Year 6 Decimals: worked examples and questions with answers

Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).

Method. Count the thousandths: there are 1000 of them in each whole, so multiply by 1000.

1. Write 0.37 with 3 decimal places. Zeros at the end do not change the value, but they show the place.
   0.370
2. Find how many thousandths make one whole. Each whole one has 1000 thousandths.
   1 × 1000 = 1000
3. Count the thousandths in 0.37. Multiply by 1000, because each whole has 1000 of them.
   0.37 × 1000 = 370
4. Write the answer. The answer is a count, so it is a whole number.
   370

Check. 370 thousandths are 370 × 0.001 = 0.370. ✓

Watch out. Each thousandth is 0.001, so a count of 370 is worth 0.370.

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Each digit moves 3 places to the left. The decimal point does not move.

1. Count how many places the digits move. 1000 is 3 zeros, so each digit moves 3 places to the left.
   1.275 × 1000
2. Move the digits. The decimal point stays where it is. Use zeros as place holders where a column is empty.
   1.275 moves 3 places to the left
3. Write the answer. Leave out any extra zeros at the end of a decimal, but keep the zeros needed for the place value.
   1275

Check. Undo it: 1275 ÷ 1000 gives 1.275 again. ✓

Watch out. Do not move the decimal point. Move the digits, and use zeros to fill the gaps.

Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).

Method. Build the number one column at a time, starting from the ones.

1. Write the ones. The ones come before the decimal point.
   4
2. Add the tenths. The first place after the point is tenths.
   4.3
3. Add the hundredths, using a zero to hold the place. The 0 keeps the 6 in the thousandths column.
   4.30
4. Add the thousandths. The third place after the point.
   4.306

Check. 4.306 = 4 + 0.3 + 0.006. ✓

Watch out. Do not write 4.36. The zero in the hundredths column must be there.

Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).

Method. Use the digits from the top of the number. The biggest and smallest possible numbers come from putting the biggest digit first.

1. List the arrangements. With 3 digits, there are 6 ways to put them in order.
   258, 285, 528, 582, 825, 852
2. Keep only the numbers less than 1 with 3 decimal places. The digits make 0.258, 0.285, 0.528, 0.582, 0.825 and 0.852.
   All six are less than 1
3. Find the biggest. The biggest starts with the biggest digit, then the next biggest.
   0.852 is bigger than 0.825
4. Write the answer.
   0.852

Check. 0.852 is the largest of the six. ✓

Watch out. The biggest digit goes in the tenths column, because that column is worth the most.

Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).

Method. Read the number carefully. Look at the digits in the question, and the order they are written in.

1. Write the question as a decimal. Read the first digit of 0.407 and 0.470 in the same column.
   0.407 and 0.470
2. Compare the tenths. Both have 4 tenths, so keep going.
   4 tenths = 4 tenths
3. Compare the hundredths. Hundredths: 0 in 0.407, 7 in 0.470.
   0 hundredths < 7 hundredths
4. Decide who is right. The number with 7 hundredths is bigger, so 0.47 is the bigger number.
   No. 0.47 is bigger.

Check. Ali looked at 407 and 47 without the decimal points. That does not compare the numbers. ✓

Watch out. Do not compare the digits without the decimal point. Compare each place from the left.

Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).

Method. Zeros at the end of a decimal do not change its value. Check with place value.

1. Write Mia's numbers with the same places. Add zeros to the end so both have three places.
   0.8 = 0.800; 0.08 = 0.080
2. Compare the places. Both numbers have 8 tenths or 8 hundredths in the same places.
   0.800 and 0.8 have the same digits
3. Decide. Adding zeros at the end does not change the value.
   No, she is wrong.

Check. 0.8 = 8/10 and 0.800 = 800/1000 = 8/10. ✓

Watch out. Zeros at the end of a decimal are free. Zeros at the start are not: 0.8 and 0.08 are not the same.

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Each digit moves 3 places to the left. The decimal point does not move.

1. Count how many places the digits move. 1000 is 3 zeros, so each digit moves 3 places to the left.
   0.035 × 1000
2. Move the digits. The decimal point stays where it is. Use zeros as place holders where a column is empty.
   0.035 moves 3 places to the left
3. Write the answer. Leave out any extra zeros at the end of a decimal, but keep the zeros needed for the place value.
   35

Check. Undo it: 35 ÷ 1000 gives 0.035 again. ✓

Watch out. Do not move the decimal point. Move the digits, and use zeros to fill the gaps.

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Each digit moves 2 places to the left. The decimal point does not move.

1. Count how many places the digits move. 100 is 2 zeros, so each digit moves 2 places to the left.
   0.45 × 100
2. Move the digits. The decimal point stays where it is. Use zeros as place holders where a column is empty.
   0.45 moves 2 places to the left
3. Write the answer. Leave out any extra zeros at the end of a decimal, but keep the zeros needed for the place value.
   45

Check. Undo it: 45 ÷ 100 gives 0.45 again. ✓

Watch out. Do not move the decimal point. Move the digits, and use zeros to fill the gaps.

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Each digit moves 2 places to the left. The decimal point does not move.

1. Count how many places the digits move. 100 is 2 zeros, so each digit moves 2 places to the left.
   3.7 × 100
2. Move the digits. The decimal point stays where it is. Use zeros as place holders where a column is empty.
   3.7 moves 2 places to the left
3. Write the answer. Leave out any extra zeros at the end of a decimal, but keep the zeros needed for the place value.
   370

Check. Undo it: 370 ÷ 100 gives 3.7 again. ✓

Watch out. Do not move the decimal point. Move the digits, and use zeros to fill the gaps.

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Think about whether the number gets bigger or smaller, then test the rule with an example.

1. Test an example. Pick a number bigger than 0 and divide it by 10.
   5 ÷ 10 = 0.5, and 0.5 is smaller than 5
2. Test another example. Try a decimal and a bigger number too.
   0.4 ÷ 10 = 0.04, and 12.5 ÷ 10 = 1.25
3. Decide. Each time the answer is smaller, so the rule is always true for numbers bigger than 0.
   Always

Check. Dividing by 10 moves every digit one place to the right, so the number gets smaller. ✓

Watch out. Zero is a special case: 0 ÷ 10 = 0, which is not smaller. So check the question for "numbers bigger than 0".

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Each digit moves 3 places to the left. The decimal point does not move.

1. Count how many places the digits move. 1000 is 3 zeros, so each digit moves 3 places to the left.
   0.3 × 1000
2. Move the digits. The decimal point stays where it is. Use zeros as place holders where a column is empty.
   0.3 moves 3 places to the left
3. Write the answer. Leave out any extra zeros at the end of a decimal, but keep the zeros needed for the place value.
   300

Check. Undo it: 300 ÷ 1000 gives 0.3 again. ✓

Watch out. Do not move the decimal point. Move the digits, and use zeros to fill the gaps.

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Each digit moves 2 places to the left. The decimal point does not move.

1. Count how many places the digits move. 100 is 2 zeros, so each digit moves 2 places to the left.
   0.07 × 100
2. Move the digits. The decimal point stays where it is. Use zeros as place holders where a column is empty.
   0.07 moves 2 places to the left
3. Write the answer. Leave out any extra zeros at the end of a decimal, but keep the zeros needed for the place value.
   7

Check. Undo it: 7 ÷ 100 gives 0.07 again. ✓

Watch out. Do not move the decimal point. Move the digits, and use zeros to fill the gaps.

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Each digit moves 1 place to the right. The decimal point does not move.

1. Count how many places the digits move. 10 is 1 zero, so each digit moves 1 place to the right.
   7 ÷ 10
2. Move the digits. The decimal point stays where it is. Use zeros as place holders where a column is empty.
   7 moves 1 place to the right
3. Write the answer. Leave out any extra zeros at the end of a decimal, but keep the zeros needed for the place value.
   0.7

Check. Undo it: 0.7 × 10 gives 7 again. ✓

Watch out. Dividing makes the number smaller, so the digits move to the right. Keep a zero in the ones place when the answer is less than 1.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

   0.85
    × 6
   ----
   5.10

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   5 × 6 = 30, write 0 and carry 3
   8 × 6 = 48 + 3 carried = 51, write 1 and carry 5
   write 5
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   85 × 6 = 510, and 510 → 5.10

Check. About 1 × 6 = 6, and 5.10 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

   2.35
    × 4
   ----
   9.40

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   5 × 4 = 20, write 0 and carry 2
   3 × 4 = 12 + 2 carried = 14, write 4 and carry 1
   2 × 4 = 8 + 1 carried = 9, write 9
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   235 × 4 = 940, and 940 → 9.40

Check. About 2 × 4 = 8, and 9.40 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

   1.25
    × 7
   ----
   8.75

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   5 × 7 = 35, write 5 and carry 3
   2 × 7 = 14 + 3 carried = 17, write 7 and carry 1
   1 × 7 = 7 + 1 carried = 8, write 8
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   125 × 7 = 875, and 875 → 8.75

Check. About 1 × 7 = 7, and 8.75 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

   0.75
    × 9
   ----
   6.75

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   5 × 9 = 45, write 5 and carry 4
   7 × 9 = 63 + 4 carried = 67, write 7 and carry 6
   write 6
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   75 × 9 = 675, and 675 → 6.75

Check. About 1 × 9 = 9, and 6.75 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

   1.45
    × 5
   ----
   7.25

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   5 × 5 = 25, write 5 and carry 2
   4 × 5 = 20 + 2 carried = 22, write 2 and carry 2
   1 × 5 = 5 + 2 carried = 7, write 7
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   145 × 5 = 725, and 725 → 7.25

Check. About 1 × 5 = 5, and 7.25 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. First find the total cost. Then take it away from the £10 note.

1. Write £10 with two decimal places. This makes the subtraction easy to line up.
   10.00 − 7.25
2. Subtract, column by column. Borrow from the column on the left when you need to.
   10.00 − 7.25 = 2.75
3. Write the answer. The change is the amount left over.
   2.75

Check. 7.25 + 2.75 = 10.00. ✓

Watch out. Line up the decimal points before you subtract.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            4.6
       6 ) 27.6
           0
           -
           27
           24
           --
             36
             36
             --
              0

2. Start with the first digit, 2. 6 does not go into 2, so this digit of the answer is 0.
   2 ÷ 6 = 0, remainder 2
3. Bring down the next digit, so we have 27. How many 6s go into 27? 4, because 4 × 6 = 24.
   27 ÷ 6 = 4, remainder 3
4. Bring down the next digit, so we have 36. Write the decimal point in the answer now, above the point. How many 6s go into 36? 6, because 6 × 6 = 36.
   36 ÷ 6 = 6, remainder 0
5. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   4.6

Check. 4.6 × 6 = 27.6. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            3.4
       4 ) 13.6
           0
           -
           13
           12
           --
             16
             16
             --
              0

2. Start with the first digit, 1. 4 does not go into 1, so this digit of the answer is 0.
   1 ÷ 4 = 0, remainder 1
3. Bring down the next digit, so we have 13. How many 4s go into 13? 3, because 3 × 4 = 12.
   13 ÷ 4 = 3, remainder 1
4. Bring down the next digit, so we have 16. Write the decimal point in the answer now, above the point. How many 4s go into 16? 4, because 4 × 4 = 16.
   16 ÷ 4 = 4, remainder 0
5. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   3.4

Check. 3.4 × 4 = 13.6. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 1 decimal place.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

    3.6
    × 5
   ----
   18.0

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   6 × 5 = 30, write 0 and carry 3
   3 × 5 = 15 + 3 carried = 18, write 8 and carry 1
   write 1
3. Count the decimal places and put the point back. The decimal number has 1 decimal place, so the answer has 1 decimal place.
   36 × 5 = 180, and 180 → 18.0

Check. About 4 × 5 = 20, and 18.0 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Estimate first, then check the working.

1. Estimate. 3.46 is a little more than 3, and 3 × 7 = 21.
   3 × 7 = 21
2. Compare with the answer. The answer 24.22 is a little more than 21, which matches the estimate.
   24.22 is a little more than 21
3. Decide. The answer is the right size.
   Yes, 24.22 is sensible.

Check. 3.46 × 7 is about 3.5 × 7 = 24.5, which is close to 24.22. ✓

Watch out. An estimate tells you the size of the answer. It does not give the exact answer.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

   0.45
    × 8
   ----
   3.60

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   5 × 8 = 40, write 0 and carry 4
   4 × 8 = 32 + 4 carried = 36, write 6 and carry 3
   write 3
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   45 × 8 = 360, and 360 → 3.60

Check. About 0 × 8 = 0, and 3.60 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

    0.36
    × 60
   -----
   21.60

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   6 × 60 = 360, write 0 and carry 36
   3 × 60 = 180 + 36 carried = 216, write 6 and carry 21
   write 21
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   36 × 60 = 2160, and 2160 → 21.60

Check. About 0 × 60 = 0, and 21.60 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            4.25
       4 ) 17.00
           0
           -
           17
           16
           --
             10
              8
             --
              20
              20
              --
               0

2. Start with the first digit, 1. 4 does not go into 1, so this digit of the answer is 0.
   1 ÷ 4 = 0, remainder 1
3. Bring down the next digit, so we have 17. How many 4s go into 17? 4, because 4 × 4 = 16.
   17 ÷ 4 = 4, remainder 1
4. Bring down a 0, so we have 10. Write the decimal point in the answer now, above the point. How many 4s go into 10? 2, because 2 × 4 = 8.
   10 ÷ 4 = 2, remainder 2
5. Bring down a 0, so we have 20. How many 4s go into 20? 5, because 5 × 4 = 20.
   20 ÷ 4 = 5, remainder 0
6. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   4.25

Check. 4.25 × 4 = 17. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Use the inverse. Multiply the answer by 4 to find the number we started with.

1. Write the division as a multiplication. If ? ÷ 4 = 1.75, then 1.75 × 4 = ?.
   ? = 1.75 × 4
2. Multiply. Use the column method and keep the two decimal places.
   1.75 × 4 = 7.00
3. Write the answer. The missing number is 7.
   7.00

Check. 7.00 ÷ 4 = 1.75. ✓

Watch out. Multiplying is the opposite of dividing, so it undoes the division.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            1.65
       8 ) 13.20
           0
           -
           13
            8
           --
             52
             48
             --
              40
              40
              --
               0

2. Start with the first digit, 1. 8 does not go into 1, so this digit of the answer is 0.
   1 ÷ 8 = 0, remainder 1
3. Bring down the next digit, so we have 13. How many 8s go into 13? 1, because 1 × 8 = 8.
   13 ÷ 8 = 1, remainder 5
4. Bring down the next digit, so we have 52. Write the decimal point in the answer now, above the point. How many 8s go into 52? 6, because 6 × 8 = 48.
   52 ÷ 8 = 6, remainder 4
5. Bring down a 0, so we have 40. How many 8s go into 40? 5, because 5 × 8 = 40.
   40 ÷ 8 = 5, remainder 0
6. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   1.65

Check. 1.65 × 8 = 13.2. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           0.6
       5 ) 3.0
           0
           -
            30
            30
            --
             0

2. Start with the first digit, 3. 5 does not go into 3, so this digit of the answer is 0.
   3 ÷ 5 = 0, remainder 3
3. Bring down a 0, so we have 30. Write the decimal point in the answer now, above the point. How many 5s go into 30? 6, because 6 × 5 = 30.
   30 ÷ 5 = 6, remainder 0
4. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   0.6

Check. 0.6 × 5 = 3. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            1.80
       8 ) 14.40
           0
           -
           14
            8
           --
             64
             64
             --
               0
               0
               -
               0

2. Start with the first digit, 1. 8 does not go into 1, so this digit of the answer is 0.
   1 ÷ 8 = 0, remainder 1
3. Bring down the next digit, so we have 14. How many 8s go into 14? 1, because 1 × 8 = 8.
   14 ÷ 8 = 1, remainder 6
4. Bring down the next digit, so we have 64. Write the decimal point in the answer now, above the point. How many 8s go into 64? 8, because 8 × 8 = 64.
   64 ÷ 8 = 8, remainder 0
5. Bring down the next digit, so we have 0. 8 does not go into 0, so this digit of the answer is 0.
   0 ÷ 8 = 0, remainder 0
6. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   1.80

Check. 1.80 × 8 = 14.40. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           1.8
       4 ) 7.2
           4
           -
            32
            32
            --
             0

2. Start with the first digit, 7. How many 4s go into 7? 1, because 1 × 4 = 4.
   7 ÷ 4 = 1, remainder 3
3. Bring down the next digit, so we have 32. Write the decimal point in the answer now, above the point. How many 4s go into 32? 8, because 8 × 4 = 32.
   32 ÷ 4 = 8, remainder 0
4. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   1.8

Check. 1.8 × 4 = 7.2. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            4.20
       3 ) 12.60
           0
           -
           12
           12
           --
              6
              6
              -
               0
               0
               -
               0

2. Start with the first digit, 1. 3 does not go into 1, so this digit of the answer is 0.
   1 ÷ 3 = 0, remainder 1
3. Bring down the next digit, so we have 12. How many 3s go into 12? 4, because 4 × 3 = 12.
   12 ÷ 3 = 4, remainder 0
4. Bring down the next digit, so we have 6. Write the decimal point in the answer now, above the point. How many 3s go into 6? 2, because 2 × 3 = 6.
   6 ÷ 3 = 2, remainder 0
5. Bring down the next digit, so we have 0. 3 does not go into 0, so this digit of the answer is 0.
   0 ÷ 3 = 0, remainder 0
6. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   4.20

Check. 4.20 × 3 = 12.60. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           0.75
       4 ) 3.00
           0
           -
            30
            28
            --
             20
             20
             --
              0

2. Start with the first digit, 3. 4 does not go into 3, so this digit of the answer is 0.
   3 ÷ 4 = 0, remainder 3
3. Bring down a 0, so we have 30. Write the decimal point in the answer now, above the point. How many 4s go into 30? 7, because 7 × 4 = 28.
   30 ÷ 4 = 7, remainder 2
4. Bring down a 0, so we have 20. How many 4s go into 20? 5, because 5 × 4 = 20.
   20 ÷ 4 = 5, remainder 0
5. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   0.75

Check. 0.75 × 4 = 3. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           0.25
       8 ) 2.00
           0
           -
            20
            16
            --
             40
             40
             --
              0

2. Start with the first digit, 2. 8 does not go into 2, so this digit of the answer is 0.
   2 ÷ 8 = 0, remainder 2
3. Bring down a 0, so we have 20. Write the decimal point in the answer now, above the point. How many 8s go into 20? 2, because 2 × 8 = 16.
   20 ÷ 8 = 2, remainder 4
4. Bring down a 0, so we have 40. How many 8s go into 40? 5, because 5 × 8 = 40.
   40 ÷ 8 = 5, remainder 0
5. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   0.25

Check. 0.25 × 8 = 2. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           1.5
       6 ) 9.0
           6
           -
            30
            30
            --
             0

2. Start with the first digit, 9. How many 6s go into 9? 1, because 1 × 6 = 6.
   9 ÷ 6 = 1, remainder 3
3. Bring down a 0, so we have 30. Write the decimal point in the answer now, above the point. How many 6s go into 30? 5, because 5 × 6 = 30.
   30 ÷ 6 = 5, remainder 0
4. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   1.5

Check. 1.5 × 6 = 9. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Look at how many decimal places each answer has, then decide.

1. Write down the decimal places. Count the digits after the point in each answer.
   3 ÷ 4 = 0.75 (2 places); 2 ÷ 8 = 0.25 (2 places); 9 ÷ 6 = 1.5 (1 place)
2. Choose the two that have exactly two decimal places. The third answer only has one decimal place.
   a and b

Check. 9 ÷ 6 = 1.5, which has one decimal place. ✓

Watch out. Do not count the places in the question. Count the places in the answer.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           0.45
       8 ) 3.60
           0
           -
            36
            32
            --
             40
             40
             --
              0

2. Start with the first digit, 3. 8 does not go into 3, so this digit of the answer is 0.
   3 ÷ 8 = 0, remainder 3
3. Bring down the next digit, so we have 36. Write the decimal point in the answer now, above the point. How many 8s go into 36? 4, because 4 × 8 = 32.
   36 ÷ 8 = 4, remainder 4
4. Bring down a 0, so we have 40. How many 8s go into 40? 5, because 5 × 8 = 40.
   40 ÷ 8 = 5, remainder 0
5. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   0.45

Check. 0.45 × 8 = 3.6. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Multiply the answer by 8 to check it.

1. Multiply the answer by the divisor. If the division is right, we get the number we started with.
   0.45 × 8 = 3.60
2. Compare with the number in the question. 3.60 and 3.6 are the same number.
   3.60 = 3.6
3. Write the answer. The division is correct.
   0.45

Check. 0.45 × 8 = 3.60, which is 3.6. ✓

Watch out. Write the number with the same number of decimal places as you start with.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            14.285
       7 ) 100.000
           0
           -
           10
            7
           --
            30
            28
            --
              20
              14
              --
               60
               56
               --
                40
                35
                --
                 5

2. Start with the first digit, 1. 7 does not go into 1, so this digit of the answer is 0.
   1 ÷ 7 = 0, remainder 1
3. Bring down the next digit, so we have 10. How many 7s go into 10? 1, because 1 × 7 = 7.
   10 ÷ 7 = 1, remainder 3
4. Bring down the next digit, so we have 30. How many 7s go into 30? 4, because 4 × 7 = 28.
   30 ÷ 7 = 4, remainder 2
5. Bring down a 0, so we have 20. Write the decimal point in the answer now, above the point. How many 7s go into 20? 2, because 2 × 7 = 14.
   20 ÷ 7 = 2, remainder 6
6. Bring down a 0, so we have 60. How many 7s go into 60? 8, because 8 × 7 = 56.
   60 ÷ 7 = 8, remainder 4
7. Bring down a 0, so we have 40. How many 7s go into 40? 5, because 5 × 7 = 35.
   40 ÷ 7 = 5, remainder 5
8. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   14.285

Check. Multiply back: 14.285 × 7 is close to 100.

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. Find the place, look at the digit to its right, then round up or stay the same.

1. Find the place you are rounding to. We want 2 decimal places, so look at the hundredths column.
   14.285: the digit in the hundredths column is 8
2. Look at the digit to its right. This digit decides whether the number goes up or stays the same.
   The digit to the right is 5
3. Decide whether to round up or down. If the digit is 5 or more, round up. If it is less than 5, keep the same digit.
   5 is 5 or more, so round up
4. Write the answer. Keep the zeros after the point so the accuracy is clear.
   14.29

Check. 14.285 lies between 14.28 and 14.29, and 14.29 is the nearer one. ✓

Watch out. Look only at the digit to the right of the place you are rounding to. Do not round the digits one at a time.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            5.875
       8 ) 47.000
           0
           -
           47
           40
           --
             70
             64
             --
              60
              56
              --
               40
               40
               --
                0

2. Start with the first digit, 4. 8 does not go into 4, so this digit of the answer is 0.
   4 ÷ 8 = 0, remainder 4
3. Bring down the next digit, so we have 47. How many 8s go into 47? 5, because 5 × 8 = 40.
   47 ÷ 8 = 5, remainder 7
4. Bring down a 0, so we have 70. Write the decimal point in the answer now, above the point. How many 8s go into 70? 8, because 8 × 8 = 64.
   70 ÷ 8 = 8, remainder 6
5. Bring down a 0, so we have 60. How many 8s go into 60? 7, because 7 × 8 = 56.
   60 ÷ 8 = 7, remainder 4
6. Bring down a 0, so we have 40. How many 8s go into 40? 5, because 5 × 8 = 40.
   40 ÷ 8 = 5, remainder 0
7. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   5.875

Check. 5.875 × 8 = 47. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. A bus holds 8 children. Round up, because 5 buses would leave some children without a seat.

1. Read the answer. 5.875 means 5 full buses and a bit more.
   5.875 buses
2. Decide the degree of accuracy. We cannot have part of a bus, so round up to the next whole number.
   5.875 buses is not enough, so 6
3. Write the answer. Six buses are needed.
   6 buses

Check. 6 buses hold 6 × 8 = 48 children, which is enough for 47. ✓

Watch out. In a problem about people or things, look at the context. Rounding down can leave someone out.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            4.166
       3 ) 12.500
           0
           -
           12
           12
           --
              5
              3
              -
              20
              18
              --
               20
               18
               --
                2

2. Start with the first digit, 1. 3 does not go into 1, so this digit of the answer is 0.
   1 ÷ 3 = 0, remainder 1
3. Bring down the next digit, so we have 12. How many 3s go into 12? 4, because 4 × 3 = 12.
   12 ÷ 3 = 4, remainder 0
4. Bring down the next digit, so we have 5. Write the decimal point in the answer now, above the point. How many 3s go into 5? 1, because 1 × 3 = 3.
   5 ÷ 3 = 1, remainder 2
5. Bring down the next digit, so we have 20. How many 3s go into 20? 6, because 6 × 3 = 18.
   20 ÷ 3 = 6, remainder 2
6. Bring down a 0, so we have 20. How many 3s go into 20? 6, because 6 × 3 = 18.
   20 ÷ 3 = 6, remainder 2
7. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   4.166

Check. Multiply back: 4.166 × 3 is close to 12.50.

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. Find the place, look at the digit to its right, then round up or stay the same.

1. Find the place you are rounding to. We want 2 decimal places, so look at the hundredths column.
   4.166: the digit in the hundredths column is 6
2. Look at the digit to its right. This digit decides whether the number goes up or stays the same.
   The digit to the right is 6
3. Decide whether to round up or down. If the digit is 5 or more, round up. If it is less than 5, keep the same digit.
   6 is 5 or more, so round up
4. Write the answer. Keep the zeros after the point so the accuracy is clear.
   4.17

Check. 4.166 lies between 4.16 and 4.17, and 4.17 is the nearer one. ✓

Watch out. Look only at the digit to the right of the place you are rounding to. Do not round the digits one at a time.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           0.42
       7 ) 3.00
           0
           -
            30
            28
            --
             20
             14
             --
              6

2. Start with the first digit, 3. 7 does not go into 3, so this digit of the answer is 0.
   3 ÷ 7 = 0, remainder 3
3. Bring down a 0, so we have 30. Write the decimal point in the answer now, above the point. How many 7s go into 30? 4, because 4 × 7 = 28.
   30 ÷ 7 = 4, remainder 2
4. Bring down a 0, so we have 20. How many 7s go into 20? 2, because 2 × 7 = 14.
   20 ÷ 7 = 2, remainder 6
5. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   0.42

Check. Multiply back: 0.42 × 7 is close to 3.

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. Find the place, look at the digit to its right, then round up or stay the same.

1. Find the place you are rounding to. We want 1 decimal place, so look at the tenths column.
   0.42: the digit in the tenths column is 4
2. Look at the digit to its right. This digit decides whether the number goes up or stays the same.
   The digit to the right is 2
3. Decide whether to round up or down. If the digit is 5 or more, round up. If it is less than 5, keep the same digit.
   2 is less than 5, so round down
4. Write the answer. Keep the zeros after the point so the accuracy is clear.
   0.4

Check. 0.42 lies between 0.4 and 0.5, and 0.4 is the nearer one. ✓

Watch out. Look only at the digit to the right of the place you are rounding to. Do not round the digits one at a time.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            33.333
       3 ) 100.000
           0
           -
           10
            9
           --
            10
             9
            --
              10
               9
              --
               10
                9
               --
                10
                 9
                --
                 1

2. Start with the first digit, 1. 3 does not go into 1, so this digit of the answer is 0.
   1 ÷ 3 = 0, remainder 1
3. Bring down the next digit, so we have 10. How many 3s go into 10? 3, because 3 × 3 = 9.
   10 ÷ 3 = 3, remainder 1
4. Bring down the next digit, so we have 10. How many 3s go into 10? 3, because 3 × 3 = 9.
   10 ÷ 3 = 3, remainder 1
5. Bring down a 0, so we have 10. Write the decimal point in the answer now, above the point. How many 3s go into 10? 3, because 3 × 3 = 9.
   10 ÷ 3 = 3, remainder 1
6. Bring down a 0, so we have 10. How many 3s go into 10? 3, because 3 × 3 = 9.
   10 ÷ 3 = 3, remainder 1
7. Bring down a 0, so we have 10. How many 3s go into 10? 3, because 3 × 3 = 9.
   10 ÷ 3 = 3, remainder 1
8. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   33.333

Check. Multiply back: 33.333 × 3 is close to 100.

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. Find the place, look at the digit to its right, then round up or stay the same.

1. Find the place you are rounding to. We want 2 decimal places, so look at the hundredths column.
   33.333: the digit in the hundredths column is 3
2. Look at the digit to its right. This digit decides whether the number goes up or stays the same.
   The digit to the right is 3
3. Decide whether to round up or down. If the digit is 5 or more, round up. If it is less than 5, keep the same digit.
   3 is less than 5, so round down
4. Write the answer. Keep the zeros after the point so the accuracy is clear.
   33.33

Check. 33.333 lies between 33.33 and 33.34, and 33.33 is the nearer one. ✓

Watch out. Look only at the digit to the right of the place you are rounding to. Do not round the digits one at a time.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. Find the smallest number with two decimal places that rounds up to 4.0.

1. Look at the number to be rounded. To round to 4.0, the number must be at least 3.95.
   Rounding to one decimal place: 3.95 → 4.0
2. Check the number below it. 3.94 is less than 3.95, so it rounds to 3.9.
   3.94 → 3.9
3. Write the answer. The smallest number is 3.95.
   3.95

Check. 3.95 rounds up to 4.0, and 3.94 does not. ✓

Watch out. Half-way numbers round up, so 3.95 is the first number that rounds up to 4.0.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. Write the rounding rule out, then apply it to 6.5.

1. Find the place you are rounding to. The nearest whole number has no decimal places.
   Look at the tenths digit: 5
2. Look at the digit to the right. The digit 5 is 5 or more.
   5 is 5 or more, so round up
3. Decide. Round up to the next whole number.
   6 + 1 = 7

Check. 6.5 is half-way between 6 and 7. Half-way numbers round up. ✓

Watch out. Mo looked at 6 and stopped. He needed the digit after the 6 to decide.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. Say what the answer means.

1. Write the answer. 6.5 rounds up to 7.
   No, 6.5 rounds up to 7.

Watch out. The answer 6 is what you get when you round down.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. Find the whole numbers that round to 40 when rounded to the nearest 10.

1. Find the half-way numbers. Numbers from 35 up to 44 are closer to 40 than to 30 or 50. 35 rounds up to 40.
   35 → 40, 44 → 40
2. Check the next numbers. 34 rounds down to 30, and 45 rounds up to 50.
   34 → 30, 45 → 50
3. Count them. Each number from 35 to 44 counts once.
   35, 36, 37, 38, 39, 40, 41, 42, 43, 44
4. Write the answer. There are ten of them.
   10

Check. 10 numbers, from 35 to 44. ✓

Watch out. The range is 5 below and 4 above, because 45 rounds up to 50.

Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).

Method. Line up the decimal points, then add each column from the right.

1. Write the numbers under each other. Line up the decimal points in one column. Fill the gaps with zeros.

     1.25
   + 4.80
   + 0.95
   ------
     7.00

2. Add column by column, from the right. Carry any ten into the next column to the left.
   1.25 + 4.80 + 0.95 = 7.00
3. Write the answer with its decimal point in the same column. The decimal points line up in the question and in the answer.
   7.00

Check. Add the numbers in a different order and you get 7.00 again. ✓

Watch out. Line up the decimal points, not the last digits. Fill the gaps with zeros.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Work out the change from £10.00.

1. Write £10 with two decimal places. This lines up the subtraction.
   10.00 − 7.00
2. Subtract. Borrow from the column on the left if you need to.
   10.00 − 7.00 = 3.00
3. Write the answer. This is the change.
   3.00

Check. 7.00 + 3.00 = 10.00. ✓

Watch out. Check that the total plus the change makes the amount paid.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           0.75
       6 ) 4.50
           0
           -
            45
            42
            --
             30
             30
             --
              0

2. Start with the first digit, 4. 6 does not go into 4, so this digit of the answer is 0.
   4 ÷ 6 = 0, remainder 4
3. Bring down the next digit, so we have 45. Write the decimal point in the answer now, above the point. How many 6s go into 45? 7, because 7 × 6 = 42.
   45 ÷ 6 = 7, remainder 3
4. Bring down a 0, so we have 30. How many 6s go into 30? 5, because 5 × 6 = 30.
   30 ÷ 6 = 5, remainder 0
5. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   0.75

Check. 0.75 × 6 = 4.5. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

    3.85
     × 5
   -----
   19.25

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   5 × 5 = 25, write 5 and carry 2
   8 × 5 = 40 + 2 carried = 42, write 2 and carry 4
   3 × 5 = 15 + 4 carried = 19, write 9 and carry 1
   write 1
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   385 × 5 = 1925, and 1925 → 19.25

Check. About 4 × 5 = 20, and 19.25 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           0.65
       4 ) 2.60
           0
           -
            26
            24
            --
             20
             20
             --
              0

2. Start with the first digit, 2. 4 does not go into 2, so this digit of the answer is 0.
   2 ÷ 4 = 0, remainder 2
3. Bring down the next digit, so we have 26. Write the decimal point in the answer now, above the point. How many 4s go into 26? 6, because 6 × 4 = 24.
   26 ÷ 4 = 6, remainder 2
4. Bring down a 0, so we have 20. How many 4s go into 20? 5, because 5 × 4 = 20.
   20 ÷ 4 = 5, remainder 0
5. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   0.65

Check. 0.65 × 4 = 2.6. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

    2.40
     × 6
   -----
   14.40

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   0 × 6 = 0, write 0
   4 × 6 = 24, write 4 and carry 2
   2 × 6 = 12 + 2 carried = 14, write 4 and carry 1
   write 1
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   240 × 6 = 1440, and 1440 → 14.40

Check. About 2 × 6 = 12, and 14.40 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            9.75
       2 ) 19.50
           0
           -
           19
           18
           --
             15
             14
             --
              10
              10
              --
               0

2. Start with the first digit, 1. 2 does not go into 1, so this digit of the answer is 0.
   1 ÷ 2 = 0, remainder 1
3. Bring down the next digit, so we have 19. How many 2s go into 19? 9, because 9 × 2 = 18.
   19 ÷ 2 = 9, remainder 1
4. Bring down the next digit, so we have 15. Write the decimal point in the answer now, above the point. How many 2s go into 15? 7, because 7 × 2 = 14.
   15 ÷ 2 = 7, remainder 1
5. Bring down a 0, so we have 10. How many 2s go into 10? 5, because 5 × 2 = 10.
   10 ÷ 2 = 5, remainder 0
6. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   9.75

Check. 9.75 × 2 = 19.5. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. A bottle holds 2 litres. Decide whether to round up or down.

1. Read the answer. 9.75 bottles would be 9 full bottles and part of one.
   9.75 bottles
2. Decide. The school needs all the juice, so an extra bottle is needed. Round up.
   9.75 → 10 bottles
3. Write the answer. Ten bottles are needed.
   10 bottles

Check. 10 bottles hold 10 × 2 = 20 litres, which is enough for 19.5 litres. ✓

Watch out. Rounding down would leave some juice out. Read the context before rounding.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

   2.65
    × 3
   ----
   7.95

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   5 × 3 = 15, write 5 and carry 1
   6 × 3 = 18 + 1 carried = 19, write 9 and carry 1
   2 × 3 = 6 + 1 carried = 7, write 7
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   265 × 3 = 795, and 795 → 7.95

Check. About 3 × 3 = 9, and 7.95 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Take the cost of the cakes from the £20.00 Mia has.

1. Write £20 with two decimal places. This lines up the subtraction.
   20.00 − 7.95
2. Subtract. Borrow when you need to.
   20.00 − 7.95 = 12.05
3. Write the answer. This is the money left.
   12.05

Check. 7.95 + 12.05 = 20.00. ✓

Watch out. The answer must be less than £20 because she spent money.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           0.60
       8 ) 4.80
           0
           -
            48
            48
            --
              0
              0
              -
              0

2. Start with the first digit, 4. 8 does not go into 4, so this digit of the answer is 0.
   4 ÷ 8 = 0, remainder 4
3. Bring down the next digit, so we have 48. Write the decimal point in the answer now, above the point. How many 8s go into 48? 6, because 6 × 8 = 48.
   48 ÷ 8 = 6, remainder 0
4. Bring down the next digit, so we have 0. 8 does not go into 0, so this digit of the answer is 0.
   0 ÷ 8 = 0, remainder 0
5. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   0.60

Check. 0.60 × 8 = 4.80. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 1 decimal place.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

    2.6
    × 4
   ----
   10.4

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   6 × 4 = 24, write 4 and carry 2
   2 × 4 = 8 + 2 carried = 10, write 0 and carry 1
   write 1
3. Count the decimal places and put the point back. The decimal number has 1 decimal place, so the answer has 1 decimal place.
   26 × 4 = 104, and 104 → 10.4

Check. About 3 × 4 = 12, and 10.4 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           0.9
       6 ) 5.4
           0
           -
            54
            54
            --
             0

2. Start with the first digit, 5. 6 does not go into 5, so this digit of the answer is 0.
   5 ÷ 6 = 0, remainder 5
3. Bring down the next digit, so we have 54. Write the decimal point in the answer now, above the point. How many 6s go into 54? 9, because 9 × 6 = 54.
   54 ÷ 6 = 9, remainder 0
4. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   0.9

Check. 0.9 × 6 = 5.4. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Multiply the answer by 6 to check it.

1. Multiply the answer by the divisor. If the division is correct, the answer times 6 is 5.4.
   0.9 × 6 = 5.4
2. Compare. The result matches the number being divided.
   5.4 = 5.4
3. Decide. Tom is right.
   Yes, 0.9 × 6 = 5.4.

Check. 0.9 × 6 = 5.4. ✓

Watch out. Multiplying is the way to check a division.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Test some examples, then decide whether the rule always works.

1. Test an example. Pick a number bigger than 0 and multiply it by 2.
   0.4 × 2 = 0.8, and 0.8 is bigger than 0.4
2. Test another example. Try a number with two decimal places.
   2.75 × 2 = 5.5, and 5.5 is bigger than 2.75
3. Decide. Multiplying a number bigger than 0 by 2 always gives a bigger number.
   Always

Check. Doubling a number bigger than 0 always makes it bigger. ✓

Watch out. A number bigger than 0 gets bigger when you double it. The question only asks about numbers bigger than 0.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Divide the length by the piece length, and look at what is left over.

1. Find how many 0.5 m pieces fit. Each piece is 0.5 m, so dividing by 0.5 tells us how many pieces fit.
   3.75 ÷ 0.5 = 7.5
2. Count the full pieces. Only whole pieces count, so take the whole number part.
   7 full pieces
3. Find what is left. The leftover is 3.75 minus 7 pieces of 0.5 m.
   7 × 0.5 = 3.5, and 3.75 − 3.5 = 0.25 m left
4. Write the answer. Ben cuts 7 full pieces.
   7

Check. 7 × 0.5 = 3.5, and 3.5 + 0.25 = 3.75. ✓

Watch out. Count only the full pieces, not the 0.25 m that is left over.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

   2.35
    × 3
   ----
   7.05

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   5 × 3 = 15, write 5 and carry 1
   3 × 3 = 9 + 1 carried = 10, write 0 and carry 1
   2 × 3 = 6 + 1 carried = 7, write 7
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   235 × 3 = 705, and 705 → 7.05

Check. About 2 × 3 = 6, and 7.05 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).

Method. Line up the decimal points, then add each column from the right.

1. Write the numbers under each other. Line up the decimal points in one column. Fill the gaps with zeros.

     7.05
   + 0.40
   ------
     7.45

2. Add column by column, from the right. Carry any ten into the next column to the left.
   7.05 + 0.40 = 7.45
3. Write the answer with its decimal point in the same column. The decimal points line up in the question and in the answer.
   7.45

Check. Add the numbers in a different order and you get 7.45 again. ✓

Watch out. Line up the decimal points, not the last digits. Fill the gaps with zeros.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply first, then add. The order matters.

1. Work out the multiplication first. Multiplication comes before addition.
   2.35 × 3 = 7.05
2. Add. Then add 0.4.
   7.05 + 0.4 = 7.45
3. Write the answer.
   7.45

Check. Adding first would give 2.75 × 3, which is wrong. ✓

Watch out. Do the multiplication before the addition, as in the order of operations.

Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).

Method. Find the column of the digit, then multiply the digit by the value of that column.

1. Find the column of the digit. Count the places from the decimal point.
   The 7 is in the hundredths column
2. Find what one of that column is worth. Each hundredth is worth 0.01.
   1 hundredth = 0.01
3. Multiply the digit by that value. The digit tells us how many of them there are.
   7 × 0.01 = 0.07
4. Write the answer. Keep the zeros that show the place.
   0.07

Check. 5.472 = 5 + 0.4 + 0.07 + 0.002. ✓

Watch out. The 0 digits still count as places. The digit 7 is worth 0.07, not 7.

Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).

Method. Count the hundredths: there are 100 of them in each whole, so multiply by 100.

1. Write 4.07 with 2 decimal places. Zeros at the end do not change the value, but they show the place.
   4.07
2. Find how many hundredths make one whole. Each whole one has 100 hundredths.
   1 × 100 = 100
3. Count the hundredths in 4.07. Multiply by 100, because each whole has 100 of them.
   4.07 × 100 = 407
4. Write the answer. The answer is a count, so it is a whole number.
   407

Check. 407 hundredths are 407 × 0.01 = 4.07. ✓

Watch out. Each hundredth is 0.01, so a count of 407 is worth 4.07.

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Each digit moves 2 places to the left. The decimal point does not move.

1. Count how many places the digits move. 100 is 2 zeros, so each digit moves 2 places to the left.
   6.4 × 100
2. Move the digits. The decimal point stays where it is. Use zeros as place holders where a column is empty.
   6.4 moves 2 places to the left
3. Write the answer. Leave out any extra zeros at the end of a decimal, but keep the zeros needed for the place value.
   640

Check. Undo it: 640 ÷ 100 gives 6.4 again. ✓

Watch out. Do not move the decimal point. Move the digits, and use zeros to fill the gaps.

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Each digit moves 1 place to the right. The decimal point does not move.

1. Count how many places the digits move. 10 is 1 zero, so each digit moves 1 place to the right.
   0.09 ÷ 10
2. Move the digits. The decimal point stays where it is. Use zeros as place holders where a column is empty.
   0.09 moves 1 place to the right
3. Write the answer. Leave out any extra zeros at the end of a decimal, but keep the zeros needed for the place value.
   0.009

Check. Undo it: 0.009 × 10 gives 0.09 again. ✓

Watch out. Dividing makes the number smaller, so the digits move to the right. Keep a zero in the ones place when the answer is less than 1.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

    4.36
     × 7
   -----
   30.52

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   6 × 7 = 42, write 2 and carry 4
   3 × 7 = 21 + 4 carried = 25, write 5 and carry 2
   4 × 7 = 28 + 2 carried = 30, write 0 and carry 3
   write 3
3. Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.
   436 × 7 = 3052, and 3052 → 30.52

Check. About 4 × 7 = 28, and 30.52 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 1 decimal place.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

    5.4
    × 8
   ----
   43.2

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   4 × 8 = 32, write 2 and carry 3
   5 × 8 = 40 + 3 carried = 43, write 3 and carry 4
   write 4
3. Count the decimal places and put the point back. The decimal number has 1 decimal place, so the answer has 1 decimal place.
   54 × 8 = 432, and 432 → 43.2

Check. About 5 × 8 = 40, and 43.2 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

           0.9
       4 ) 3.6
           0
           -
            36
            36
            --
             0

2. Start with the first digit, 3. 4 does not go into 3, so this digit of the answer is 0.
   3 ÷ 4 = 0, remainder 3
3. Bring down the next digit, so we have 36. Write the decimal point in the answer now, above the point. How many 4s go into 36? 9, because 9 × 4 = 36.
   36 ÷ 4 = 9, remainder 0
4. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   0.9

Check. 0.9 × 4 = 3.6. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.

Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.

1. Set out the layout. The number being divided goes inside the bracket. The answer goes above it.

            2.3
       5 ) 11.5
           0
           -
           11
           10
           --
             15
             15
             --
              0

2. Start with the first digit, 1. 5 does not go into 1, so this digit of the answer is 0.
   1 ÷ 5 = 0, remainder 1
3. Bring down the next digit, so we have 11. How many 5s go into 11? 2, because 2 × 5 = 10.
   11 ÷ 5 = 2, remainder 1
4. Bring down the next digit, so we have 15. Write the decimal point in the answer now, above the point. How many 5s go into 15? 3, because 3 × 5 = 15.
   15 ÷ 5 = 3, remainder 0
5. Write the answer. Keep going until the remainder is 0, or until you have the places you need.
   2.3

Check. 2.3 × 5 = 11.5. ✓

Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. Find the place, look at the digit to its right, then round up or stay the same.

1. Find the place you are rounding to. We want 2 decimal places, so look at the hundredths column.
   6.4375: the digit in the hundredths column is 3
2. Look at the digit to its right. This digit decides whether the number goes up or stays the same.
   The digit to the right is 7
3. Decide whether to round up or down. If the digit is 5 or more, round up. If it is less than 5, keep the same digit.
   7 is 5 or more, so round up
4. Write the answer. Keep the zeros after the point so the accuracy is clear.
   6.44

Check. 6.4375 lies between 6.43 and 6.44, and 6.44 is the nearer one. ✓

Watch out. Look only at the digit to the right of the place you are rounding to. Do not round the digits one at a time.

Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.

Method. Find the place, look at the digit to its right, then round up or stay the same.

1. Find the place you are rounding to. We want 1 decimal place, so look at the tenths column.
   27.46: the digit in the tenths column is 4
2. Look at the digit to its right. This digit decides whether the number goes up or stays the same.
   The digit to the right is 6
3. Decide whether to round up or down. If the digit is 5 or more, round up. If it is less than 5, keep the same digit.
   6 is 5 or more, so round up
4. Write the answer. Keep the zeros after the point so the accuracy is clear.
   27.5

Check. 27.46 lies between 27.4 and 27.5, and 27.5 is the nearer one. ✓

Watch out. Look only at the digit to the right of the place you are rounding to. Do not round the digits one at a time.

Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).

Method. Find the column of the digit, then multiply the digit by the value of that column.

1. Find the column of the digit. Count the places from the decimal point.
   The 6 is in the thousandths column
2. Find what one of that column is worth. Each thousandth is worth 0.001.
   1 thousandth = 0.001
3. Multiply the digit by that value. The digit tells us how many of them there are.
   6 × 0.001 = 0.006
4. Write the answer. Keep the zeros that show the place.
   0.006

Check. 3.206 = 3 + 0.2 + 0.006. ✓

Watch out. The 0 digits still count as places. The digit 6 is worth 0.006, not 6.

Key words. The decimal point does not move when you multiply or divide by 10, 100 or 1000. The digits move instead. A digit that moves one place to the left is ten times bigger.

Method. Each digit moves 3 places to the left. The decimal point does not move.

1. Count how many places the digits move. 1000 is 3 zeros, so each digit moves 3 places to the left.
   0.37 × 1000
2. Move the digits. The decimal point stays where it is. Use zeros as place holders where a column is empty.
   0.37 moves 3 places to the left
3. Write the answer. Leave out any extra zeros at the end of a decimal, but keep the zeros needed for the place value.
   370

Check. Undo it: 370 ÷ 1000 gives 0.37 again. ✓

Watch out. Do not move the decimal point. Move the digits, and use zeros to fill the gaps.

Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.

Method. Multiply as if there were no decimal point. Then put the point back so the answer has 1 decimal place.

1. Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.

    8.6
    × 7
   ----
   60.2

2. Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.
   6 × 7 = 42, write 2 and carry 4
   8 × 7 = 56 + 4 carried = 60, write 0 and carry 6
   write 6
3. Count the decimal places and put the point back. The decimal number has 1 decimal place, so the answer has 1 decimal place.
   86 × 7 = 602, and 602 → 60.2

Check. About 9 × 7 = 63, and 60.2 is close to that. ✓

Watch out. Count the decimal places in the question. Do not count the zeros you carry.