Add and subtract unlike fractions

Name both amounts in the same parts

For 2/3 + 1/4, twelfths work for both denominators:

The numerator and denominator of each original fraction change by the same factor. Adding numerators before matching units would combine thirds and quarters as if they were identical pieces.

For 5/6-1/4, use twelfths again:

You do not need the product of the denominators every time. For 1/6 + 1/4, twelve is already a multiple of both six and four.

A common denominator can be greater than 12. For 1/8 + 1/12, use twenty-fourths: 3/24 + 2/24 = 5/24. The limit of 12 concerns the given denominators, not the denominator you choose for equivalent fractions.

Whole amounts can appear in the result

The sum exceeds one whole. Check its size: both original fractions exceed one half, so together they exceed one. If a mixed number is requested, write 15/12. If either a fraction or a mixed number is allowed, 17/12 names the same amount.

An improper fraction can also be given: 7/6-1/4 = 14/12-3/12 = 11/12. Match the parts in the same way; do not discard the whole contained in 7/6.

Check your understanding 1 Choose your own common unit

Find 2/5 + 3/10 and 7/8-1/4. Explain why you can use a given denominator in each case.

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Use tenths for the first: 4/10 + 3/10 = 7/10. Use eighths for the second: 7/8-2/8 = 5/8. Ten is a multiple of five; eight is a multiple of four.

Read a multi-step situation

A ribbon is 3/4 m long. A piece 1/4 m long is cut off, then a piece 1/8 m long is cut off. There are only two different given denominators: four and eight.

Use eighths: 3/4 = 6/8 m and 1/4 = 2/8 m. The remainder is

Check that the two cut pieces and the remaining ribbon total the starting length. Operations starting with mixed numbers are taught later; here you are matching fractional units.

Check your understanding 2 Find an unknown starting length

After cutting 1/6 m from a ribbon, 3/4 m remains. What was the original length? Explain why this question calls for addition.

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The original consists of the cut piece and the remainder. Using twelfths: 2/12 + 9/12 = 11/12 m. Subtracting the cut piece from the remainder would find a different length.

Check your understanding 3 Keep track of three amounts

Find 3/4 + 2/3-1/4. Give your answer as a mixed number in simplest form. Show one common unit throughout, then check by grouping the two amounts written in quarters first.

Your mixed number and two methods

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Using twelfths, 9/12 + 8/12-3/12 = 14/12 = 11/6.

Alternatively, 3/4-1/4 = 1/2, and 1/2 + 2/3 = 3/6 + 4/6 = 7/6 = 11/6. Both methods add two thirds and remove one quarter from three quarters. Keep the subtraction attached to the one quarter when regrouping.

The given denominators are four and three: three amounts can still use only two different given denominators. 7/6 has the right value, but the requested mixed-number form is 11/6.