Operate with fraction units

Add quantities in equal-sized units

For 2/3 + 1/4, the pieces are different sizes. Counting “two thirds and one quarter” as “three sevenths” changes the unit and loses the amounts.

Use the same whole and rename both fractions in twelfths:

Eight twelfths and three twelfths make eleven twelfths. The denominator stays twelve because the unit stays one twelfth. As a check, the answer is just less than one whole.

For 11/2-3/4, write the first amount in quarters:

Six quarter-units minus three quarter-units leaves three. You can check by adding 3/4 back to the result: 3/4 + 3/4 = 11/2.

Check your understanding 1 Repair a changed denominator

A learner writes 2/5 + 1/5 = 3/10. Explain what the denominator should describe, then find the sum.

Your corrected sum and explanation

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The unit is still one fifth of the same whole. Two fifths and one fifth make three fifths: 3/5. Splitting the whole into ten pieces would require renaming the quantities too: 4/10 + 2/10 = 6/10.

Now try 1/6 + 1/3 without the example: rename the third as two sixths, giving 3/6 = 1/2.

Multiply to find a fraction of an amount

Three quarters of 12 means split 12 into four equal groups and take three groups. Each group is 12÷4 = 3, so the amount is 3 × 3 = 9:

Multiplication by 3/4 makes a positive amount smaller. Multiplication does not always make an amount bigger: the scale factor matters.

For two thirds of three quarters of a strip, first take three quarters of the original strip. Split that chosen amount into three equal parts and take two. Each of those parts is one quarter of the original strip, so two make one half:

The numerator products count selected small units; the denominator products name how many of those units fit in the original whole. This argument uses positive fractions of the same named whole.

Use improper fractions for mixed-number products

Find the total length of three pieces of ribbon, each 12/3 m long. There are three equal groups of 5/3 m:

Another valid method keeps the mixed number: three lots of 1 m give 3 m, and three lots of 2/3 m give 2 m. The total is 5 m. You do not need a bar model when you can explain a valid calculation.

For 7/4 × 6/5, cancel common factors in the product, not added terms:

Check the size: multiplying 7/4 by 6/5, which exceeds one, should make the first positive number larger.

Check your understanding 2 Choose the operation

A bottle contains 13/4 litres. You pour out 2/5 of its contents. How much do you pour out? How much remains?

Your equations, amounts and units

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The poured amount is 2/5 × 7/4 = 7/10 litre. The remainder is 7/4-7/10 = 35/20-14/20 = 21/20 = 11/20 litres. The two amounts add back to 13/4 litres. Subtracting 2/5 litre directly would confuse a fraction of the contents with a volume.

Before you move on

Explain why 21/4 + 11/2 and 21/4 × 1/2 need different decisions. Then calculate both: 33/4 and 11/8. Decide which result should exceed 21/4 before you calculate.