Maths

Addition, subtraction, multiplication and division - Higher

You should have already revised Fraction basics. Now we are going to look at more difficult calculations involving mixed fractions.

Adding and subtracting

To add or subtract fractions, they have to have the same denominator (bottom number). Once this is the case, adding and subtracting are easy.

Here is a typical question:

Question

Work out: 1/5 + 1/2

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Answer

We need both the bottom numbers to be the same. We need to find the smallest number which both 5 and 2 can divide into - that is, the lowest common multiple of 5 and 2. This is 10. So we change the denominators into 10:

Remember to multiply both the denominator (bottom number) and numerator (top number).

1 over 5 + 1 over 2 = 1 x 2 over 5 x 2 + 1 x 5 over 2 x 5 = 2 over 10 + 5 over 10 = 7 over 10

The common denominator is not always the two denominators multiplied together. For instance, the lowest common multiple of 4 and 6 is 12.

If you do just multiply the denominators together then it will always work. It might be that you have to cancel your answer down at the end.

Here is another question:

Question

Work out: 7/9 - 1/6

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Answer

The lowest common multiple of 9 and 6 is 18, but we could use 9 x 6 = 54. Here are both ways, showing that we get the same answer:

Using 18

7 over 9 - 1 over 6 = 7 x 2 over 9 x 2 - 1 x 3 over 6 x 3 = 14 over 18 - 3 over 18 = 11 over 18

Using 54

7 over 9 - 1 over 6 = 7 x 6 over 9 x 6 - 1 x 9 over 6 x 9 = 42 over 54 - 9 over 54 = 33 over 54

But 33/54 can be cancelled to 11/18 by dividing the top and bottom by 3:

33 over 54 = 33 divided by 3 over 54 divided by 3 = 11 over 18

Remember

If the question allows you to use a calculator, most scientific calculators have a fraction button (ab/c), so you can use this to check your answers and see if you have cancelled down correctly.

Here is one for you to try:

Question

Work out: 3/8 + 2/5

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Answer

Did you get 31/40?

If you had problems, remember that the common denominator is 40, as this is the lowest common multiple of 5 and 8. See if you can fill in the blanks here:

3 over 8 + 2 over 5 = ? x 5 over 8 x 5 + ? x 8 over 5 x 8 = ? over 40 + 16 over 40 = 31 over 40

Fractions and whole numbers

2 3/4 is a mixed number. 2 is a whole number, and 3/4 is a fraction.

You can change this into a top-heavy fraction by writing 2 x 4 + 3 over 4 = 11 over 4

To change a top-heavy fraction into a mixed number, you first divide the top of the fraction (numerator) by the bottom (denominator).

For example 13/4 = 3 1/4, because 4 divides into 13 three times, with one left over.

Question

Calculate: 3/5 + 3/4

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Answer

3/5 + 3/4 = 12/20 + 15/20 = 27/20

Notice that this is a top heavy fraction.

= 1 7/20

Subtracting fractions with mixed numbers

Again, to subtract fractions you must make the denominators equal.

Question

Calculate: 2 2/3 - 1 4/5

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Answer

2 2/3 - 1 4/5

= 8/3 - 9/5

= 40/15 - 27/15

= 13/15

Multiplying fractions

Multiplying fractions is easy - we just multiply the numerators (top numbers) and multiply the denominators (bottom numbers). This will probably mean that you can cancel down the answer at the end. Or you can do it before you start.

You can try to find common factors between any of the numerators and any of the denominators. You can cancel down diagonally - they do not have to belong to the same fraction.

Question

Work out: 2/5 × 7/8

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Answer

We can cancel the top 2 with the bottom 8, so this 2 over 5 x 7 over 8 becomes this 1 over 5 x 7 over 4 = 7 over 20

Question

Work out:3/8 × 4/5

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Answer

The answer is 3/10?

You can cancel the top 4 with the bottom 8.

If you had problems, see if you can fill in the blanks here:

3/8 × 4/5

= 3/? × 1/5

= 3/10

Multiplying with mixed numbers

You will sometimes need to multiply fractions with mixed numbers. You must always remember to turn any mixed numbers into top-heavy fractions before you multiply.

Dividing fractions

The basic rule when dividing two fractions is to turn the second fraction upside down, and multiply.

So to work out 2/3 ÷ 3/4 we work out 2/3 × 4/3 = 2×4/3×3 = 8/9

Remember we only turn the fraction we are dividing by (the second one) upside down.

Here is another question:

Question

Work out: 3/5 ÷ 6/7

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Answer

3/5 ÷ 6/7

= 3/5 × 7/6

= 1/5 × 7/2    (the top 3 and bottom 6 cancel)

= 1×7/5×2

= 7/10

Try this one:

Question

Work out: 4/9 ÷ 3/5

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Answer

Did you get 20/27?

If not, check that you only turned the second fraction upside down. See if you can fill in the blanks here:

4/9 ÷ 3/5 = 4/9 × ?/3 = 20/27

Dividing mixed numbers

Again, turn any mixed numbers into top-heavy fractions at the start. Then turn the fraction you are dividing by upside down, and multiply. Here is an example:

3 1/3 ÷ 2 2/3

= 10/3 ÷ 8/3

= 10/3 × 3/8

= 5/4

= 1 1/4

Reciprocals

The reciprocal of a fraction is the fraction turned upside down.

So the reciprocal of 3/4 is 4/3

The reciprocal of a number is 1 divided by that number. For instance, to work out the reciprocal of 4:

We can think of 4 as 4/1

So 4 turned upside down is 1/4

Now try a Test Bite

 

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