Operations with Decimals (Cambridge International IGCSE Mathematics)

Operations with decimals

How do I add or subtract decimals without a calculator?

• If the numbers involve decimals, the column methods for addition and subtraction can be used

  • Line the place value columns up carefully

  • Make sure the decimal points are all in the same column

    • Writing zeros (often called place value holders) can help keep everything in line

  • e.g. 2.145 + 13.02 would be written as equation

Worked Example Johnny has £32.50 and spends £1.74.

Calculate how much money Johnny has left.

Answer:

This is a subtraction question as Johnny has spent money

Make a quick estimate

33 - 2 = 31

Align the digits by place value, ensuring £32.50 is the top number and using the decimal points as a starting point Using place value holding zeros is optional

equation

The column furthest right is the hundredths column 0 is smaller than 4 so borrow from the next (tenths) column

equation* *

Next is the tenths column, but again 4 is smaller than 7 so borrow 10 from the ones column

equation* *

14 - 7 = 7 in the tenths column and continue working 'right to left'

equation

Check the final answer is similar to the estimate £31 and £30.76 are reasonably close

Final answer: Johnny has £30.76 left

How do I multiply decimals without a calculator?

• The standard methods for multiplication can be easily adapted for use with decimal numbers

  • E.g. column method, lattice method and grid method

• You can make a problem easier by converting to integer values then undoing the action to the answer

  • Multiply each value by a power of 10 to make it an integer

  • Perform the easier multiplication

  • Undo the initial action by dividing by the same powers of 10

  • E.g. 2.5 x 4.01

    • Multiply 2.5 by 10 to give 25 and 4.01 by 100 to give 401
    • 25 x 401 = 10 025
    • Divide the answer by 10 and 100 to undo the initial changes
    • So the answer is 10.025

• An alternative method is to ignore the decimal point whilst multiplying

  • then put it back in the correct place for the final answer using estimation

  • E.g. 1.3 × 2.3

    • Ignoring the decimals this is 13 × 23, which works out to 299
    • 1.3 × 2.3 is approximately 1.5 × 2, which is 3
    • So we know the answer should be close to 3, rather than 0.3 or 30
    • So the answer is 2.99

Exam Hint A good way to check your answer without a calculator is to estimate it by rounding everything to 1 or 2 significant figures.

Worked Example Without using a calculator, find 1.57 × 0.78 .

Answer:

Method 1

Multiply both values by powers of 10 to create a calculation using integer values

1.57 x 100 = 157 0.78 x 100 = 78

157 x 78

Using the grid method

	100	50	7
70	7000	3500	490
8	800	400	56

charalign center stackalign right end 7000 | + 3500 end | 490 800 400 56 horizontal line 12246 end

157 × 78 = 12 246

Undo the initial action by dividing the answer by 10 000 (This is dividing by 100 and then 100 again)

12 246 ÷ 10 000

Final answer: 1.2246

Method 2

Ignore decimal point to form a different, non-decimal calculation

157 × 78

Using the grid method

	100	50	7
70	7000	3500	490
8	800	400	56

charalign center stackalign right end 7000 | + 3500 end | 490 800 400 56 horizontal line 12246 end

157 × 78 = 12 246

Estimate the original calculation to write this as the correct answer

1.57 × 0.78 is approximately 2 × 1 = 2

So the correct answer is between 1 and 10

Final answer: 1.2246

How do I divide a decimal without a calculator?

• Use a similar approach to when multiplying decimals

• Make a problem easier by converting to integer values then undoing the action to the answer

  • Write the division as a fraction

  • Multiply each value by a power of 10 to make it an integer

  • Perform the easier division

  • Undo the initial action

    • divide by the same power of 10 the **numerator **was multiplied by
    • multiply by the** same power of 10 the denominator** was multiplied by

  • E.g. 37.5 ÷ 0.25

    • $37.5\div0.25\rightarrow\frac{37.5}{0.25}$
    • $\frac{37.5\times10}{0.25\times100}=\frac{375}{25}$
    • equation$\frac{375}{25}=15$
    • Undo the initial actions $\frac{15\times100}{10}$
    • So the answer is 150

• Alternatively, ignore the decimal point whilst carrying out the division

  • then put it back in the correct place for the final answer using estimation

  • E.g. 4.68 ÷ 6

    • 4.68 ÷ 6 is approximately 5 ÷ 6, which is between 1 and 0.5
    • So we know the answer should be 0.78, rather than 7.8 or 0.078

Worked Example Without using a calculator, find 69.02 ÷ 0.7 .

Answer:

Method 1

Write as a fraction

$\frac{69.02}{0.7}$

Multiply both numbers by powers of 10 to create a calculation with integer values

$\frac{69.02\times100}{0.7\times10}=\frac{6902}{7}$

Use short division (bus stop method)

long division by 7 yields 0986 pile 6 ^ 6 9 ^ 6 0 ^ 4 2 end pile end long division

$\frac{6902}{7}=986$

Convert the answer Divide by 100 to cancel out 69.02 being multiplied by 100 Multiply by 10 to cancel out 0.7 being multiplied by 10

$\frac{986\times10}{100}$

Final answer: 98.6

Method 2

Ignore the decimal point to form a different, non-decimal calculation

6902 ÷ 7

Use short division (bus stop method)

long division by 7 yields 0986 pile 6 ^ 6 9 ^ 6 0 ^ 4 2 end pile end long division

6902 ÷ 7 = 986

Estimate the original calculation to write this as the correct answer

69.02 ÷ 0.7 is approximately 70 ÷ 1 = 70

So the correct answer is between 10 and 100

Final answer: 98.6