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Decimal operations

Adding, subtracting, multiplying and dividing decimals.

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You already know what each column after the decimal point is worth (Lesson 1.2). Now you will calculate with decimals: money, lengths and weights all use them.

In this lesson you will learn how to:

  • add and subtract decimals by lining up the points
  • multiply a decimal by a whole number
  • multiply a decimal by a decimal
  • divide a decimal by a whole number
  • divide by a decimal
Dark blue: whole numbers Blue: decimals, the parts of a whole Red dot: the decimal point

1. Adding and subtracting: line up the points

Key idea Write the numbers with their decimal points in a line, one under the other. Then every digit is in its right column, and you add or take away just as you did in Lesson 2.1.

How to add or subtract decimals

  1. Write the numbers with the points in a line
  2. Fill any empty places at the end with 0
    3.7 becomes 3.70. A whole number like 6 becomes 6.00. These zeros change nothing (Lesson 1.2), but they stop you missing a column.
  3. Work from the right, column by column
    Carry and exchange exactly as with whole numbers. You can carry or exchange across the point.
  4. Put the point in the answer straight below the others

Example: 4.56 + 3.7

  • Hundredths: 6 + 0 = 6.
  • Tenths: 5 + 7 = 12. Write 2, carry 1 into the ones.
  • Ones: 4 + 3 + 1 = 8.
Answer8.26

Example: 6 − 2.35

  • Write 6 as 6.00, so the top number has hundredths to take from.
  • Hundredths: 0 − 5? Too small, and the tenths have nothing to give. Go to the ones: 6 becomes 5, the tenths get 10, give 1 away and become 9, and the hundredths become 10. 10 − 5 = 5.
  • Tenths: 9 − 3 = 6. Ones: 5 − 2 = 3.
Answer3.65

Check with the inverse: 3.65 + 2.35 = 6.

Watch out Do not line up the last digits, the way you would with whole numbers. Set out like that, 4.56 + 3.7 puts the 7 under the 6 and gives 4.93, which is wrong.

2. Multiplying a decimal by a whole number

Key idea Ignore the point and multiply the digits as whole numbers. Then put the point back so the answer has the same number of decimal places as the question.

Example: 0.3 × 4

1 whole2 tenths

0.3 is 3 tenths. 3 tenths × 4 = 12 tenths. 10 tenths make 1 whole, so 12 tenths is 1.2.

The quick way: 3 × 4 = 12. The question has 1 decimal place, so the answer has 1: 1.2.

Example: 2.45 × 3

  1. Ignore the point and multiply (Lesson 2.2).245 × 3 = 735
  2. Count the decimal places in the question: 2.45 has 2.so the answer has 2
  3. Put the point back, 2 places from the end.7.35
Answer2.45 × 3 = 7.35

Check with an estimate: 2.45 is about 2, and 2 × 3 = 6. So 7.35 makes sense; 73.5 would not.

Watch out Count the places after the point, from the end of the answer. 0.3 × 4 is 1.2, not 0.12.

3. Multiplying a decimal by a decimal

Key idea Multiply the digits as whole numbers. Add up the decimal places in both numbers. The answer has that many decimal places.

Example: 0.3 × 0.4

0.3 of the square 0.4 of that

The dark part is 12 of the 100 small squares: 12 hundredths, which is 0.12.

The quick way

  • 0.3 × 0.4: 3 × 4 = 12. Decimal places: 1 + 1 = 2. So 0.12.
  • 1.2 × 0.3: 12 × 3 = 36. Decimal places: 1 + 1 = 2. So 0.36.
  • 0.04 × 0.3: 4 × 3 = 12. Decimal places: 2 + 1 = 3. You need 3 digits after the point, so put in a zero: 0.012.
  • 0.5 × 0.4: 5 × 4 = 20. Decimal places: 2. So 0.20, which is 0.2. Put the point back before you take off the end zero.
Did you notice? 0.3 × 0.4 = 0.12 is smaller than both numbers. Multiplying by a number less than 1 always makes a number smaller, because you are taking only part of it.

4. Dividing a decimal by a whole number

Key idea Use the bus stop from Lesson 2.3. The point in the answer goes straight above the point in the number. If something is left over at the end, write a 0 after the last digit and keep going.

Example: 7.56 ÷ 3

  • 7 ÷ 3 = 2, remainder 1. Carry the 1.
  • Put the point in the answer, straight above.
  • 15 ÷ 3 = 5. Then 6 ÷ 3 = 2.
Answer2.52

Example: 7.4 ÷ 4

  • 7 ÷ 4 = 1 r 3. Point straight up. 34 ÷ 4 = 8 r 2.
  • No digits left, but 2 is left over. Write 7.4 as 7.40: 20 ÷ 4 = 5.
Answer1.85
  • With decimals you do not write a remainder. Add zeros instead until it comes out exactly.
  • A whole number works the same way: 9 ÷ 4 is 9.00 ÷ 4 = 2.25.
  • If the first digit is too small, write a 0 before the point: 2.16 ÷ 4 = 0.54.

5. Dividing by a decimal

Key idea Make the number you divide by a whole number first. Multiply both numbers by 10 (or by 100). The answer stays the same.

Example: 3.6 ÷ 0.4

× 10 both
  1. 0.4 is not whole. Multiply both numbers by 10.3.6 ÷ 0.4 = 36 ÷ 4
  2. Now divide.36 ÷ 4 = 9
Answer9

Check: 9 × 0.4 = 3.6.

Why does it work? 3.6 ÷ 0.4 asks: how many 4 tenths fit into 36 tenths? That is the same as asking how many 4s fit into 36.
  • 2.4 ÷ 0.08: 0.08 needs × 100 to be whole. 240 ÷ 8 = 30.
  • 6 ÷ 0.2: × 10 both. 60 ÷ 2 = 30.
Watch out Multiply both numbers, never just one. And dividing by a number less than 1 gives a bigger answer: 3.6 ÷ 0.4 is 9, not 0.9.

The mistakes that cost marks

  1. Lining up the last digits. 4.56 + 3.7 set out with the 7 under the 6 gives 4.93. line up the points: 4.56 + 3.70 = 8.26.
  2. Bringing the decimals straight down. 6 − 2.35 written as 4.35. write 6 as 6.00 and exchange: 6.00 − 2.35 = 3.65.
  3. Counting the decimal places wrongly. 0.3 × 4 written as 0.12. 3 × 4 = 12, and the question has 1 decimal place: 1.2.
  4. Forgetting to add up the places in both numbers. 0.3 × 0.4 written as 1.2. 1 + 1 = 2 decimal places: 0.12.
  5. Putting the point in the wrong place in a division. 7.56 ÷ 3 written as 25.2. the point in the answer goes straight above the point in 7.56: 2.52.
  6. Writing a remainder. 7.4 ÷ 4 written as 1.8 r 2. write a 0 after the last digit and keep going: 7.40 ÷ 4 = 1.85.
  7. Multiplying only one number. 3.6 ÷ 0.4 worked out as 3.6 ÷ 4 = 0.9. multiply both by 10: 36 ÷ 4 = 9.

Quick checklist

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