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GCSE Maths · EdexcelChapter 4 · Decimals, rounding and estimation
GCSE Maths · Decimals, Rounding and Estimation · Foundation · Lesson 4.3

GCSE Maths: Multiplying decimals (Foundation)

Part of the GCSE Maths decimals, rounding and estimation chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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4.3

Multiplying decimals

Edexcel 1MA1: N2
Grade 2-4

What is 0.3 × 0.4? Think of a big square with side 1 whose area is 1. Cut it into 10 rows and 10 columns: that makes 100 small squares, and each one is worth one hundredth (0.01). A rectangle that is 0.3 high and 0.4 wide covers 3 rows and 4 columns, which is 3 × 4 = 12 small squares. Twelve hundredths is 0.12. So 0.3 × 0.4 = 0.12.

0.30.411
The shaded rectangle is 0.3 by 0.4. It covers 12 of the 100 small squares, so 0.3 × 0.4 = 0.12.

This shows why the method works. Tenths times tenths make hundredths: 0.3 = 310 and 0.4 = 410, and 310 × 410 = 12100 = 0.12 (multiply tops, multiply bottoms, as in Chapter 3). So we can multiply the digits as if they were whole numbers (3 × 4 = 12), and then the decimal points tell us where to put the decimal point in the answer.

The method

  1. Count the decimal places in each number and add them up. (The total is the number of decimal places the answer will have.)
  2. Ignore the decimal points and multiply the numbers as whole numbers. Use long multiplication if you need to.
  3. Put the decimal point back so that the answer has the total number of decimal places. If there are not enough digits, add zeros at the front.
  4. Check with an estimate: round each number to 1 significant figure and multiply (see section 4.7).

Notice that you do not line up the decimal points when you multiply. That rule is only for adding and subtracting.

Using a fact you already know

If you know 23 × 45 = 1035, you can write down 2.3 × 4.5 without any new multiplying. 2.3 is 23 divided by 10, and 4.5 is 45 divided by 10, so the answer is 1035 divided by 10 and divided by 10 again: 10.35. The digits stay the same; only the position of the decimal point changes.

Multiplying by a number less than 1

Multiplying by a number between 0 and 1 gives a smaller answer. For example 8 × 0.5 = 4: this is half of 8. Do not expect multiplication to always make things bigger.

Key ideas

  • Count the total number of decimal places in the question. The answer has that many.
  • Multiply the digits as whole numbers, then put the decimal point back.
  • Do not line up the decimal points when multiplying.
  • Use a known fact: if you multiply a number by 10, the answer is 10 times bigger; if you divide a number by 10, the answer is 10 times smaller.
  • Multiplying by a number between 0 and 1 makes the answer smaller.
Worked example 4.3.1
Work out 0.3 × 0.4.

Key words. A decimal place is a digit after the decimal point. The product is the answer to a multiplication.

Method. Multiply decimals like whole numbers, then put the decimal point back: the answer has as many decimal places as the two numbers together.

  1. Ignore the decimal points for now and count the decimal places. Multiplying decimals is the same as multiplying whole numbers, then fixing the decimal point at the end. First count how many digits come after the decimal point in each number.
    0.3 has 1 decimal place and 0.4 has 1 decimal place.
    Total decimal places = 1 + 1 = 2
  2. Write the numbers without the decimal points and multiply them as whole numbers. Removing the points turns the question into a whole-number multiplication.
    3 × 4
  3. Multiply 3 by 4, one digit at a time, from the right. Multiply each digit by the multiplier; when the answer is 10 or more, write the units digit and carry the tens.
    3 × 4 = 12, write 12
    = 12
  4. Put the decimal point in the answer: count 2 places from the right. The number of decimal places in the answer equals the total number of decimal places in the question. Add zeros at the front if there are not enough digits.
    12 with 2 decimal places = 0.12

Check. Divide back: 0.12 ÷ 0.4 = 0.3. ✓

Watch out. Do not line up the decimal points when multiplying. Count the total decimal places instead. Also check the size of the answer with an estimate.

Worked example 4.3.2
Work out 4.36 × 7.

Key words. A decimal place is a digit after the decimal point. The product is the answer to a multiplication.

Method. Multiply decimals like whole numbers, then put the decimal point back: the answer has as many decimal places as the two numbers together.

  1. Ignore the decimal points for now and count the decimal places. Multiplying decimals is the same as multiplying whole numbers, then fixing the decimal point at the end. First count how many digits come after the decimal point in each number.
    4.36 has 2 decimal places and 7 has 0 decimal places.
    Total decimal places = 2 + 0 = 2
  2. Write the numbers without the decimal points and multiply them as whole numbers. Removing the points turns the question into a whole-number multiplication.
    436 × 7
  3. Multiply 436 by 7, one digit at a time, from the right. Multiply each digit by the multiplier; when the answer is 10 or more, write the units digit and carry the tens.
    6 × 7 = 42, write 2, carry 4
    3 × 7 + 4 = 25, write 5, carry 2
    4 × 7 + 2 = 30, write 30
    = 3052
  4. Put the decimal point in the answer: count 2 places from the right. The number of decimal places in the answer equals the total number of decimal places in the question. Add zeros at the front if there are not enough digits.
    3052 with 2 decimal places = 30.52

Check. Estimate by rounding to 1 significant figure: 4 × 7 = 28, which is close to 30.52. ✓

Watch out. Do not line up the decimal points when multiplying. Count the total decimal places instead. Also check the size of the answer with an estimate.

Worked example 4.3.3
Work out 3.6 × 0.24.

Key words. A decimal place is a digit after the decimal point. The product is the answer to a multiplication.

Method. Multiply decimals like whole numbers, then put the decimal point back: the answer has as many decimal places as the two numbers together.

  1. Ignore the decimal points for now and count the decimal places. Multiplying decimals is the same as multiplying whole numbers, then fixing the decimal point at the end. First count how many digits come after the decimal point in each number.
    3.6 has 1 decimal place and 0.24 has 2 decimal places.
    Total decimal places = 1 + 2 = 3
  2. Write the numbers without the decimal points and multiply them as whole numbers. Removing the points turns the question into a whole-number multiplication.
    36 × 24
  3. Multiply 36 by each digit of 24. Long multiplication: multiply by the units digit, then by the tens digit (the digit in the tens column is worth ten times as much, so put a 0 on the end of that line), and so on.
    36 × 4
    6 × 4 = 24, write 4, carry 2
    3 × 4 + 2 = 14, write 14
    = 144

    36 × 2: the 2 is in the tens column, so it really means 20, and we work out 36 × 2 then put 1 zero on the end
    6 × 2 = 12, write 2, carry 1
    3 × 2 + 1 = 7
    72 with 1 zero on the end = 720
  4. Add the partial answers. The full product is the sum of the lines above.
    144 + 720 = 864
  5. Put the decimal point in the answer: count 3 places from the right. The number of decimal places in the answer equals the total number of decimal places in the question. Add zeros at the front if there are not enough digits.
    864 with 3 decimal places = 0.864

Check. Estimate by rounding to 1 significant figure: 4 × 0.2 = 0.8, which is close to 0.864. ✓

Watch out. Do not line up the decimal points when multiplying. Count the total decimal places instead. Also check the size of the answer with an estimate.

Worked example 4.3.4
Given that 23 × 45 = 1035, work out 2.3 × 0.45.

Method. Use the fact you know. Work out how each number has been multiplied or divided by a power of 10, and make the same changes to the known answer.

  1. Compare the first numbers. See what has happened to each number compared to the one in the fact you know.
    2.3 = 23 ÷ 10
  2. Compare the second numbers. Do the same for the other number.
    0.45 = 45 ÷ 100
  3. Make the same changes to the known answer. If a number is made 10 times smaller, the product is 10 times smaller. Start from the known answer 23 × 45 = 1035.
    1035 ÷ 10 ÷ 100 = 1.035

Check. Estimate: 2 × 0.5 = 1, close to 1.035. ✓

Watch out. If you divide one number by 10 and the other by 100, the product is divided by 10 and then by 100: that is 1000 in total.

Worked example 4.3.5
Imran buys 12 pens at £1.35 each. How much does he pay in total?

Method. Cost of 12 pens = price of one pen multiplied by 12.

  1. Write the calculation. Each pen costs £1.35 and Imran buys 12.
    1.35 × 12
  2. Multiply the price of one pen by the number of pens. Ignore the decimal points and multiply as whole numbers. Then put the decimal point back, so the answer has as many decimal places as the two numbers together.
    1.35 × 12: ignore the points, 135 × 12 = 1620
    2 + 0 = 2 decimal places in total, so the answer is 16.2
  3. Write the answer as money. Amounts of money are written with two decimal places.
    £16.20

Check. Estimate: £1 × 12 = £12 and £1.35 is a bit more, so about £16 is sensible. ✓

Watch out. The answer is money, so write it with 2 decimal places (£16.20, not £16.2).

Watch out

The number of decimal places in the answer is the total for both numbers, not the number of decimal places in one of them. 0.2 × 0.3 has 1 + 1 = 2 decimal places, so the answer is 0.06 (not 0.6).

Exercise 4C  Multiplying decimals

Non-calculator
1
FWork out
(a)0.4 × 0.2
(b)0.5 × 0.3
(c)0.6 × 7
(d)1.2 × 3
(e)0.07 × 0.9
(f)2.5 × 0.4
2
FThe diagram shows a square of side 1 split into 100 equal small squares. The shaded rectangle is 0.3 high and 0.5 wide. How many small squares are shaded, and what is 0.3 × 0.5?
0.30.511
3
FWork out
(a)3.4 × 5
(b)2.15 × 4
(c)6.3 × 0.8
(d)0.25 × 12
(e)1.8 × 1.5
(f)0.45 × 0.06
4
FWork out
(a)2.4 × 1.3
(b)0.36 × 0.15
(c)1.07 × 0.35
(d)5.2 × 0.07
(e)3.14 × 2.5
(f)0.9 × 0.004
5
FUse the fact that 36 × 47 = 1692 to work out
(a)3.6 × 47
(b)0.36 × 4.7
(c)360 × 0.47
(d)0.036 × 0.47
6
PSMaria buys 8 books that each cost £6.75. Work out the total cost.
7
PSGran buys 12 packs of rice. Each pack has a mass of 0.45 kg. Work out the total mass of rice.
8
PSA rectangle has length 6.5 cm and width 3.2 cm. Work out its area.
6.5 cm3.2 cm
9
PSA fence is made from 15 planks laid end to end. Each plank is 2.4 m long. How long is the fence?
10
RTick the correct statement and give a reason: “Multiplying a number by 0.5 always makes it bigger.” True or false?
11
RSam says: “0.2 × 0.3 = 0.6 because 2 × 3 = 6.” Explain what is wrong and give the correct answer.
12
RWithout working it out, is 0.7 × 0.8 bigger than 0.7 or smaller than 0.7? Give a reason, then work out 0.7 × 0.8.
13
RFind the missing digit:   3.□ × 0.4 = 1.36

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