Compare the digits from the left, one column at a time. The first column where the digits are different decides which number is bigger. Compare whole numbers first, then tenths, then hundredths.
   0.8 and 0.8
3. Decide. All the digits match.
   0.8 = 0.8

Check. Both numbers have the same digits in every column.

Watch out. Do not compare decimals by their length. 0.5 is bigger than 0.45, even though 45 is bigger than 5.

Method. Compare numbers column by column starting from the biggest column on the left. Make decimals the same length first.

1. Give both numbers the same number of decimal places by adding zeros on the end. Adding a zero after the last decimal digit does not change the value, but it makes the columns match so we can compare digit by digit.
   12.5 = 12.50
   12.45 = 12.45
2. Compare the digits from the left, one column at a time. The first column where the digits are different decides which number is bigger. Compare whole numbers first, then tenths, then hundredths.
   12.50 and 12.45
3. Find the first column where the digits are different. In that column one number has 5 and the other has 4.
   5 > 4, so 12.5 > 12.45

Check. Subtract: 12.5 − 12.45 = 0.05. A positive result confirms which is bigger. ✓

Watch out. Do not compare decimals by their length. 0.5 is bigger than 0.45, even though 45 is bigger than 5.

Method. Compare numbers column by column starting from the biggest column on the left. Make decimals the same length first.

1. Compare the number of digits. A whole number with more digits is bigger, because its first digit is in a higher column.
   4509 has 4 digits; 4590 has 4 digits.
2. Compare the digits from the left. The first column where they differ decides it.
   4509 and 4590
3. Find the first column where the digits are different. In that column one number has 0 and the other has 9.
   0 < 9, so 4509 < 4590

Check. Subtract: 4590 − 4509 = 81. A positive result confirms which is bigger. ✓

Watch out. Do not compare decimals by their length. 0.5 is bigger than 0.45, even though 45 is bigger than 5.

Method. Compare numbers column by column starting from the biggest column on the left. Make decimals the same length first.

1. Give both numbers the same number of decimal places by adding zeros on the end. Adding a zero after the last decimal digit does not change the value, but it makes the columns match so we can compare digit by digit.
   6.05 = 6.050
   6.005 = 6.005
2. Compare the digits from the left, one column at a time. The first column where the digits are different decides which number is bigger. Compare whole numbers first, then tenths, then hundredths.
   6.050 and 6.005
3. Find the first column where the digits are different. In that column one number has 5 and the other has 0.
   5 > 0, so 6.05 > 6.005

Check. Subtract: 6.05 − 6.005 = 0.045. A positive result confirms which is bigger. ✓

Watch out. Do not compare decimals by their length. 0.5 is bigger than 0.45, even though 45 is bigger than 5.

Method. Compare numbers column by column starting from the biggest column on the left. Make decimals the same length first.

1. Give both numbers the same number of decimal places by adding zeros on the end. Adding a zero after the last decimal digit does not change the value, but it makes the columns match so we can compare digit by digit.
   0.31 = 0.310
   0.299 = 0.299
2. Compare the digits from the left, one column at a time. The first column where the digits are different decides which number is bigger. Compare whole numbers first, then tenths, then hundredths.
   0.310 and 0.299
3. Find the first column where the digits are different. In that column one number has 3 and the other has 2.
   3 > 2, so 0.31 > 0.299

Check. Subtract: 0.31 − 0.299 = 0.011. A positive result confirms which is bigger. ✓

Watch out. Do not compare decimals by their length. 0.5 is bigger than 0.45, even though 45 is bigger than 5.

Method. Make the numbers easy to compare, then order them smallest to largest.

1. Count the digits in each number. A whole number with more digits is bigger, because its first digit is in a higher column.
   4065: 4 digits, 4506: 4 digits, 4056: 4 digits, 4650: 4 digits
2. Where the numbers have the same number of digits, compare column by column from the left. The first column where two numbers differ decides which is bigger.
   4065, 4506, 4056, 4650
3. Write the original numbers in order. Order smallest to largest.
   4056, 4065, 4506, 4650

Check. Check each neighbouring pair: 4056 < 4065; 4065 < 4506; 4506 < 4650. ✓

Watch out. Ordering decimals: line up the decimal places. Do not decide by the length of the number.

Method. Make the numbers easy to compare, then order them smallest to largest.

1. Write every number with the same number of decimal places, using zeros. The extra zeros do not change the values. They make the columns line up so we can compare digit by digit.
   0.6 = 0.600
   0.06 = 0.060
   0.65 = 0.650
   0.605 = 0.605
2. Compare the sizes, ignoring the decimal point and any minus signs. Now every number has 3 decimal places, so we can compare them as if they were whole numbers.
   0.6: 600, 0.06: 60, 0.65: 650, 0.605: 605
3. Write the original numbers in order. Order smallest to largest.
   0.06, 0.6, 0.605, 0.65

Check. Check each neighbouring pair: 0.06 < 0.6; 0.6 < 0.605; 0.605 < 0.65. ✓

Watch out. Ordering decimals: line up the decimal places. Do not decide by the length of the number.

Method. Make the numbers easy to compare, then order them smallest to largest.

1. Write every number with the same number of decimal places, using zeros. The extra zeros do not change the values. They make the columns line up so we can compare digit by digit.
   3.4 = 3.400
   3.04 = 3.040
   3.44 = 3.440
   3.404 = 3.404
   3.044 = 3.044
2. Compare the sizes, ignoring the decimal point and any minus signs. Now every number has 3 decimal places, so we can compare them as if they were whole numbers.
   3.4: 3400, 3.04: 3040, 3.44: 3440, 3.404: 3404, 3.044: 3044
3. Write the original numbers in order. Order smallest to largest.
   3.04, 3.044, 3.4, 3.404, 3.44

Check. Check each neighbouring pair: 3.04 < 3.044; 3.044 < 3.4; 3.4 < 3.404; 3.404 < 3.44. ✓

Watch out. Ordering decimals: line up the decimal places. Do not decide by the length of the number.

Method. Multiply by 10 by moving every digit 1 place to the left.

1. Remember what happens when we multiply or divide by 10, 100, 1000. Multiplying by 10 makes every digit worth ten times as much, so it moves one column to the left. Dividing by 10 moves every digit one column to the right. The decimal point does not move; the digits do.
   × 10 → digits move 1 place to the left
2. Show the digits before and after. Each digit moves 1 column to the left. Empty columns are filled with zeros.
   Before: 6.4
   After: 64
   (the digits moved 1 place left)
3. Write the answer. Tidy up: remove any leading zeros, and include the decimal point if there are digits after it.
   6.4 × 10 = 64

Check. Reverse it: 64 ÷ 10 = 6.4. ✓

Watch out. The digits move, not the decimal point (even though people often say "move the point"). Fill the gaps with zeros.

Method. Multiply by 100 by moving every digit 2 places to the left.

1. Remember what happens when we multiply or divide by 10, 100, 1000. Multiplying by 10 makes every digit worth ten times as much, so it moves one column to the left. Dividing by 10 moves every digit one column to the right. The decimal point does not move; the digits do.
   × 100 → digits move 2 places to the left
2. Show the digits before and after. Each digit moves 2 columns to the left. Empty columns are filled with zeros.
   Before: 0.35
   After: 35
   (the digits moved 2 places left)
3. Write the answer. Tidy up: remove any leading zeros, and include the decimal point if there are digits after it.
   0.35 × 100 = 35

Check. Reverse it: 35 ÷ 100 = 0.35. ✓

Watch out. The digits move, not the decimal point (even though people often say "move the point"). Fill the gaps with zeros.

Method. Divide by 100 by moving every digit 2 places to the right.

1. Remember what happens when we multiply or divide by 10, 100, 1000. Multiplying by 10 makes every digit worth ten times as much, so it moves one column to the left. Dividing by 10 moves every digit one column to the right. The decimal point does not move; the digits do.
   ÷ 100 → digits move 2 places to the right
2. Show the digits before and after. Each digit moves 2 columns to the right. Empty columns are filled with zeros.
   Before: 2900
   After: 29
   (the digits moved 2 places right)
3. Write the answer. Tidy up: remove any leading zeros, and include the decimal point if there are digits after it.
   2900 ÷ 100 = 29

Check. Reverse it: 29 × 100 = 2900. ✓

Watch out. The digits move, not the decimal point (even though people often say "move the point"). Fill the gaps with zeros.

Method. Divide by 10 by moving every digit 1 place to the right.

1. Remember what happens when we multiply or divide by 10, 100, 1000. Multiplying by 10 makes every digit worth ten times as much, so it moves one column to the left. Dividing by 10 moves every digit one column to the right. The decimal point does not move; the digits do.
   ÷ 10 → digits move 1 place to the right
2. Show the digits before and after. Each digit moves 1 column to the right. Empty columns are filled with zeros.
   Before: 45.6
   After: 4.56
   (the digits moved 1 place right)
3. Write the answer. Tidy up: remove any leading zeros, and include the decimal point if there are digits after it.
   45.6 ÷ 10 = 4.56

Check. Reverse it: 4.56 × 10 = 45.6. ✓

Watch out. The digits move, not the decimal point (even though people often say "move the point"). Fill the gaps with zeros.

Method. Multiply by 1000 by moving every digit 3 places to the left.

1. Remember what happens when we multiply or divide by 10, 100, 1000. Multiplying by 10 makes every digit worth ten times as much, so it moves one column to the left. Dividing by 10 moves every digit one column to the right. The decimal point does not move; the digits do.
   × 1 000 → digits move 3 places to the left
2. Show the digits before and after. Each digit moves 3 columns to the left. Empty columns are filled with zeros.
   Before: 0.072
   After: 72
   (the digits moved 3 places left)
3. Write the answer. Tidy up: remove any leading zeros, and include the decimal point if there are digits after it.
   0.072 × 1000 = 72

Check. Reverse it: 72 ÷ 1000 = 0.072. ✓

Watch out. The digits move, not the decimal point (even though people often say "move the point"). Fill the gaps with zeros.

Method. Divide by 1000 by moving every digit 3 places to the right.

1. Remember what happens when we multiply or divide by 10, 100, 1000. Multiplying by 10 makes every digit worth ten times as much, so it moves one column to the left. Dividing by 10 moves every digit one column to the right. The decimal point does not move; the digits do.
   ÷ 1 000 → digits move 3 places to the right
2. Show the digits before and after. Each digit moves 3 columns to the right. Empty columns are filled with zeros.
   Before: 83
   After: 0.083
   (the digits moved 3 places right)
3. Write the answer. Tidy up: remove any leading zeros, and include the decimal point if there are digits after it.
   83 ÷ 1000 = 0.083

Check. Reverse it: 0.083 × 1000 = 83. ✓

Watch out. The digits move, not the decimal point (even though people often say "move the point"). Fill the gaps with zeros.

Method. Write both with two decimal places and compare.

1. Write 0.5 with two decimal places. Adding a zero on the end of a decimal does not change its value.
   0.5 = 0.50
2. Compare the hundredths. 50 hundredths is more than 45 hundredths.
   0.50 > 0.45

Check. 0.50 − 0.45 = 0.05, which is positive. ✓

Method. The symbol ≥ means “greater than or equal to”.

1. Compare the two numbers. They are the same, so 7 is equal to 7.
   7 = 7
2. Check the symbol allows equal. ≥ is true when the numbers are equal or the first is bigger.
   So 7 ≥ 7 is true.

Check. Also 7 ≤ 7 is true for the same reason. ✓

Method. Decide whether the two numbers have the same value.

1. Remove the zero on the end of 0.30. A zero at the end of the decimal places does not change the value.
   0.30 = 0.3
2. Decide. The two numbers are equal, so they are not different.
   0.30 = 0.3, so the statement 0.30 ≠ 0.3 is false.

Check. 0.30 is 30 hundredths = 3 tenths. ✓

Method. Compare the number of digits first.

1. Count the digits. A whole number with more digits is bigger.
   1000 has 4 digits and 999 has 3 digits.
2. Decide. 1000 is bigger than 999, so 1000 < 999 is false.
   1000 > 999

Check. 999 + 1 = 1000, so 1000 is 1 more. ✓

Method. To make the largest number, put the digits in order from biggest to smallest.

1. Put the biggest digit in the biggest column. The left-hand column is worth the most, so it gets the biggest digit, then the next biggest, and so on.
   Digits in order, biggest first: 9, 6, 3, 1
2. Write the number.
   9631

Check. It uses each of the digits 1, 3, 6, 9 exactly once. ✓

Method. To make the smallest number, put the digits in order from smallest to biggest (but never start with 0).

1. Put the smallest digit in the biggest column. The left-hand column is worth the most, so it gets the smallest digit.
   Smallest first: 1, 3, 6, 9
2. Write the number.
   1369

Check. It uses each of the digits 1, 3, 6, 9 exactly once. ✓

Method. The last digit decides whether the number is even or odd. Try each possible last digit, then choose the biggest.

1. Decide what the last digit must be. A number is even only when its last digit is 0, 2, 4, 6 or 8.
   Possible last digits: 6
2. For each choice of last digit, arrange the other digits to make the biggest number. The digit in the last column is fixed, so we use the same idea as before for the other columns.
   Last digit 6: the other digits 9, 3, 1 → 9316
3. Choose the biggest of these numbers.
   9316

Check. It uses each of the digits 1, 3, 6, 9 exactly once. ✓

Method. Think about the leftmost column.

1. Compare the leftmost columns. The extra digit is in a higher column than any digit of the shorter number. For example the smallest 4-digit number is 1000.
   1000 is bigger than the biggest 3-digit number, 999.
2. Decide. This works for every pair of whole numbers (with no leading zeros).
   Always true.

Check. Try 10 and 9: 10 has more digits and is bigger. ✓

Method. Find one example that is true and one that is false.

1. Look for an example where it is true. 0.52 has more digits than 0.5, and 0.52 > 0.50.
   0.52 > 0.5 (true)
2. Look for an example where it is false. 0.35 has more digits than 0.4, but it is smaller.
   0.35 < 0.4 (false)
3. Decide. It is true for some pairs and false for others.
   Sometimes true.

Check. Two different examples give two different results. ✓

Method. Think about zero as well as ordinary numbers.

1. Try an ordinary positive number. 3 × 10 = 30, which is bigger.
   30 > 3
2. Try zero. 0 × 10 = 0, which is not bigger.
   0 = 0
3. Decide. It makes positive numbers bigger but not zero (and later we will see that negative numbers get smaller).
   Sometimes true.

Check. 0 × 10 = 0. ✓

Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.

Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.

1. Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and so on.
   345 + 278 -----
2. Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.
   Units: 5 + 8 = 13. Write 3 and carry 1 into the tens column.
   Tens: 4 + 7 + 1 (carried) = 12. Write 2 and carry 1 into the hundreds column.
   Hundreds: 3 + 2 + 1 (carried) = 6. Write 6.
3. Write the answer. Read the digits of the bottom row. The small numbers on the bottom row show the carries.
   345 + 278 ----- 623 11
   345 + 278 = 623

Check. Check with an inverse (opposite) operation: 623 − 278 = 345, which is the first number. ✓

Watch out. Do not forget to add the carried digit into the next column.

Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.

Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.

1. Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and so on.
   4806 + 1975 ------
2. Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.
   Units: 6 + 5 = 11. Write 1 and carry 1 into the tens column.
   Tens: 0 + 7 + 1 (carried) = 8. Write 8.
   Hundreds: 8 + 9 = 17. Write 7 and carry 1 into the thousands column.
   Thousands: 4 + 1 + 1 (carried) = 6. Write 6.
3. Write the answer. Read the digits of the bottom row. The small numbers on the bottom row show the carries.
   4806 + 1975 ------ 6781 1 1
   4806 + 1975 = 6781

Check. Check with an inverse (opposite) operation: 6781 − 1975 = 4806, which is the first number. ✓

Watch out. Do not forget to add the carried digit into the next column.

Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.

Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.

1. Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and decimal points under decimal points. Fill any gaps with zeros.
   73.40 + 8.95 -------
2. Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.
   Hundredths: 0 + 5 = 5. Write 5.
   Tenths: 4 + 9 = 13. Write 3 and carry 1 into the units column.
   Units: 3 + 8 + 1 (carried) = 12. Write 2 and carry 1 into the tens column.
   Tens: 7 + 0 + 1 (carried) = 8. Write 8.
3. Put the decimal point in the answer. The point goes directly under the other decimal points.
   73.40 + 8.95 ------- 82.35 11
   73.4 + 8.95 = 82.35

Check. Check with an inverse (opposite) operation: 82.35 − 8.95 = 73.4, which is the first number. ✓

Watch out. Line up the decimal points, not the right-hand ends of the numbers. 5.2 + 0.37 needs 5.20 + 0.37.

Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.

Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.

1. Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and so on.
   256 389 + 47 -----
2. Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.
   Units: 6 + 9 + 7 = 22. Write 2 and carry 2 into the tens column.
   Tens: 5 + 8 + 4 + 2 (carried) = 19. Write 9 and carry 1 into the hundreds column.
   Hundreds: 2 + 3 + 0 + 1 (carried) = 6. Write 6.
3. Write the answer. Read the digits of the bottom row. The small numbers on the bottom row show the carries.
   256 389 + 47 ----- 692 12
   256 + 389 + 47 = 692

Check. Check with an inverse (opposite) operation: 692 − 47 = 645 = 645, which is 256 + 389. ✓

Watch out. Do not forget to add the carried digit into the next column.

Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.

Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.

1. Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and decimal points under decimal points. Fill any gaps with zeros.
   0.68 + 5.70 ------
2. Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.
   Hundredths: 8 + 0 = 8. Write 8.
   Tenths: 6 + 7 = 13. Write 3 and carry 1 into the units column.
   Units: 0 + 5 + 1 (carried) = 6. Write 6.
3. Put the decimal point in the answer. The point goes directly under the other decimal points.
   0.68 + 5.70 ------ 6.38 1
   0.68 + 5.7 = 6.38

Check. Check with an inverse (opposite) operation: 6.38 − 5.7 = 0.68, which is the first number. ✓

Watch out. Line up the decimal points, not the right-hand ends of the numbers. 5.2 + 0.37 needs 5.20 + 0.37.

Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.

Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.

1. Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and so on.
   12480 + 9735 -------
2. Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.
   Units: 0 + 5 = 5. Write 5.
   Tens: 8 + 3 = 11. Write 1 and carry 1 into the hundreds column.
   Hundreds: 4 + 7 + 1 (carried) = 12. Write 2 and carry 1 into the thousands column.
   Thousands: 2 + 9 + 1 (carried) = 12. Write 2 and carry 1 into the ten thousands column.
   Ten thousands: 1 + 0 + 1 (carried) = 2. Write 2.
3. Write the answer. Read the digits of the bottom row. The small numbers on the bottom row show the carries.
   12480 + 9735 ------- 22215 111
   12 480 + 9735 = 22 215

Check. Check with an inverse (opposite) operation: 22 215 − 9735 = 12 480, which is the first number. ✓

Watch out. Do not forget to add the carried digit into the next column.

Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.

Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.

1. Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens.
   802 − 457 -----
2. Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.
   Units: 2 is smaller than 7, so we must exchange. The tens column is 0, so there is nothing to take there. Go left to the hundreds column. Take 1 hundred (8 becomes 7) and exchange it for 10 tens. Take 1 of those tens (leaving 9) and exchange it for 10 units. 2 becomes 12. Now 12 − 7 = 5.
   Tens: 9 − 5 = 4.
   Hundreds: 7 − 4 = 3.
3. Write the answer. Read the bottom row.
   802 − 457 ----- 345
   802 − 457 = 345

Check. Check with the inverse (opposite) operation, addition: 345 + 457 = 802. ✓

Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.

Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.

Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.

1. Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens.
   6000 − 2348 ------
2. Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.
   Units: 0 is smaller than 8, so we must exchange. The tens and hundreds columns are 0, so there is nothing to take there. Go left to the thousands column. Take 1 thousand (6 becomes 5) and exchange it for 10 hundreds. Take 1 of those hundreds (leaving 9) and exchange it for 10 tens. Take 1 of those tens (leaving 9) and exchange it for 10 units. 0 becomes 10. Now 10 − 8 = 2.
   Tens: 9 − 4 = 5.
   Hundreds: 9 − 3 = 6.
   Thousands: 5 − 2 = 3.
3. Write the answer. Read the bottom row.
   6000 − 2348 ------ 3652
   6000 − 2348 = 3652

Check. Check with the inverse (opposite) operation, addition: 3652 + 2348 = 6000. ✓

Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.

Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.

Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.

1. Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens, decimal points under decimal points. Fill gaps with zeros.
   15.20 − 7.85 -------
2. Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.
   Hundredths: 0 is smaller than 5, so we cannot take 5 from 0. Exchange: take 1 tenth from the tenths column (2 becomes 1) and add it to the hundredths column as 10 hundredths. 0 becomes 10. Now 10 − 5 = 5.
   Tenths: 1 is smaller than 8, so we cannot take 8 from 1. Exchange: take 1 one from the units column (5 becomes 4) and add it to the tenths column as 10 tenths. 1 becomes 11. Now 11 − 8 = 3.
   Units: 4 is smaller than 7, so we cannot take 7 from 4. Exchange: take 1 ten from the tens column (1 becomes 0) and add it to the units column as 10 units. 4 becomes 14. Now 14 − 7 = 7.
   Tens: 0 − 0 = 0.
3. Write the answer. Put the decimal point in line with the others.
   15.20 − 7.85 ------- 07.35
   15.2 − 7.85 = 7.35

Check. Check with the inverse (opposite) operation, addition: 7.35 + 7.85 = 15.2. ✓

Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.

Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.

Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.

1. Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens, decimal points under decimal points. Fill gaps with zeros.
   9.00 − 3.64 ------
2. Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.
   Hundredths: 0 is smaller than 4, so we must exchange. The tenths column is 0, so there is nothing to take there. Go left to the units column. Take 1 one (9 becomes 8) and exchange it for 10 tenths. Take 1 of those tenths (leaving 9) and exchange it for 10 hundredths. 0 becomes 10. Now 10 − 4 = 6.
   Tenths: 9 − 6 = 3.
   Units: 8 − 3 = 5.
3. Write the answer. Put the decimal point in line with the others.
   9.00 − 3.64 ------ 5.36
   9 − 3.64 = 5.36

Check. Check with the inverse (opposite) operation, addition: 5.36 + 3.64 = 9. ✓

Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.

Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.

Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.

1. Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens.
   30105 − 8967 -------
2. Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.
   Units: 5 is smaller than 7, so we must exchange. The tens column is 0, so there is nothing to take there. Go left to the hundreds column. Take 1 hundred (1 becomes 0) and exchange it for 10 tens. Take 1 of those tens (leaving 9) and exchange it for 10 units. 5 becomes 15. Now 15 − 7 = 8.
   Tens: 9 − 6 = 3.
   Hundreds: 0 is smaller than 9, so we must exchange. The thousands column is 0, so there is nothing to take there. Go left to the ten thousands column. Take 1 ten thousand (3 becomes 2) and exchange it for 10 thousands. Take 1 of those thousands (leaving 9) and exchange it for 10 hundreds. 0 becomes 10. Now 10 − 9 = 1.
   Thousands: 9 − 8 = 1.
   Ten thousands: 2 − 0 = 2.
3. Write the answer. Read the bottom row.
   30105 − 8967 ------- 21138
   30 105 − 8967 = 21 138

Check. Check with the inverse (opposite) operation, addition: 21 138 + 8967 = 30 105. ✓

Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.

Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.

Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.

1. Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens, decimal points under decimal points. Fill gaps with zeros.
   4.200 − 0.375 -------
2. Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.
   Thousandths: 0 is smaller than 5, so we must exchange. The hundredths column is 0, so there is nothing to take there. Go left to the tenths column. Take 1 tenth (2 becomes 1) and exchange it for 10 hundredths. Take 1 of those hundredths (leaving 9) and exchange it for 10 thousandths. 0 becomes 10. Now 10 − 5 = 5.
   Hundredths: 9 − 7 = 2.
   Tenths: 1 is smaller than 3, so we cannot take 3 from 1. Exchange: take 1 one from the units column (4 becomes 3) and add it to the tenths column as 10 tenths. 1 becomes 11. Now 11 − 3 = 8.
   Units: 3 − 0 = 3.
3. Write the answer. Put the decimal point in line with the others.
   4.200 − 0.375 ------- 3.825
   4.2 − 0.375 = 3.825

Check. Check with the inverse (opposite) operation, addition: 3.825 + 0.375 = 4.2. ✓

Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.

Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.

Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.

1. Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and decimal points under decimal points. Fill any gaps with zeros.
   3.45 12.99 + 0.85 -------
2. Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.
   Hundredths: 5 + 9 + 5 = 19. Write 9 and carry 1 into the tenths column.
   Tenths: 4 + 9 + 8 + 1 (carried) = 22. Write 2 and carry 2 into the units column.
   Units: 3 + 2 + 0 + 2 (carried) = 7. Write 7.
   Tens: 0 + 1 + 0 = 1. Write 1.
3. Put the decimal point in the answer. The point goes directly under the other decimal points.
   3.45 12.99 + 0.85 ------- 17.29 2 1
   3.45 + 12.99 + 0.85 = 17.29

Check. Check with an inverse (opposite) operation: 17.29 − 0.85 = 16.44 = 16.44, which is 3.45 + 12.99. ✓

Watch out. Line up the decimal points, not the right-hand ends of the numbers. 5.2 + 0.37 needs 5.20 + 0.37.

Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.

Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.

1. Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens, decimal points under decimal points. Fill gaps with zeros.
   20.00 − 17.29 -------
2. Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.
   Hundredths: 0 is smaller than 9, so we must exchange. The tenths and units columns are 0, so there is nothing to take there. Go left to the tens column. Take 1 ten (2 becomes 1) and exchange it for 10 units. Take 1 of those units (leaving 9) and exchange it for 10 tenths. Take 1 of those tenths (leaving 9) and exchange it for 10 hundredths. 0 becomes 10. Now 10 − 9 = 1.
   Tenths: 9 − 2 = 7.
   Units: 9 − 7 = 2.
   Tens: 1 − 1 = 0.
3. Write the answer. Put the decimal point in line with the others.
   20.00 − 17.29 ------- 02.71
   20 − 17.29 = 2.71

Check. Check with the inverse (opposite) operation, addition: 2.71 + 17.29 = 20. ✓

Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.

Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.

1. Split 20 into 2 and 10. 20 is 2 lots of 10. Multiplying by a number like 20 is the same as multiplying by 2 and then by 10.
   20 = 2 × 10
2. Multiply by 2. This is an easy times-table style calculation.
   34 × 2 = 68
3. Multiply by 10 by moving the digits 1 place to the left. Each place the digits move left makes the number ten times bigger.
   68 × 10 = 680

Check. Reverse: 680 ÷ 20 = 34. ✓

Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.

Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.

1. Split 30 into 3 and 10. 30 is 3 lots of 10. Multiplying by a number like 30 is the same as multiplying by 3 and then by 10.
   30 = 3 × 10
2. Multiply by 3. This is an easy times-table style calculation.
   250 × 3 = 750
3. Multiply by 10 by moving the digits 1 place to the left. Each place the digits move left makes the number ten times bigger.
   750 × 10 = 7500

Check. Reverse: 7500 ÷ 30 = 250. ✓

Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.

Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.

1. Split 200 into 2 and 100. 200 is 2 lots of 100. Multiplying by a number like 200 is the same as multiplying by 2 and then by 100.
   200 = 2 × 100
2. Multiply by 2. This is an easy times-table style calculation.
   46 × 2 = 92
3. Multiply by 100 by moving the digits 2 places to the left. Each place the digits move left makes the number ten times bigger.
   92 × 100 = 9200

Check. Reverse: 9200 ÷ 200 = 46. ✓

Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.

Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.

1. Split 40 into 4 and 10. 40 is 4 lots of 10. Multiplying by a number like 40 is the same as multiplying by 4 and then by 10.
   40 = 4 × 10
2. Multiply by 4. This is an easy times-table style calculation.
   7.5 × 4 = 30
3. Multiply by 10 by moving the digits 1 place to the left. Each place the digits move left makes the number ten times bigger.
   30 × 10 = 300

Check. Reverse: 300 ÷ 40 = 7.5. ✓

Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.

Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.

1. Split 600 into 6 and 100. 600 is 6 lots of 100. Multiplying by a number like 600 is the same as multiplying by 6 and then by 100.
   600 = 6 × 100
2. Multiply by 6. This is an easy times-table style calculation.
   0.8 × 6 = 4.8
3. Multiply by 100 by moving the digits 2 places to the left. Each place the digits move left makes the number ten times bigger.
   4.8 × 100 = 480

Check. Reverse: 480 ÷ 600 = 0.8. ✓

Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.

Method. Multiply by the non-zero digit first, then by 10, 100 or 1000 by moving the digits.

1. Split 4000 into 4 and 1000. 4000 is 4 lots of 1000. Multiplying by a number like 4000 is the same as multiplying by 4 and then by 1000.
   4000 = 4 × 1000
2. Multiply by 4. This is an easy times-table style calculation.
   125 × 4 = 500
3. Multiply by 1000 by moving the digits 3 places to the left. Each place the digits move left makes the number ten times bigger.
   500 × 1000 = 500 000

Check. Reverse: 500 000 ÷ 4000 = 125. ✓

Watch out. Do not just stick zeros on the end of a decimal. 0.4 × 10 = 4, not 0.40.

Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.

Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.

1. Set out the multiplication. Write 245 on top and 6 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.
   245 × 6 -----
2. Multiply 245 by 6. Work from the right. Multiply each digit of the top number by 6. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   5 × 6 = 30: write 0, carry 3
   4 × 6 + 3 (carried) = 27: write 7, carry 2
   2 × 6 + 2 (carried) = 14: write 4, carry 1
   Write the last carry: 1
   So 245 × 6 = 1470.

Check. Estimate by rounding each number to 1 significant figure: 200 × 6 = 1200. This is close to 1470, so the size is sensible. ✓

Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.

Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.

Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.

1. Set out the multiplication. Write 1308 on top and 7 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.
   1308 × 7 ------
2. Multiply 1308 by 7. Work from the right. Multiply each digit of the top number by 7. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   8 × 7 = 56: write 6, carry 5
   0 × 7 + 5 (carried) = 5: write 5
   3 × 7 = 21: write 1, carry 2
   1 × 7 + 2 (carried) = 9: write 9
   So 1308 × 7 = 9156.

Check. Estimate by rounding each number to 1 significant figure: 1000 × 7 = 7000. This is close to 9156, so the size is sensible. ✓

Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.

Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.

Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.

1. Set out the multiplication. Write 86 on top and 9 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.
   86 × 9 ----
2. Multiply 86 by 9. Work from the right. Multiply each digit of the top number by 9. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   6 × 9 = 54: write 4, carry 5
   8 × 9 + 5 (carried) = 77: write 7, carry 7
   Write the last carry: 7
   So 86 × 9 = 774.

Check. Estimate by rounding each number to 1 significant figure: 90 × 9 = 810. This is close to 774, so the size is sensible. ✓

Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.

Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.

Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.

1. Set out the multiplication. Write 4125 on top and 8 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.
   4125 × 8 ------
2. Multiply 4125 by 8. Work from the right. Multiply each digit of the top number by 8. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   5 × 8 = 40: write 0, carry 4
   2 × 8 + 4 (carried) = 20: write 0, carry 2
   1 × 8 + 2 (carried) = 10: write 0, carry 1
   4 × 8 + 1 (carried) = 33: write 3, carry 3
   Write the last carry: 3
   So 4125 × 8 = 33 000.

Check. Estimate by rounding each number to 1 significant figure: 4000 × 8 = 32 000. This is close to 33 000, so the size is sensible. ✓

Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.

Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.

Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.

1. Set out the multiplication. Write 47 on top and 36 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.
   47 × 36 ----
2. Multiply 47 by the units digit, 6. Work from the right. Multiply each digit of the top number by 6. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   7 × 6 = 42: write 2, carry 4
   4 × 6 + 4 (carried) = 28: write 8, carry 2
   Write the last carry: 2
   So 47 × 6 = 282.
3. Multiply 47 by the tens digit, 3 (which means 30). The 3 is in the tens column, so it is really 30. First write 1 zero on the right as place holder, then multiply by 3.
   7 × 3 = 21: write 1, carry 2
   4 × 3 + 2 (carried) = 14: write 4, carry 1
   Write the last carry: 1
   47 × 3 = 141, then add 1 zero: 47 × 30 = 1410.
4. Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.
   282 + 1410 = 1692
   47 × 36 ---- 282 1410 ----- 1692

Check. Estimate by rounding each number to 1 significant figure: 50 × 40 = 2000. This is close to 1692, so the size is sensible. ✓

Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.

Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.

Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.

1. Set out the multiplication. Write 308 on top and 25 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.
   308 × 25 -----
2. Multiply 308 by the units digit, 5. Work from the right. Multiply each digit of the top number by 5. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   8 × 5 = 40: write 0, carry 4
   0 × 5 + 4 (carried) = 4: write 4
   3 × 5 = 15: write 5, carry 1
   Write the last carry: 1
   So 308 × 5 = 1540.
3. Multiply 308 by the tens digit, 2 (which means 20). The 2 is in the tens column, so it is really 20. First write 1 zero on the right as place holder, then multiply by 2.
   8 × 2 = 16: write 6, carry 1
   0 × 2 + 1 (carried) = 1: write 1
   3 × 2 = 6: write 6
   308 × 2 = 616, then add 1 zero: 308 × 20 = 6160.
4. Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.
   1540 + 6160 = 7700
   308 × 25 ----- 1540 6160 ----- 7700

Check. Estimate by rounding each number to 1 significant figure: 300 × 30 = 9000. This is close to 7700, so the size is sensible. ✓

Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.

Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.

Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.

1. Set out the multiplication. Write 726 on top and 48 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.
   726 × 48 -----
2. Multiply 726 by the units digit, 8. Work from the right. Multiply each digit of the top number by 8. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   6 × 8 = 48: write 8, carry 4
   2 × 8 + 4 (carried) = 20: write 0, carry 2
   7 × 8 + 2 (carried) = 58: write 8, carry 5
   Write the last carry: 5
   So 726 × 8 = 5808.
3. Multiply 726 by the tens digit, 4 (which means 40). The 4 is in the tens column, so it is really 40. First write 1 zero on the right as place holder, then multiply by 4.
   6 × 4 = 24: write 4, carry 2
   2 × 4 + 2 (carried) = 10: write 0, carry 1
   7 × 4 + 1 (carried) = 29: write 9, carry 2
   Write the last carry: 2
   726 × 4 = 2904, then add 1 zero: 726 × 40 = 29 040.
4. Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.
   5808 + 29 040 = 34 848
   726 × 48 ----- 5808 29040 ------ 34848

Check. Estimate by rounding each number to 1 significant figure: 700 × 50 = 35 000. This is close to 34 848, so the size is sensible. ✓

Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.

Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.

Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.

1. Set out the multiplication. Write 1204 on top and 63 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.
   1204 × 63 ------
2. Multiply 1204 by the units digit, 3. Work from the right. Multiply each digit of the top number by 3. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   4 × 3 = 12: write 2, carry 1
   0 × 3 + 1 (carried) = 1: write 1
   2 × 3 = 6: write 6
   1 × 3 = 3: write 3
   So 1204 × 3 = 3612.
3. Multiply 1204 by the tens digit, 6 (which means 60). The 6 is in the tens column, so it is really 60. First write 1 zero on the right as place holder, then multiply by 6.
   4 × 6 = 24: write 4, carry 2
   0 × 6 + 2 (carried) = 2: write 2
   2 × 6 = 12: write 2, carry 1
   1 × 6 + 1 (carried) = 7: write 7
   1204 × 6 = 7224, then add 1 zero: 1204 × 60 = 72 240.
4. Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.
   3612 + 72 240 = 75 852
   1204 × 63 ------ 3612 72240 ------ 75852

Check. Estimate by rounding each number to 1 significant figure: 1000 × 60 = 60 000. This is close to 75 852, so the size is sensible. ✓

Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.

Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.

Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.

1. Ignore the decimal points and multiply as whole numbers. 0.6 is 6 divided by 10, and 0.8 is 8 divided by 10. So we can multiply 6 by 8, then fix the size at the end.
   0.6 × 0.8 → 6 × 8
2. Multiply 8 by 6. Work from the right. Multiply each digit of the top number by 6. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   8 × 6 = 48: write 8, carry 4
   Write the last carry: 4
   So 8 × 6 = 48.
3. Put the decimal point back. Count the decimal places in the question: 0.6 has 1 and 0.8 has 1, so the answer has 1 + 1 = 2 decimal places. Count that many digits in from the right.
   48 with 2 decimal places: 0.48

Check. Estimate: 0.6 × 0.8 = 0.48. 0.48 is close to 0.48. ✓

Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.

Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.

Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.

1. Ignore the decimal points and multiply as whole numbers. 2.5 is 25 divided by 10, and 4.2 is 42 divided by 10. So we can multiply 25 by 42, then fix the size at the end.
   2.5 × 4.2 → 25 × 42
2. Multiply 42 by the units digit, 5. Work from the right. Multiply each digit of the top number by 5. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   2 × 5 = 10: write 0, carry 1
   4 × 5 + 1 (carried) = 21: write 1, carry 2
   Write the last carry: 2
   So 42 × 5 = 210.
3. Multiply 42 by the tens digit, 2 (which means 20). The 2 is in the tens column, so it is really 20. First write 1 zero on the right as place holder, then multiply by 2.
   2 × 2 = 4: write 4
   4 × 2 = 8: write 8
   42 × 2 = 84, then add 1 zero: 42 × 20 = 840.
4. Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.
   210 + 840 = 1050
   42 × 25 ---- 210 840 ----- 1050
5. Put the decimal point back. Count the decimal places in the question: 2.5 has 1 and 4.2 has 1, so the answer has 1 + 1 = 2 decimal places. Count that many digits in from the right.
   1050 with 2 decimal places: 10.5

Check. Estimate: 3 × 4 = 12. 10.5 is close to 12. ✓

Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.

Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.

Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.

1. Ignore the decimal points and multiply as whole numbers. 12.4 is 124 divided by 10, and 0.35 is 35 divided by 100. So we can multiply 124 by 35, then fix the size at the end.
   12.4 × 0.35 → 124 × 35
2. Multiply 124 by the units digit, 5. Work from the right. Multiply each digit of the top number by 5. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   4 × 5 = 20: write 0, carry 2
   2 × 5 + 2 (carried) = 12: write 2, carry 1
   1 × 5 + 1 (carried) = 6: write 6
   So 124 × 5 = 620.
3. Multiply 124 by the tens digit, 3 (which means 30). The 3 is in the tens column, so it is really 30. First write 1 zero on the right as place holder, then multiply by 3.
   4 × 3 = 12: write 2, carry 1
   2 × 3 + 1 (carried) = 7: write 7
   1 × 3 = 3: write 3
   124 × 3 = 372, then add 1 zero: 124 × 30 = 3720.
4. Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.
   620 + 3720 = 4340
   124 × 35 ----- 620 3720 ----- 4340
5. Put the decimal point back. Count the decimal places in the question: 12.4 has 1 and 0.35 has 2, so the answer has 1 + 2 = 3 decimal places. Count that many digits in from the right.
   4340 with 3 decimal places: 4.34

Check. Estimate: 10 × 0.4 = 4. 4.34 is close to 4. ✓

Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.

Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.

Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.

1. Ignore the decimal points and multiply as whole numbers. 0.09 is 9 divided by 100, and 0.7 is 7 divided by 10. So we can multiply 9 by 7, then fix the size at the end.
   0.09 × 0.7 → 9 × 7
2. Multiply 9 by 7. Work from the right. Multiply each digit of the top number by 7. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   9 × 7 = 63: write 3, carry 6
   Write the last carry: 6
   So 9 × 7 = 63.
3. Put the decimal point back. Count the decimal places in the question: 0.09 has 2 and 0.7 has 1, so the answer has 2 + 1 = 3 decimal places. Count that many digits in from the right.
   63 with 3 decimal places: 0.063

Check. Estimate: 0.09 × 0.7 = 0.063. 0.063 is close to 0.063. ✓

Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.

Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.

Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.

1. Ignore the decimal points and multiply as whole numbers. 6.8 is 68 divided by 10, and 5 is 5 divided by 1. So we can multiply 68 by 5, then fix the size at the end.
   6.8 × 5 → 68 × 5
2. Multiply 68 by 5. Work from the right. Multiply each digit of the top number by 5. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   8 × 5 = 40: write 0, carry 4
   6 × 5 + 4 (carried) = 34: write 4, carry 3
   Write the last carry: 3
   So 68 × 5 = 340.
3. Put the decimal point back. Count the decimal places in the question: 6.8 has 1 and 5 has 0, so the answer has 1 + 0 = 1 decimal places. Count that many digits in from the right.
   340 with 1 decimal places: 34

Check. Estimate: 7 × 5 = 35. 34 is close to 35. ✓

Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.

Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.

Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.

1. Ignore the decimal points and multiply as whole numbers. 0.25 is 25 divided by 100, and 0.12 is 12 divided by 100. So we can multiply 25 by 12, then fix the size at the end.
   0.25 × 0.12 → 25 × 12
2. Multiply 25 by the units digit, 2. Work from the right. Multiply each digit of the top number by 2. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   5 × 2 = 10: write 0, carry 1
   2 × 2 + 1 (carried) = 5: write 5
   So 25 × 2 = 50.
3. Multiply 25 by the tens digit, 1 (which means 10). The 1 is in the tens column, so it is really 10. First write 1 zero on the right as place holder, then multiply by 1.
   5 × 1 = 5: write 5
   2 × 1 = 2: write 2
   25 × 1 = 25, then add 1 zero: 25 × 10 = 250.
4. Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.
   50 + 250 = 300
   25 × 12 ---- 50 250 ---- 300
5. Put the decimal point back. Count the decimal places in the question: 0.25 has 2 and 0.12 has 2, so the answer has 2 + 2 = 4 decimal places. Count that many digits in from the right.
   300 with 4 decimal places: 0.03

Check. Estimate: 0.3 × 0.1 = 0.03. 0.03 is close to 0.03. ✓

Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 7 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 56 under the "bus stop" and 7 outside it. The answer will be written on top, one digit above each digit we divide.
   7 ) 56
2. Start with the first 2 digits, 56. 7 does not go into the first digit, so we take digits from the left until we have a number that 7 can go into.
   56 ÷ 7 = 8 (since 7 × 8 = 56)
3. Write the answer. Read the digits along the top.
   8 5 7) 56
   56 ÷ 7 = 8

Check. Multiply back: 8 × 7 = 56. ✓

Watch out. If 7 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 9 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 81 under the "bus stop" and 9 outside it. The answer will be written on top, one digit above each digit we divide.
   9 ) 81
2. Start with the first 2 digits, 81. 9 does not go into the first digit, so we take digits from the left until we have a number that 9 can go into.
   81 ÷ 9 = 9 (since 9 × 9 = 81)
3. Write the answer. Read the digits along the top.
   9 8 9) 81
   81 ÷ 9 = 9

Check. Multiply back: 9 × 9 = 81. ✓

Watch out. If 9 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 12 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 132 under the "bus stop" and 12 outside it. The answer will be written on top, one digit above each digit we divide.
   12 ) 132
2. Start with the first 2 digits, 13. 12 does not go into the first digit, so we take digits from the left until we have a number that 12 can go into.
   13 ÷ 12 = 1 remainder 1 (since 12 × 1 = 12)
3. Bring down the next digit, 2, to make 12. The remainder 1 goes in front of the 2. This is the same as carrying the remainder to the next column.
   12 ÷ 12 = 1 (since 12 × 1 = 12)
4. Write the answer. Read the digits along the top.
   11 11 12) 132
   132 ÷ 12 = 11

Check. Multiply back: 11 × 12 = 132. ✓

Watch out. If 12 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 8 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 96 under the "bus stop" and 8 outside it. The answer will be written on top, one digit above each digit we divide.
   8 ) 96
2. Start with the first digit, 9. Work from the left. We ask how many times 8 goes into 9.
   9 ÷ 8 = 1 remainder 1 (since 8 × 1 = 8)
3. Bring down the next digit, 6, to make 16. The remainder 1 goes in front of the 6. This is the same as carrying the remainder to the next column.
   16 ÷ 8 = 2 (since 8 × 2 = 16)
4. Write the answer. Read the digits along the top.
   12 1 8) 96
   96 ÷ 8 = 12

Check. Multiply back: 12 × 8 = 96. ✓

Watch out. If 8 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 4 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 936 under the "bus stop" and 4 outside it. The answer will be written on top, one digit above each digit we divide.
   4 ) 936
2. Start with the first digit, 9. Work from the left. We ask how many times 4 goes into 9.
   9 ÷ 4 = 2 remainder 1 (since 4 × 2 = 8)
3. Bring down the next digit, 3, to make 13. The remainder 1 goes in front of the 3. This is the same as carrying the remainder to the next column.
   13 ÷ 4 = 3 remainder 1 (since 4 × 3 = 12)
4. Bring down the next digit, 6, to make 16. The remainder 1 goes in front of the 6. This is the same as carrying the remainder to the next column.
   16 ÷ 4 = 4 (since 4 × 4 = 16)
5. Write the answer. Read the digits along the top.
   234 11 4) 936
   936 ÷ 4 = 234

Check. Multiply back: 234 × 4 = 936. ✓

Watch out. If 4 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 6 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 1548 under the "bus stop" and 6 outside it. The answer will be written on top, one digit above each digit we divide.
   6 ) 1548
2. Start with the first 2 digits, 15. 6 does not go into the first digit, so we take digits from the left until we have a number that 6 can go into.
   15 ÷ 6 = 2 remainder 3 (since 6 × 2 = 12)
3. Bring down the next digit, 4, to make 34. The remainder 3 goes in front of the 4. This is the same as carrying the remainder to the next column.
   34 ÷ 6 = 5 remainder 4 (since 6 × 5 = 30)
4. Bring down the next digit, 8, to make 48. The remainder 4 goes in front of the 8. This is the same as carrying the remainder to the next column.
   48 ÷ 6 = 8 (since 6 × 8 = 48)
5. Write the answer. Read the digits along the top.
   258 134 6) 1548
   1548 ÷ 6 = 258

Check. Multiply back: 258 × 6 = 1548. ✓

Watch out. If 6 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 8 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 4032 under the "bus stop" and 8 outside it. The answer will be written on top, one digit above each digit we divide.
   8 ) 4032
2. Start with the first 2 digits, 40. 8 does not go into the first digit, so we take digits from the left until we have a number that 8 can go into.
   40 ÷ 8 = 5 (since 8 × 5 = 40)
3. Bring down the next digit, 3. The remainder was 0, so the new number to divide is just that digit.
   8 does not go into 3, so write 0 and carry on.
4. Bring down the next digit, 2, to make 32. The remainder 3 goes in front of the 2. This is the same as carrying the remainder to the next column.
   32 ÷ 8 = 4 (since 8 × 4 = 32)
5. Write the answer. Read the digits along the top.
   504 4 3 8) 4032
   4032 ÷ 8 = 504

Check. Multiply back: 504 × 8 = 4032. ✓

Watch out. If 8 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 5 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 6175 under the "bus stop" and 5 outside it. The answer will be written on top, one digit above each digit we divide.
   5 ) 6175
2. Start with the first digit, 6. Work from the left. We ask how many times 5 goes into 6.
   6 ÷ 5 = 1 remainder 1 (since 5 × 1 = 5)
3. Bring down the next digit, 1, to make 11. The remainder 1 goes in front of the 1. This is the same as carrying the remainder to the next column.
   11 ÷ 5 = 2 remainder 1 (since 5 × 2 = 10)
4. Bring down the next digit, 7, to make 17. The remainder 1 goes in front of the 7. This is the same as carrying the remainder to the next column.
   17 ÷ 5 = 3 remainder 2 (since 5 × 3 = 15)
5. Bring down the next digit, 5, to make 25. The remainder 2 goes in front of the 5. This is the same as carrying the remainder to the next column.
   25 ÷ 5 = 5 (since 5 × 5 = 25)
6. Write the answer. Read the digits along the top.
   1235 112 5) 6175
   6175 ÷ 5 = 1235

Check. Multiply back: 1235 × 5 = 6175. ✓

Watch out. If 5 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 6 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 487 under the "bus stop" and 6 outside it. The answer will be written on top, one digit above each digit we divide.
   6 ) 487
2. Start with the first 2 digits, 48. 6 does not go into the first digit, so we take digits from the left until we have a number that 6 can go into.
   48 ÷ 6 = 8 (since 6 × 8 = 48)
3. Bring down the next digit, 7. The remainder was 0, so the new number to divide is just that digit.
   7 ÷ 6 = 1 remainder 1 (since 6 × 1 = 6)
4. Write the answer. Read the digits along the top.
   81 4 6) 487
   487 ÷ 6 = 81 remainder 1

Check. Multiply back: 81 × 6 = 486, then add the remainder 1: 486 + 1 = 487. ✓

Watch out. If 6 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 7 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 1000 under the "bus stop" and 7 outside it. The answer will be written on top, one digit above each digit we divide.
   7 ) 1000
2. Start with the first 2 digits, 10. 7 does not go into the first digit, so we take digits from the left until we have a number that 7 can go into.
   10 ÷ 7 = 1 remainder 3 (since 7 × 1 = 7)
3. Bring down the next digit, 0, to make 30. The remainder 3 goes in front of the 0. This is the same as carrying the remainder to the next column.
   30 ÷ 7 = 4 remainder 2 (since 7 × 4 = 28)
4. Bring down the next digit, 0, to make 20. The remainder 2 goes in front of the 0. This is the same as carrying the remainder to the next column.
   20 ÷ 7 = 2 remainder 6 (since 7 × 2 = 14)
5. Write the answer. Read the digits along the top.
   142 132 7) 1000
   1000 ÷ 7 = 142 remainder 6

Check. Multiply back: 142 × 7 = 994, then add the remainder 6: 994 + 6 = 1000. ✓

Watch out. If 7 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 9 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 2345 under the "bus stop" and 9 outside it. The answer will be written on top, one digit above each digit we divide.
   9 ) 2345
2. Start with the first 2 digits, 23. 9 does not go into the first digit, so we take digits from the left until we have a number that 9 can go into.
   23 ÷ 9 = 2 remainder 5 (since 9 × 2 = 18)
3. Bring down the next digit, 4, to make 54. The remainder 5 goes in front of the 4. This is the same as carrying the remainder to the next column.
   54 ÷ 9 = 6 (since 9 × 6 = 54)
4. Bring down the next digit, 5. The remainder was 0, so the new number to divide is just that digit.
   9 does not go into 5, so write 0 and carry on.
5. Write the answer. Read the digits along the top.
   260 25 9) 2345
   2345 ÷ 9 = 260 remainder 5

Check. Multiply back: 260 × 9 = 2340, then add the remainder 5: 2340 + 5 = 2345. ✓

Watch out. If 9 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Long division: work from the left, dividing 24 into each number in turn, carrying the remainder to the next digit.

1. Write out the 24 times table up to 24 × 9. Having the multiples ready makes it quick to see how many times 24 goes into each number.
   24 × 1 = 24
   24 × 2 = 48
   24 × 3 = 72
   24 × 4 = 96
   24 × 5 = 120
   24 × 6 = 144
   24 × 7 = 168
   24 × 8 = 192
   24 × 9 = 216
2. Set out the division. Write 936 under the "bus stop" and 24 outside it. The answer will be written on top, one digit above each digit we divide.
   24 ) 936
3. Start with the first 2 digits, 93. 24 does not go into the first digit, so we take digits from the left until we have a number that 24 can go into.
   24 goes into 93 3 times: 24 × 3 = 72
   Subtract: 93 − 72 = 21
4. Bring down the next digit, 6, to make 216. The remainder 21 goes in front of the 6. This is the same as carrying the remainder to the next column.
   24 goes into 216 9 times: 24 × 9 = 216
   Subtract: 216 − 216 = 0
5. Write the answer. Read the digits along the top.
   39 24) 936 72 -- 216 216 --- 0
   936 ÷ 24 = 39

Check. Multiply back: 39 × 24 = 936. ✓

Watch out. If 24 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Long division: work from the left, dividing 34 into each number in turn, carrying the remainder to the next digit.

1. Write out the 34 times table up to 34 × 9. Having the multiples ready makes it quick to see how many times 34 goes into each number.
   34 × 1 = 34
   34 × 2 = 68
   34 × 3 = 102
   34 × 4 = 136
   34 × 5 = 170
   34 × 6 = 204
   34 × 7 = 238
   34 × 8 = 272
   34 × 9 = 306
2. Set out the division. Write 7523 under the "bus stop" and 34 outside it. The answer will be written on top, one digit above each digit we divide.
   34 ) 7523
3. Start with the first 2 digits, 75. 34 does not go into the first digit, so we take digits from the left until we have a number that 34 can go into.
   34 goes into 75 2 times: 34 × 2 = 68
   Subtract: 75 − 68 = 7
4. Bring down the next digit, 2, to make 72. The remainder 7 goes in front of the 2. This is the same as carrying the remainder to the next column.
   34 goes into 72 2 times: 34 × 2 = 68
   Subtract: 72 − 68 = 4
5. Bring down the next digit, 3, to make 43. The remainder 4 goes in front of the 3. This is the same as carrying the remainder to the next column.
   34 goes into 43 1 time: 34 × 1 = 34
   Subtract: 43 − 34 = 9
6. Write the answer. Read the digits along the top.
   221 34) 7523 68 -- 72 68 -- 43 34 -- 9
   7523 ÷ 34 = 221 remainder 9

Check. Multiply back: 221 × 34 = 7514, then add the remainder 9: 7514 + 9 = 7523. ✓

Watch out. If 34 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Long division: work from the left, dividing 28 into each number in turn, carrying the remainder to the next digit.

1. Write out the 28 times table up to 28 × 9. Having the multiples ready makes it quick to see how many times 28 goes into each number.
   28 × 1 = 28
   28 × 2 = 56
   28 × 3 = 84
   28 × 4 = 112
   28 × 5 = 140
   28 × 6 = 168
   28 × 7 = 196
   28 × 8 = 224
   28 × 9 = 252
2. Set out the division. Write 4704 under the "bus stop" and 28 outside it. The answer will be written on top, one digit above each digit we divide.
   28 ) 4704
3. Start with the first 2 digits, 47. 28 does not go into the first digit, so we take digits from the left until we have a number that 28 can go into.
   28 goes into 47 1 time: 28 × 1 = 28
   Subtract: 47 − 28 = 19
4. Bring down the next digit, 0, to make 190. The remainder 19 goes in front of the 0. This is the same as carrying the remainder to the next column.
   28 goes into 190 6 times: 28 × 6 = 168
   Subtract: 190 − 168 = 22
5. Bring down the next digit, 4, to make 224. The remainder 22 goes in front of the 4. This is the same as carrying the remainder to the next column.
   28 goes into 224 8 times: 28 × 8 = 224
   Subtract: 224 − 224 = 0
6. Write the answer. Read the digits along the top.
   168 28) 4704 28 -- 190 168 --- 224 224 --- 0
   4704 ÷ 28 = 168

Check. Multiply back: 168 × 28 = 4704. ✓

Watch out. If 28 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Long division: work from the left, dividing 16 into each number in turn, carrying the remainder to the next digit.

1. Write out the 16 times table up to 16 × 9. Having the multiples ready makes it quick to see how many times 16 goes into each number.
   16 × 1 = 16
   16 × 2 = 32
   16 × 3 = 48
   16 × 4 = 64
   16 × 5 = 80
   16 × 6 = 96
   16 × 7 = 112
   16 × 8 = 128
   16 × 9 = 144
2. Set out the division. Write 10000 under the "bus stop" and 16 outside it. The answer will be written on top, one digit above each digit we divide.
   16 ) 10000
3. Start with the first 3 digits, 100. 16 does not go into the first digit, so we take digits from the left until we have a number that 16 can go into.
   16 goes into 100 6 times: 16 × 6 = 96
   Subtract: 100 − 96 = 4
4. Bring down the next digit, 0, to make 40. The remainder 4 goes in front of the 0. This is the same as carrying the remainder to the next column.
   16 goes into 40 2 times: 16 × 2 = 32
   Subtract: 40 − 32 = 8
5. Bring down the next digit, 0, to make 80. The remainder 8 goes in front of the 0. This is the same as carrying the remainder to the next column.
   16 goes into 80 5 times: 16 × 5 = 80
   Subtract: 80 − 80 = 0
6. Write the answer. Read the digits along the top.
   625 16) 10000 96 -- 40 32 -- 80 80 -- 0
   10 000 ÷ 16 = 625

Check. Multiply back: 625 × 16 = 10 000. ✓

Watch out. If 16 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 4 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 9 under the "bus stop" and 4 outside it. The answer will be written on top, one digit above each digit we divide.
   4 ) 9
2. Start with the first digit, 9. Work from the left. We ask how many times 4 goes into 9.
   9 ÷ 4 = 2 remainder 1 (since 4 × 2 = 8)
3. There is a remainder of 1, so add a 0 on the end and bring it down to make 10. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   10 ÷ 4 = 2 remainder 2 (since 4 × 2 = 8)
4. There is a remainder of 2, so add a 0 on the end and bring it down to make 20. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   20 ÷ 4 = 5 (since 4 × 5 = 20)
5. Write the answer. Read the digits along the top. The decimal point stays in line with the point in the question.
   2.25 12 4) 9.00
   9 ÷ 4 = 2.25

Check. Multiply back: 2.25 × 4 = 9. ✓

Watch out. If 4 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 8 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 3 under the "bus stop" and 8 outside it. The answer will be written on top, one digit above each digit we divide.
   8 ) 3
2. Start with the first digit, 3. Work from the left. We ask how many times 8 goes into 3.
   8 does not go into 3, so write 0 and carry on.
3. There is a remainder of 3, so add a 0 on the end and bring it down to make 30. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   30 ÷ 8 = 3 remainder 6 (since 8 × 3 = 24)
4. There is a remainder of 6, so add a 0 on the end and bring it down to make 60. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   60 ÷ 8 = 7 remainder 4 (since 8 × 7 = 56)
5. There is a remainder of 4, so add a 0 on the end and bring it down to make 40. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   40 ÷ 8 = 5 (since 8 × 5 = 40)
6. Write the answer. Read the digits along the top. The decimal point stays in line with the point in the question.
   0.375 364 8) 3.000
   3 ÷ 8 = 0.375

Check. Multiply back: 0.375 × 8 = 3. ✓

Watch out. If 8 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Long division: work from the left, dividing 20 into each number in turn, carrying the remainder to the next digit.

1. Write out the 20 times table up to 20 × 9. Having the multiples ready makes it quick to see how many times 20 goes into each number.
   20 × 1 = 20
   20 × 2 = 40
   20 × 3 = 60
   20 × 4 = 80
   20 × 5 = 100
   20 × 6 = 120
   20 × 7 = 140
   20 × 8 = 160
   20 × 9 = 180
2. Set out the division. Write 17 under the "bus stop" and 20 outside it. The answer will be written on top, one digit above each digit we divide.
   20 ) 17
3. Start with the first 2 digits, 17. 20 does not go into the first digit, so we take digits from the left until we have a number that 20 can go into.
   20 does not go into 17, so write 0 in the answer.
4. There is a remainder of 17, so add a 0 on the end and bring it down to make 170. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   20 goes into 170 8 times: 20 × 8 = 160
   Subtract: 170 − 160 = 10
5. There is a remainder of 10, so add a 0 on the end and bring it down to make 100. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   20 goes into 100 5 times: 20 × 5 = 100
   Subtract: 100 − 100 = 0
6. Write the answer. Read the digits along the top. The decimal point stays in line with the point in the question.
   0.85 20) 17.00 160 --- 100 100 --- 0
   17 ÷ 20 = 0.85

Check. Multiply back: 0.85 × 20 = 17. ✓

Watch out. If 20 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Long division: work from the left, dividing 16 into each number in turn, carrying the remainder to the next digit.

1. Write out the 16 times table up to 16 × 9. Having the multiples ready makes it quick to see how many times 16 goes into each number.
   16 × 1 = 16
   16 × 2 = 32
   16 × 3 = 48
   16 × 4 = 64
   16 × 5 = 80
   16 × 6 = 96
   16 × 7 = 112
   16 × 8 = 128
   16 × 9 = 144
2. Set out the division. Write 45 under the "bus stop" and 16 outside it. The answer will be written on top, one digit above each digit we divide.
   16 ) 45
3. Start with the first 2 digits, 45. 16 does not go into the first digit, so we take digits from the left until we have a number that 16 can go into.
   16 goes into 45 2 times: 16 × 2 = 32
   Subtract: 45 − 32 = 13
4. There is a remainder of 13, so add a 0 on the end and bring it down to make 130. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   16 goes into 130 8 times: 16 × 8 = 128
   Subtract: 130 − 128 = 2
5. There is a remainder of 2, so add a 0 on the end and bring it down to make 20. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   16 goes into 20 1 time: 16 × 1 = 16
   Subtract: 20 − 16 = 4
6. There is a remainder of 4, so add a 0 on the end and bring it down to make 40. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   16 goes into 40 2 times: 16 × 2 = 32
   Subtract: 40 − 32 = 8
7. There is a remainder of 8, so add a 0 on the end and bring it down to make 80. Adding a zero after the last decimal place does not change the value of the number, but it lets us keep dividing to get an exact answer.
   16 goes into 80 5 times: 16 × 5 = 80
   Subtract: 80 − 80 = 0
8. Write the answer. Read the digits along the top. The decimal point stays in line with the point in the question.
   2.8125 16) 45.0000 32 -- 13 0 128 --- 20 16 -- 40 32 -- 80 80 -- 0
   45 ÷ 16 = 2.8125

Check. Multiply back: 2.8125 × 16 = 45. ✓

Watch out. If 16 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 5 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 7.5 under the "bus stop" and 5 outside it. The answer will be written on top, one digit above each digit we divide.
   5 ) 7.5
2. Start with the first digit, 7. Work from the left. We ask how many times 5 goes into 7.
   7 ÷ 5 = 1 remainder 2 (since 5 × 1 = 5)
3. The next digit is after the decimal point. Put a decimal point in the answer, then bring down 5 to make 25. The decimal point in the answer goes directly above the decimal point in the question.
   25 ÷ 5 = 5 (since 5 × 5 = 25)
4. Write the answer. Read the digits along the top. The decimal point stays in line with the point in the question.
   1.5 2 5) 7.5
   7.5 ÷ 5 = 1.5

Check. Multiply back: 1.5 × 5 = 7.5. ✓

Watch out. If 5 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 7 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 0.84 under the "bus stop" and 7 outside it. The answer will be written on top, one digit above each digit we divide.
   7 ) 0.84
2. The number is less than 1, so the answer starts "0." Put the decimal point in the answer, then look at the first digit after the point, 8. We still ask how many times 7 goes into 8.
   8 ÷ 7 = 1 remainder 1 (since 7 × 1 = 7)
3. Bring down the next digit, 4, to make 14. The remainder 1 goes in front of the 4. This is the same as carrying the remainder to the next column.
   14 ÷ 7 = 2 (since 7 × 2 = 14)
4. Write the answer. Read the digits along the top. The decimal point stays in line with the point in the question.
   0.12 1 7) 0.84
   0.84 ÷ 7 = 0.12

Check. Multiply back: 0.12 × 7 = 0.84. ✓

Watch out. If 7 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 9 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 12.6 under the "bus stop" and 9 outside it. The answer will be written on top, one digit above each digit we divide.
   9 ) 12.6
2. Start with the first 2 digits, 12. 9 does not go into the first digit, so we take digits from the left until we have a number that 9 can go into.
   12 ÷ 9 = 1 remainder 3 (since 9 × 1 = 9)
3. The next digit is after the decimal point. Put a decimal point in the answer, then bring down 6 to make 36. The decimal point in the answer goes directly above the decimal point in the question.
   36 ÷ 9 = 4 (since 9 × 4 = 36)
4. Write the answer. Read the digits along the top. The decimal point stays in line with the point in the question.
   1.4 1 3 9) 12.6
   12.6 ÷ 9 = 1.4

Check. Multiply back: 1.4 × 9 = 12.6. ✓

Watch out. If 9 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 12 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 20.4 under the "bus stop" and 12 outside it. The answer will be written on top, one digit above each digit we divide.
   12 ) 20.4
2. Start with the first 2 digits, 20. 12 does not go into the first digit, so we take digits from the left until we have a number that 12 can go into.
   20 ÷ 12 = 1 remainder 8 (since 12 × 1 = 12)
3. The next digit is after the decimal point. Put a decimal point in the answer, then bring down 4 to make 84. The decimal point in the answer goes directly above the decimal point in the question.
   84 ÷ 12 = 7 (since 12 × 7 = 84)
4. Write the answer. Read the digits along the top. The decimal point stays in line with the point in the question.
   1.7 2 8 12) 20.4
   20.4 ÷ 12 = 1.7

Check. Multiply back: 1.7 × 12 = 20.4. ✓

Watch out. If 12 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Dividing by a decimal: multiply both numbers by 10, 100... until the divisor is a whole number. Then divide. Short division: work from the left, dividing 4 into each number in turn, carrying the remainder to the next digit.

1. Make the number you are dividing by a whole number. Multiplying both numbers by 10 does not change the answer (it is like multiplying the top and bottom of a fraction by the same number). It makes the divisor a whole number, which is much easier.
   7.2 ÷ 0.4 = 72 ÷ 4
   (7.2 × 10 = 72 and 0.4 × 10 = 4)
2. Set out the division. Write 72 under the "bus stop" and 4 outside it. The answer will be written on top, one digit above each digit we divide.
   4 ) 72
3. Start with the first digit, 7. Work from the left. We ask how many times 4 goes into 7.
   7 ÷ 4 = 1 remainder 3 (since 4 × 1 = 4)
4. Bring down the next digit, 2, to make 32. The remainder 3 goes in front of the 2. This is the same as carrying the remainder to the next column.
   32 ÷ 4 = 8 (since 4 × 8 = 32)
5. Write the answer. Read the digits along the top.
   18 3 4) 72
   72 ÷ 4 = 18

Check. Multiply back: 18 × 4 = 72. ✓

Watch out. If 4 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Dividing by a decimal: multiply both numbers by 10, 100... until the divisor is a whole number. Then divide. Short division: work from the left, dividing 7 into each number in turn, carrying the remainder to the next digit.

1. Make the number you are dividing by a whole number. Multiplying both numbers by 100 does not change the answer (it is like multiplying the top and bottom of a fraction by the same number). It makes the divisor a whole number, which is much easier.
   5.6 ÷ 0.07 = 560 ÷ 7
   (5.6 × 100 = 560 and 0.07 × 100 = 7)
2. Set out the division. Write 560 under the "bus stop" and 7 outside it. The answer will be written on top, one digit above each digit we divide.
   7 ) 560
3. Start with the first 2 digits, 56. 7 does not go into the first digit, so we take digits from the left until we have a number that 7 can go into.
   56 ÷ 7 = 8 (since 7 × 8 = 56)
4. Bring down the next digit, 0. The remainder was 0, so the new number to divide is just that digit.
   7 does not go into 0, so write 0 and carry on.
5. Write the answer. Read the digits along the top.
   80 5 7) 560
   560 ÷ 7 = 80

Check. Multiply back: 80 × 7 = 560. ✓

Watch out. If 7 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Dividing by a decimal: multiply both numbers by 10, 100... until the divisor is a whole number. Then divide. Short division: work from the left, dividing 3 into each number in turn, carrying the remainder to the next digit.

1. Make the number you are dividing by a whole number. Multiplying both numbers by 100 does not change the answer (it is like multiplying the top and bottom of a fraction by the same number). It makes the divisor a whole number, which is much easier.
   0.9 ÷ 0.03 = 90 ÷ 3
   (0.9 × 100 = 90 and 0.03 × 100 = 3)
2. Set out the division. Write 90 under the "bus stop" and 3 outside it. The answer will be written on top, one digit above each digit we divide.
   3 ) 90
3. Start with the first digit, 9. Work from the left. We ask how many times 3 goes into 9.
   9 ÷ 3 = 3 (since 3 × 3 = 9)
4. Bring down the next digit, 0. The remainder was 0, so the new number to divide is just that digit.
   3 does not go into 0, so write 0 and carry on.
5. Write the answer. Read the digits along the top.
   30 3) 90
   90 ÷ 3 = 30

Check. Multiply back: 30 × 3 = 90. ✓

Watch out. If 3 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Dividing by a decimal: multiply both numbers by 10, 100... until the divisor is a whole number. Then divide. Long division: work from the left, dividing 15 into each number in turn, carrying the remainder to the next digit.

1. Make the number you are dividing by a whole number. Multiplying both numbers by 10 does not change the answer (it is like multiplying the top and bottom of a fraction by the same number). It makes the divisor a whole number, which is much easier.
   4.5 ÷ 1.5 = 45 ÷ 15
   (4.5 × 10 = 45 and 1.5 × 10 = 15)
2. Write out the 15 times table up to 15 × 9. Having the multiples ready makes it quick to see how many times 15 goes into each number.
   15 × 1 = 15
   15 × 2 = 30
   15 × 3 = 45
   15 × 4 = 60
   15 × 5 = 75
   15 × 6 = 90
   15 × 7 = 105
   15 × 8 = 120
   15 × 9 = 135
3. Set out the division. Write 45 under the "bus stop" and 15 outside it. The answer will be written on top, one digit above each digit we divide.
   15 ) 45
4. Start with the first 2 digits, 45. 15 does not go into the first digit, so we take digits from the left until we have a number that 15 can go into.
   15 goes into 45 3 times: 15 × 3 = 45
   Subtract: 45 − 45 = 0
5. Write the answer. Read the digits along the top.
   3 15) 45 45 -- 0
   45 ÷ 15 = 3

Check. Multiply back: 3 × 15 = 45. ✓

Watch out. If 15 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Dividing by a decimal: multiply both numbers by 10, 100... until the divisor is a whole number. Then divide. Long division: work from the left, dividing 25 into each number in turn, carrying the remainder to the next digit.

1. Make the number you are dividing by a whole number. Multiplying both numbers by 100 does not change the answer (it is like multiplying the top and bottom of a fraction by the same number). It makes the divisor a whole number, which is much easier.
   18 ÷ 0.25 = 1800 ÷ 25
   (18 × 100 = 1800 and 0.25 × 100 = 25)
2. Write out the 25 times table up to 25 × 9. Having the multiples ready makes it quick to see how many times 25 goes into each number.
   25 × 1 = 25
   25 × 2 = 50
   25 × 3 = 75
   25 × 4 = 100
   25 × 5 = 125
   25 × 6 = 150
   25 × 7 = 175
   25 × 8 = 200
   25 × 9 = 225
3. Set out the division. Write 1800 under the "bus stop" and 25 outside it. The answer will be written on top, one digit above each digit we divide.
   25 ) 1800
4. Start with the first 3 digits, 180. 25 does not go into the first digit, so we take digits from the left until we have a number that 25 can go into.
   25 goes into 180 7 times: 25 × 7 = 175
   Subtract: 180 − 175 = 5
5. Bring down the next digit, 0, to make 50. The remainder 5 goes in front of the 0. This is the same as carrying the remainder to the next column.
   25 goes into 50 2 times: 25 × 2 = 50
   Subtract: 50 − 50 = 0
6. Write the answer. Read the digits along the top.
   72 25) 1800 175 --- 50 50 -- 0
   1800 ÷ 25 = 72

Check. Multiply back: 72 × 25 = 1800. ✓

Watch out. If 25 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Short division: work from the left, dividing 12 into each number in turn, carrying the remainder to the next digit.

1. Set out the division. Write 9.84 under the "bus stop" and 12 outside it. The answer will be written on top, one digit above each digit we divide.
   12 ) 9.84
2. Start with the first digit, 9. Work from the left. We ask how many times 12 goes into 9.
   12 does not go into 9, so write 0 and carry on.
3. The next digit is after the decimal point. Put a decimal point in the answer, then bring down 8 to make 98. The decimal point in the answer goes directly above the decimal point in the question.
   98 ÷ 12 = 8 remainder 2 (since 12 × 8 = 96)
4. Bring down the next digit, 4, to make 24. The remainder 2 goes in front of the 4. This is the same as carrying the remainder to the next column.
   24 ÷ 12 = 2 (since 12 × 2 = 24)
5. Write the answer. Read the digits along the top. The decimal point stays in line with the point in the question.
   0.82 92 12) 9.84
   9.84 ÷ 12 = 0.82

Check. Multiply back: 0.82 × 12 = 9.84. ✓

Watch out. If 12 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. A decimal place is a digit after the decimal point. 3.45 has 2 decimal places.

Method. Multiply decimals as whole numbers first, then put the decimal point back using the total number of decimal places in the question.

1. Ignore the decimal points and multiply as whole numbers. 0.82 is 82 divided by 100, and 15 is 15 divided by 1. So we can multiply 82 by 15, then fix the size at the end.
   0.82 × 15 → 82 × 15
2. Multiply 82 by the units digit, 5. Work from the right. Multiply each digit of the top number by 5. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   2 × 5 = 10: write 0, carry 1
   8 × 5 + 1 (carried) = 41: write 1, carry 4
   Write the last carry: 4
   So 82 × 5 = 410.
3. Multiply 82 by the tens digit, 1 (which means 10). The 1 is in the tens column, so it is really 10. First write 1 zero on the right as place holder, then multiply by 1.
   2 × 1 = 2: write 2
   8 × 1 = 8: write 8
   82 × 1 = 82, then add 1 zero: 82 × 10 = 820.
4. Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.
   410 + 820 = 1230
   82 × 15 ---- 410 820 ----- 1230
5. Put the decimal point back. Count the decimal places in the question: 0.82 has 2 and 15 has 0, so the answer has 2 + 0 = 2 decimal places. Count that many digits in from the right.
   1230 with 2 decimal places: 12.3

Check. Estimate: 0.8 × 20 = 16. 12.3 is close to 16. ✓

Watch out. Count decimal places in both numbers and add them. The answer has as many as the two numbers together.

Method. Make the numbers easy to compare, then order them smallest to largest.

1. Sort the negative numbers from the positive ones. Every negative number is smaller than every positive number or zero, because it lies to the left of zero on the number line.
   Negative: −5, −8, −1
   Not negative: 3, 0
2. Count the digits in each number. Compare the sizes first, ignoring the minus signs. More digits means a bigger size.
   5: 1 digit, 3: 1 digit, 8: 1 digit, 0: 1 digit, 1: 1 digit
3. Where the numbers have the same number of digits, compare column by column from the left. The first column where two numbers differ decides which is bigger.
   5, 3, 8, 0, 1
4. Use the signs to finish. Among negative numbers, the one with the bigger size is the smaller number (it is further left of zero).
   −8, −5, −1, 0, 3

Check. Check each neighbouring pair: −8 < −5; −5 < −1; −1 < 0; 0 < 3. ✓

Watch out. For negative numbers, the number with the bigger digits is the smaller number.

Method. Make the numbers easy to compare, then order them smallest to largest.

1. Sort the negative numbers from the positive ones. Every negative number is smaller than every positive number or zero, because it lies to the left of zero on the number line.
   Negative: −0.5, −0.45, −0.55
   Not negative: 0.4
2. Write every number with the same number of decimal places, using zeros. The extra zeros do not change the values. They make the columns line up so we can compare digit by digit.
   −0.5 = −0.50
   −0.45 = −0.45
   −0.55 = −0.55
   0.4 = 0.40
3. Compare the sizes, ignoring the decimal point and any minus signs. Now every number has 2 decimal places, so we can compare them as if they were whole numbers. (The minus signs are dealt with next.)
   0.5: 50, 0.45: 45, 0.55: 55, 0.4: 40
4. Use the signs to finish. Among negative numbers, the one with the bigger size is the smaller number (it is further left of zero).
   −0.55, −0.5, −0.45, 0.4

Check. Check each neighbouring pair: −0.55 < −0.5; −0.5 < −0.45; −0.45 < 0.4. ✓

Watch out. For negative numbers, the number with the bigger digits is the smaller number.

Method. Make the numbers easy to compare, then order them largest to smallest.

1. Sort the negative numbers from the positive ones. Every negative number is smaller than every positive number or zero, because it lies to the left of zero on the number line.
   Negative: −12, −2, −21
   Not negative: 7
2. Count the digits in each number. Compare the sizes first, ignoring the minus signs. More digits means a bigger size.
   12: 2 digits, 2: 1 digit, 7: 1 digit, 21: 2 digits
3. Where the numbers have the same number of digits, compare column by column from the left. The first column where two numbers differ decides which is bigger.
   12, 2, 7, 21
4. Use the signs to finish. Among negative numbers, the one with the bigger size is the smaller number (it is further left of zero).
   7, −2, −12, −21

Check. Check each neighbouring pair: 7 > −2; −2 > −12; −12 > −21. ✓

Watch out. For negative numbers, the number with the bigger digits is the smaller number.

Method. For two negative numbers, the one closer to zero is bigger. Compare the sizes, then reverse the symbol.

1. Both numbers are negative: ignore the minus signs for a moment. We compare how far each is from zero.
   3 and 5
2. Compare the sizes. The number further from zero is the smaller number when it is negative.
   3 < 5 so the order reverses for negatives
3. Write the final comparison. Reverse the symbol.
   −3 > −5

Check. On a number line the bigger number is to the right. ✓

Watch out. −7 is smaller than −2, even though 7 is bigger than 2.

Method. Negative numbers are below zero, so they are smaller than positive numbers.

1. Look at the signs. A negative number is always smaller than a positive number or zero, because it is below zero on the number line.
   −7 is negative, 2 is not.
2. Decide. The negative one is smaller.
   −7 < 2

Check. On a number line −7 is to the left of 2. ✓

Method. For two negative numbers, the one closer to zero is bigger. Compare the sizes, then reverse the symbol.

1. Both numbers are negative: ignore the minus signs for a moment. We compare how far each is from zero.
   0.4 and 0.04
2. Compare the sizes. The number further from zero is the smaller number when it is negative.
   0.4 > 0.04 so the order reverses for negatives
3. Write the final comparison. Reverse the symbol.
   −0.4 < −0.04

Check. On a number line the bigger number is to the right. ✓

Watch out. −7 is smaller than −2, even though 7 is bigger than 2.

Method. For two negative numbers, the one closer to zero is bigger. Compare the sizes, then reverse the symbol.

1. Both numbers are negative: ignore the minus signs for a moment. We compare how far each is from zero.
   12 and 21
2. Compare the sizes. The number further from zero is the smaller number when it is negative.
   12 < 21 so the order reverses for negatives
3. Write the final comparison. Reverse the symbol.
   −12 > −21

Check. On a number line the bigger number is to the right. ✓

Watch out. −7 is smaller than −2, even though 7 is bigger than 2.

Method. Find 0 on the scale, then count the marks below it.

1. Find 0 on the scale. Marks above 0 are positive and marks below 0 are negative.
   0°C
2. Count down to A. A is 7 small marks below 0, so it is a negative temperature.
   A = −7°C

Check. 7 below zero is −7. ✓

Method. Find 0 on the scale, then count the marks above it.

1. Find 0 on the scale.
   0°C
2. Count up to B. B is 4 small marks above 0, so it is positive.
   B = 4°C

Check. 4 above zero is 4. ✓

Method. The colder place has the lower temperature, which is the lower point on the thermometer.

1. Compare the two temperatures. A negative number is always less than a positive number.
   −7 < 4
2. Decide. The lower temperature is colder.
   A is colder.

Check. A is below zero but B is above zero. ✓

Method. The difference is the higher temperature minus the lower temperature.

1. Write the subtraction. Higher minus lower.
   4 − (−7)
2. Deal with the two signs next to each other: subtracting a negative. Subtracting a negative cancels it out: if someone takes away a debt of 7, you are 7 better off. So "− (−7)" is the same as "+ 7".
   4 − (−7) = 4 + 7
3. Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.
   Start at 4 on the number line and move 7 steps to the right (adding moves up).
   4 + 7 = 11

Check. From −7 up to 0 is 7 degrees, and from 0 to 4 is 4 degrees: 7 + 4 = 11. ✓

Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).

Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.

1. Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.
   −3 + 5
2. Look at the signs. The number being added is positive, so there are no double signs to fix.
   −3 + 5
3. Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.
   Start at −3 on the number line and move 5 steps to the right (adding moves up).
   3 steps take you to 0, then 2 more steps take you to 2.
   −3 + 5 = 2

Check. Reverse the calculation: 2 − 5 = −3. ✓

Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.

Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).

Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.

1. Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.
   4 − 9
2. Look at the signs. The number being subtracted is positive, so there are no double signs to fix.
   4 − 9
3. Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.
   Start at 4 on the number line and move 9 steps to the left (subtracting moves down).
   4 steps take you down to 0, then 5 more steps take you to −5.
   4 − 9 = −5

Check. Reverse the calculation: −5 + 9 = 4. ✓

Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.

Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).

Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.

1. Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.
   −7 − 2
2. Look at the signs. The number being subtracted is positive, so there are no double signs to fix.
   −7 − 2
3. Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.
   Start at −7 on the number line and move 2 steps to the left (subtracting moves down).
   You start at or below zero and go further down, so the answer is more negative: 7 + 2 = 9, and with a minus sign that is −9.
   −7 − 2 = −9

Check. Reverse the calculation: −9 + 2 = −7. ✓

Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.

Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).

Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.

1. Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.
   6 − (−4)
2. Deal with the two signs next to each other: subtracting a negative. Subtracting a negative cancels it out: if someone takes away a debt of 4, you are 4 better off. So "− (−4)" is the same as "+ 4".
   6 − (−4) = 6 + 4
3. Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.
   Start at 6 on the number line and move 4 steps to the right (adding moves up).
   6 + 4 = 10

Check. Reverse the calculation: 10 + (−4) = 6. ✓

Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.

Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).

Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.

1. Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.
   −5 + (−3)
2. Deal with the two signs next to each other: adding a negative. Adding a negative number takes away: if you add a debt of 3 you are 3 worse off. So "+ (−3)" is the same as "− 3".
   −5 + (−3) = −5 − 3
3. Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.
   Start at −5 on the number line and move 3 steps to the left (subtracting moves down).
   You start at or below zero and go further down, so the answer is more negative: 5 + 3 = 8, and with a minus sign that is −8.
   −5 − 3 = −8

Check. Reverse the calculation: −8 − (−3) = −5. ✓

Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.

Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).

Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.

1. Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.
   −2 − (−8)
2. Deal with the two signs next to each other: subtracting a negative. Subtracting a negative cancels it out: if someone takes away a debt of 8, you are 8 better off. So "− (−8)" is the same as "+ 8".
   −2 − (−8) = −2 + 8
3. Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.
   Start at −2 on the number line and move 8 steps to the right (adding moves up).
   2 steps take you to 0, then 6 more steps take you to 6.
   −2 + 8 = 6

Check. Reverse the calculation: 6 + (−8) = −2. ✓

Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.