SumNinja

Add and subtract fractions with equal-sized pieces

About 3 min

Add or subtract fractions by giving their pieces one shared size. Then count the pieces you gain or lose.

You have a third of a pizza and add another quarter of the same-sized pizza. That does not make two thirds or two quarters: the pieces have different sizes. Split both amounts into twelfths. Now you can count the total without mixing unlike pieces.

Rename the shares without changing them

A third covers four twelfths. A quarter covers three twelfths. Together they cover seven twelfths of a whole.

Put the two shares into twelfths

Find 1/3 + 1/4.

  1. Choose pieces that fit both shares

    13=412;14=312\frac13=\frac4{12};\quad\frac14=\frac3{12}

    Twelve appears in both times tables. Split each third into four and each quarter into three. Keep the amount of each share fixed.

  2. Count the new pieces

    412+312=712\frac4{12}+\frac3{12}=\frac7{12}

    Four twelfths and three twelfths make seven twelfths. The piece size stays the same, so the bottom stays twelve.

  3. Use the same idea to take a share away

    34−16=912−212=712\frac34-\frac16=\frac9{12}-\frac2{12}=\frac7{12}

    Start with nine twelfths. Take away two twelfths. Seven remain. Subtraction changes the piece count, not the piece size.

A shared bottom number need not be large

The denominator is the bottom number. It names the piece size. If one bottom number is a multiple of the other, change just one fraction. For 1/3 + 1/6, use 2/6 + 1/6 = 1/2.

You can always multiply the two bottom numbers to find a shared size. A smaller common multiple may save work. Use it if you can spot one, then simplify the answer if needed.

A whole also has a fraction name. To find 1 − 3/8, write the whole as 8/8. Taking away three eighths leaves five eighths.

Try it yourself

Work out an answer and explain your reasoning. Then check the explanation.

Add, then simplify if needed

12+38\frac12+\frac38

Find the total. Choose a piece size that lets you combine the two shares.

Show the answerHide the answer
78\frac78

Rename one half as four eighths. Four eighths plus three eighths is seven eighths.

Subtract unlike fractions

56−14\frac56-\frac14

Find the difference and explain how you chose a shared bottom number.

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712\frac7{12}

Use twelfths: 10/12 − 3/12 = 7/12.

Keep the piece size fixed

13+13=26\frac13+\frac13=\frac26

The learner has changed the size of the pieces as well as their count. What is the correct sum?

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23\frac23

The pieces are still thirds. Add the piece counts, not the bottom numbers.

Choose the piece size yourself

Try addition or subtraction. Give each pair one shared piece size, then combine the counts. Reduce the answer if you can.

2-minute practice · keep going after mistakes · opens in a new tab

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