Fractions · Grades 4, 5

Subtracting Fractions: Like and Unlike Denominators

Quick answer

To subtract fractions, give them a common denominator, then subtract the top numbers and keep the bottom number. Three quarters minus one sixth becomes nine twelfths minus two twelfths, which is seven twelfths. With mixed numbers, borrow one whole from the whole-number part when the fraction you are taking away is the bigger one.

What you'll learn

  • Subtract fractions with the same bottom number
  • Subtract fractions with different bottom numbers
  • Borrow one whole when subtracting mixed numbers

The same setup as adding

Subtracting needs the parts to be the same size, for the same reason adding does. You cannot take sixths away from quarters until both are cut to a shared size.

When the bottom numbers already match, take the tops away and keep the bottom.

Five eighths take away two eighths Two bars of equal length, each split into eight parts. The top bar has five parts shaded and the bottom bar has two. 58 28
Five eighths take away two eighths
58−28=38\frac{5}{8} - \frac{2}{8} = \frac{3}{8}

Five eighths take away two eighths leaves three eighths, the same way five apples take away two apples leaves three.

Unlike bottom numbers

Three steps, and the first two are identical to adding.

  1. Find a common denominator.
  2. Rewrite both fractions over it.
  3. Subtract the tops, keeping the bottom, then simplify.

Work out 34−16\tfrac{3}{4} - \tfrac{1}{6}. The LCD of 44 and 66 is 1212.

34=91216=212\frac{3}{4} = \frac{9}{12} \qquad \frac{1}{6} = \frac{2}{12} 912−212=712\frac{9}{12} - \frac{2}{12} = \frac{7}{12}

77 and 1212 share no factors, so 712\tfrac{7}{12} is the answer.

Mixed numbers, without borrowing

Take the whole numbers apart from the fractions.

334−114=(3−1)+(34−14)=224=2123\tfrac{3}{4} - 1\tfrac{1}{4} = (3 - 1) + \left(\frac{3}{4} - \frac{1}{4}\right) = 2\tfrac{2}{4} = 2\tfrac{1}{2}

This works whenever the first fraction is the bigger one.

Why borrowing works

Sometimes the fraction you are taking away is bigger than the one you have. Look at 415−1354\tfrac{1}{5} - 1\tfrac{3}{5}. You cannot take 35\tfrac{3}{5} from 15\tfrac{1}{5} and stay positive.

So borrow one whole from the 44 and turn it into fifths.

415=3+1+15=3+55+15=3654\tfrac{1}{5} = 3 + 1 + \frac{1}{5} = 3 + \frac{5}{5} + \frac{1}{5} = 3\tfrac{6}{5}

Nothing was created or lost. One whole became 55\tfrac{5}{5}, which is the same amount written in the units you need.

Now the subtraction goes through:

365−135=2353\tfrac{6}{5} - 1\tfrac{3}{5} = 2\tfrac{3}{5}

This is the same borrowing you already do with whole numbers. Taking 77 from 5252 borrows ten ones from the tens column; here you borrow one whole as five fifths.

Worked examples

Common mistakes

Checking your answer

Add your answer back to the number you took away. You should land on the number you started with.

For Example 4: 212+2342\tfrac{1}{2} + 2\tfrac{3}{4}. The whole numbers give 44, and 24+34=54=114\tfrac{2}{4} + \tfrac{3}{4} = \tfrac{5}{4} = 1\tfrac{1}{4}, so the total is 5145\tfrac{1}{4} — the number the problem started with.

Practice problems

  1. Work out 57−27\tfrac{5}{7} - \tfrac{2}{7}.

    Answer

    37\tfrac{3}{7}

    Full solution

    The bottom numbers match: 5−2=35 - 2 = 3, giving 37\tfrac{3}{7}.

  2. Work out 78−12\tfrac{7}{8} - \tfrac{1}{2}.

    Hint

    88 is a multiple of 22.

    Answer

    38\tfrac{3}{8}

    Full solution

    12=48\tfrac{1}{2} = \tfrac{4}{8}, so 78−48=38\tfrac{7}{8} - \tfrac{4}{8} = \tfrac{3}{8}.

  3. Work out 23−14\tfrac{2}{3} - \tfrac{1}{4}.

    Answer

    512\tfrac{5}{12}

    Full solution

    The LCD is 1212: 812−312=512\tfrac{8}{12} - \tfrac{3}{12} = \tfrac{5}{12}.

  4. Work out 56−14\tfrac{5}{6} - \tfrac{1}{4}.

    Answer

    712\tfrac{7}{12}

    Full solution

    The LCD of 66 and 44 is 1212: 1012−312=712\tfrac{10}{12} - \tfrac{3}{12} = \tfrac{7}{12}.

  5. Work out 358−1183\tfrac{5}{8} - 1\tfrac{1}{8}.

    Answer

    2122\tfrac{1}{2}

    Full solution

    Whole numbers: 3−1=23 - 1 = 2. Fractions: 58−18=48\tfrac{5}{8} - \tfrac{1}{8} = \tfrac{4}{8}.

    Simplifying gives 2122\tfrac{1}{2}.

  6. Work out 613−2236\tfrac{1}{3} - 2\tfrac{2}{3}.

    Hint

    You will need to borrow.

    Answer

    3233\tfrac{2}{3}

    Full solution

    23\tfrac{2}{3} is bigger than 13\tfrac{1}{3}, so borrow: 613=5436\tfrac{1}{3} = 5\tfrac{4}{3}.

    Then 543−223=3235\tfrac{4}{3} - 2\tfrac{2}{3} = 3\tfrac{2}{3}.

  7. Work out 5−385 - \tfrac{3}{8}.

    Answer

    4584\tfrac{5}{8}

    Full solution

    Write 55 as 4884\tfrac{8}{8}, then 488−38=4584\tfrac{8}{8} - \tfrac{3}{8} = 4\tfrac{5}{8}.

  8. Work out 414−1234\tfrac{1}{4} - 1\tfrac{2}{3}.

    Hint

    Common denominator first, then borrow.

    Answer

    27122\tfrac{7}{12}

    Full solution

    Over an LCD of 1212: 4312−18124\tfrac{3}{12} - 1\tfrac{8}{12}.

    Borrow one whole: 4312=315124\tfrac{3}{12} = 3\tfrac{15}{12}.

    Then 31512−1812=27123\tfrac{15}{12} - 1\tfrac{8}{12} = 2\tfrac{7}{12}.

  9. A jug holds 78\tfrac{7}{8} of a litre. You pour out 13\tfrac{1}{3} of a litre. How much is left?

    Answer

    1324\tfrac{13}{24} of a litre

    Full solution

    The LCD of 88 and 33 is 2424: 2124−824=1324\tfrac{21}{24} - \tfrac{8}{24} = \tfrac{13}{24}.

    That is a little over half a litre, which is a sensible amount to be left with.

  10. Omar works out 315−1453\tfrac{1}{5} - 1\tfrac{4}{5} and writes 2352\tfrac{3}{5}. Check his answer and correct it if needed.

    Hint

    Add his answer to 1451\tfrac{4}{5} and see where you land.

    Answer

    Wrong. The answer is 1251\tfrac{2}{5}.

    Full solution

    Checking: 235+145=375=4252\tfrac{3}{5} + 1\tfrac{4}{5} = 3\tfrac{7}{5} = 4\tfrac{2}{5}, which is not the 3153\tfrac{1}{5} he started with.

    Omar took the smaller top from the bigger one to dodge a negative. Borrowing instead: 315=2653\tfrac{1}{5} = 2\tfrac{6}{5}, and 265−145=1252\tfrac{6}{5} - 1\tfrac{4}{5} = 1\tfrac{2}{5}.

Frequently asked questions

How do I subtract fractions with different denominators?

Rewrite both over a common denominator, then subtract the top numbers and keep the bottom. Three quarters minus one sixth is nine twelfths minus two twelfths, which is seven twelfths.

What if the second fraction is bigger?

With mixed numbers, borrow one whole from the whole-number part and add it to the fraction as a full set of parts. Four and one fifth becomes three and six fifths.

Can the answer be negative?

Yes, if you take away more than you started with. Four fifths minus one whole is negative one fifth. In most grade school problems the answer stays positive.

Is subtracting the same work as adding?

Yes, up to the last step. You find a common denominator and rewrite exactly the same way, then subtract the top numbers instead of adding them.

What to learn next

Formulas on this page

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.4.NF.B.3aNumber and Operations—FractionsUnderstand addition and subtraction of fractions as joining and separating parts referring to the same whole.
  • CCSS.MATH.CONTENT.4.NF.B.3cNumber and Operations—FractionsAdd and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
  • CCSS.MATH.CONTENT.5.NF.A.1Number and Operations—FractionsAdd and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
For teachersTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.