GCSE Maths: Place value and ordering numbers (Higher)
Part of the GCSE Maths number basics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Place value and ordering numbers
Every digit in a number has a value that depends on its place (its column). In 4 305 the digit 3 is in the hundreds column, so it is worth 300, not 3. The 0 is a place holder: it holds the units and tens columns open so that the 3 and the 4 stay in the correct columns. Each column is worth ten times the column to its right, and one tenth of the column to its left.
Reading and writing numbers. We read big numbers in groups of three digits. 3 405 271 is “three million, four hundred and five thousand, two hundred and seventy-one”. The digits after the decimal point are decimal places: tenths, hundredths, thousandths. We read them one at a time: 0.869 is “nought point eight six nine”.
Comparing and ordering. To decide which of two numbers is bigger, compare them column by column from the left. A whole number with more digits is bigger. For decimals, do not count digits: first write both numbers with the same number of decimal places (adding zeros on the end does not change a decimal). Then 0.5 = 0.50 is bigger than 0.45, because 50 hundredths is more than 45 hundredths.
Symbols. We use these symbols to compare numbers. The wide end of < or > always faces the bigger number.
| Symbol | Meaning | Example |
|---|---|---|
| = | is equal to | 0.5 = 0.50 |
| ≠ | is not equal to | 0.5 ≠ 0.05 |
| < | is less than | 3 < 7 |
| > | is greater than | 7 > 3 |
| ≤ | is less than or equal to | n ≤ 5 means n can be 5 or smaller |
| ≥ | is greater than or equal to | n ≥ 5 means n can be 5 or bigger |
Multiplying and dividing by 10, 100, 1000. When you multiply by 10 every digit moves one column to the left (it becomes ten times bigger). When you divide by 10 every digit moves one column to the right. The digits move; the decimal point stays where it is. So 3.47 × 100 = 347 and 3.47 ÷ 10 = 0.347.
Key ideas
- Each column is worth 10 times the column on its right.
- Compare numbers column by column from the left. For decimals, add zeros so the lengths match first.
- < means “less than”, > means “greater than”, ≤ and ≥ also allow “equal to”.
- × 10 moves every digit one place left; ÷ 10 moves every digit one place right.
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.
- 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 tenths hundredths 4 0 7 5 . 3 1 - Find the column of the digit 7. Reading the headings, the 7 is in the tens column. One in this column is worth 10.The tens column is worth 10.
- Multiply the digit by the column value. There are 7 lots of 10.7 × 10 = 70
Check. In words, the digit 7 stands for 7 tens, which is 70. ✓
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.
- 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.3 million | 450 thousand | 72
- 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.3 million = three million
450 thousand = four hundred and fifty thousand
72 = seventy-two - Join the groups. Put the groups together in order, with "and" before the last part if it is less than 100.three million four hundred and fifty thousand and seventy-two
Check. Say it out loud: does "three million four hundred and fifty thousand and seventy-two" sound like 3 450 072? ✓
Watch out. Do not write "and" between every part. We say "four thousand and fifty-six", but "four thousand two hundred and six".
Method. Make the numbers easy to compare, then order them smallest to largest.
- 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 = 0.500
0.45 = 0.450
0.405 = 0.405
0.54 = 0.540 - 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.0.5: 500, 0.45: 450, 0.405: 405, 0.54: 540
- Write the original numbers in order. Order smallest to largest.0.405, 0.45, 0.5, 0.54
Check. Check each neighbouring pair: 0.405 < 0.45; 0.45 < 0.5; 0.5 < 0.54. ✓
Watch out. Ordering decimals: line up the decimal places. Do not decide by the length of the number.
Method. Find the lower limit and the upper limit, decide whether each end is included, then list the whole numbers in between.
- Read the inequality in two parts. A double inequality gives a lower limit and an upper limit. n must satisfy both at the same time.3 ≤ n means n is at least 3.
n < 8 means n is smaller than 8. - Decide whether each end is included. The symbols ≤ and ≥ include the number (the number itself is allowed). The symbols < and > do not include it.3 is included; 8 is not included.
- Find the smallest and the biggest whole number allowed. We want whole numbers (integers), so look at the whole numbers just inside the limits.Smallest: 3
Biggest: 7 - List every whole number from the smallest to the biggest. 3, 4, 5, 6, 7
Check. Test the ends: 3 is allowed, 2 is not; 7 is allowed, 8 is not. ✓
Watch out. A filled circle (or ≤ ≥) means the end number is included; an open circle (or < >) means it is not.
Method. Use the midpoint: it is always strictly between two different numbers.
- Write both numbers with the same number of decimal places, adding zeros. This lets us see the gap between them.0.7 = 0.700
0.71 = 0.710 - Add one more decimal place to make room. If the numbers are neighbours (like 0.7 and 0.71) a longer decimal fits between them.0.700 < ? < 0.710
- Choose a number that is bigger than the first and smaller than the second. The number halfway between them is always a good choice: add the two numbers and halve.(0.7 + 0.71) ÷ 2 = 1.41 ÷ 2 = 0.705
Check. 0.7 < 0.705 < 0.71. ✓ (Many other answers are correct too.)
Watch out
A longer decimal is not a bigger decimal. 0.5 is bigger than 0.45 even though 45 is bigger than 5. Write both with the same number of decimal places first (0.50 and 0.45).Exercise 1A Place value and ordering
Non-calculatorKey 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.
- 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 - 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.
- 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.
- 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 - 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.
- 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.
- 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 - 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.
- 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.
- 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 - 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.
- 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.
- 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 - 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.
- 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.
- 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 - 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.
- 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.
- 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
- 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 - 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.
- 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
- 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 - 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.
- 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
- 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 - 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.
- 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
- 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
- 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.
- 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 - 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
- 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.
- 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 - 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
- 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.
- 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 - 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
- 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.
- 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 - 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
- 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.
- 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 - 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
- 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.
- 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 - 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
- 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.
- 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 - 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
- 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.
- 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.
- Compare the digits from the left. The first column where they differ decides it.4509 and 4590
- 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.
- 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 - 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
- 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.
- 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 - 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
- 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.
- 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
- 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
- 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.
- 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 - 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
- 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.
- 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 - 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
- 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.
- 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
- 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) - 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.
- 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
- 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) - 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.
- 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
- 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) - 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.
- 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
- 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) - 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.
- 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
- 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) - 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.
- 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
- 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) - 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.
- 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
- 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”.
- Compare the two numbers. They are the same, so 7 is equal to 7.7 = 7
- 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.
- 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
- 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.
- Count the digits. A whole number with more digits is bigger.1000 has 4 digits and 999 has 3 digits.
- 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.
- 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
- 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).
- 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
- 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.
- 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
- 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
- Choose the biggest of these numbers. 9316
Check. It uses each of the digits 1, 3, 6, 9 exactly once. ✓
Method. Compare the numbers column by column from the left.
- Compare the ten thousands, then the thousands. All four numbers have 5 digits, so compare from the left. Three start 45 and one starts 44.44 999 is the smallest because 44 thousand is less than 45 thousand.
- Compare the hundreds of the other three. 45 302, 45 032 and 45 230 have 3, 0 and 2 hundreds.3 hundreds > 2 hundreds > 0 hundreds
- Write the crowds from biggest to smallest. 45 302, 45 230, 45 032, 44 999
Check. Each number is smaller than the one before it: 45 302 > 45 230 > 45 032 > 44 999. ✓
Method. Compare the two decimals using the same number of decimal places.
- Write both numbers with two decimal places. We add a zero so the columns match. This does not change the value.0.4 = 0.40 and 0.35 = 0.35
- Compare the tenths column first. 0.40 has 4 tenths and 0.35 has 3 tenths. Tenths are the biggest columns after the point.4 tenths > 3 tenths
- Say what Kasim did wrong. He compared 35 with 4 as whole numbers. But 4 tenths is 40 hundredths, and 40 > 35.So No: 0.4 is bigger than 0.35.
Check. 0.40 − 0.35 = 0.05, which is positive, so 0.4 is bigger. ✓
Watch out. A decimal is not bigger because it has more digits.
Method. Find the lower and upper limits, check which ends are included, then list.
- Read the limits. n is bigger than 12 (12 itself is not allowed) and at most 17 (17 is allowed).12 < n ≤ 17
- List the whole numbers. Start at the first whole number after 12 and stop at 17.13, 14, 15, 16, 17
Check. 13 is allowed (> 12), 12 is not; 17 is allowed (≤ 17), 18 is not. ✓
Method. Use the number halfway between them.
- Write the numbers with an extra decimal place. There are no two-decimal-place numbers between 0.31 and 0.32, so we need three decimal places.0.31 = 0.310 and 0.32 = 0.320
- Pick a number between them. Any number from 0.311 to 0.319 works. The midpoint is a good choice.(0.31 + 0.32) ÷ 2 = 0.315
Check. 0.31 < 0.315 < 0.32. ✓
Method. Change the fractions to decimals so that all four numbers can be compared.
- Change each fraction into a decimal. Divide the top by the bottom.35 = 3 ÷ 5 = 0.6
58 = 5 ÷ 8 = 0.625 - Write all four with three decimal places. Adding zeros makes the columns match.0.600, 0.550, 0.625, 0.600
- Order them. Compare from the left: 0.550 is smallest, then the two 0.600s, then 0.625.0.55 , 35 , 0.6 , 58
Check. 0.55 < 0.6 = 0.6 < 0.625. ✓
Method. Think about the leftmost column.
- 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.
- 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.
- 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)
- 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)
- 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.
- Try an ordinary positive number. 3 × 10 = 30, which is bigger.30 > 3
- Try zero. 0 × 10 = 0, which is not bigger.0 = 0
- 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. ✓