Operations with Decimals (Cambridge (CIE) IGCSE Maths)
Revision Note
Written by: Jamie Wood
Reviewed by: Dan Finlay
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
Worked Example
Johnny has £32.50 and spends £1.74.
Calculate how much money Johnny has left.
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
The column furthest right is the hundredths column
0 is smaller than 4 so borrow from the next (tenths) column
Next is the tenths column, but again 4 is smaller than 7 so borrow 10 from the ones column
14 - 7 = 7 in the tenths column and continue working 'right to left'
Check the final answer is similar to the estimate
£31 and £30.76 are reasonably close
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
Examiner Tips and Tricks
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 .
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 |
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
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 |
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
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
Undo the initial actions
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 .
Method 1
Write as a fraction
Multiply both numbers by powers of 10 to create a calculation with integer values
Use short division (bus stop method)
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
98.6
Method 2
Ignore the decimal point to form a different, non-decimal calculation
6902 ÷ 7
Use short division (bus stop method)
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
98.6
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