How do you divide fractions?

Keep, change, flip. It is the rule everybody remembers and almost nobody can justify, which is a shame, because the reason is short.

3/4 ÷ 2/5

Keep     3/4
Change   ÷  becomes  ×
Flip     2/5  becomes  5/2

3/4 × 5/2  =  (3 × 5) / (4 × 2)  =  15/8  =  1 7/8

Once the division has become a multiplication, you are just multiplying fractions: tops times tops, bottoms times bottoms. Division is the only hard part, and flipping removes it.

Why flipping the second fraction works

Dividing by a number undoes multiplying by it. Multiplying by the reciprocal undoes it too, because a fraction times its own flip is 1:

2/5 × 5/2  =  10/10  =  1

Two operations that both undo the same thing have to be the same operation. So dividing by 2/5 and multiplying by 5/2 must land on the same answer.

If you prefer it written out, stack the division as one fraction and multiply the top and bottom by 5/2. The bottom collapses to 1 and the division disappears:

  3/4       3/4 × 5/2      3/4 × 5/2
 ─────  =  ───────────  =  ─────────  =  3/4 × 5/2  =  15/8
  2/5       2/5 × 5/2          1

Nothing was invented there. The fraction was multiplied by 1, in the shape of (5/2)/(5/2), which never changes a value.

The version you can picture

Division asks how many of these fit inside that. Half a pizza, cut into eighths, gives four slices:

1/2 ÷ 1/8  =  1/2 × 8/1  =  8/2  =  4

Count it on your fingers: an eighth fits into a half four times. The arithmetic agrees, which is reassuring, because this is also the place where dividing stops behaving like people expect. Dividing by a number smaller than 1 makes the answer bigger. There is no trick in that, just a lot of small things fitting inside a larger one.

3 ÷ 1/4  =  3/1 × 4/1  =  12/1  =  12     three wholes, four quarters each

Whole numbers, both ways round

A whole number is that number over 1, and the rule does not change. Which side it sits on decides what happens:

Fraction ÷ whole      2/3 ÷ 4  =  2/3 × 1/4  =  2/12  =  1/6
Whole ÷ fraction      6 ÷ 3/4  =  6/1 × 4/3  =  24/3   =  8

The shortcut for the first line is worth keeping: to divide a fraction by a whole number, multiply the denominator by it and leave the numerator alone. Thirds cut into four give twelfths.

Mixed numbers: convert first, always

This is where marks get lost. You cannot divide the whole parts and the fraction parts separately. Turn both into improper fractions, then keep, change, flip:

2 1/2 ÷ 1 1/4

Convert    2 1/2  =  (2 × 2 + 1)/2  =  5/2
           1 1/4  =  (1 × 4 + 1)/4  =  5/4

Flip       5/2 × 4/5  =  20/10  =  2

Sanity check: 1 1/4 is 1.25, and 1.25 fits into 2.5 exactly twice.

Cancel before you multiply

Once the flip is done, every shortcut from multiplication is available, including cancelling any numerator against any denominator. Do it after the flip, never before:

6/7 ÷ 9/14  =  6/7 × 14/9

Cancel   6 and 9 share 3    ->  2 and 3
         14 and 7 share 7   ->  2 and 1

         2/1 × 2/3  =  4/3  =  1 1/3

Straight across instead:  84/63,  which reduces by 21 to 4/3.

Same answer, but one route asks you to spot that 84 and 63 share 21, and the other does not.

The rule your teacher did not mention

You can also divide fractions straight across, tops by tops and bottoms by bottoms. It is not a folk method, it is always true:

8/9 ÷ 2/3     straight across:  (8 ÷ 2) / (9 ÷ 3)  =  4/3
              keep change flip: 8/9 × 3/2 = 24/18   =  4/3

The catch is that it is only useful when both divisions come out whole, which is rare. Try it on 3/4 ÷ 2/5 and you get (3 ÷ 2)/(4 ÷ 5), a fraction stacked inside a fraction, and now you have two problems. Keep, change, flip works every time, so it is the one to learn.

Where this actually comes up

Anywhere you are portioning something out. A batch of dough takes 3/4 of a cup of flour and the bag holds 6 cups, so you get 8 batches, which is the 6 ÷ 3/4 above. Cutting a 2 1/2 metre board into 1 1/4 metre lengths gives 2 pieces. Splitting 3/4 of a tank of fuel across trips that each use 2/5 of a tank gets you 1 7/8 trips, and the useful reading of that is one full trip with most of another in reserve.

The decimal trap

It is tempting to convert to decimals and divide those instead. The problem is that plenty of fractions have no exact decimal form, and division does not just carry the rounding error, it magnifies it. A rounded number in the denominator is the worst place to put one:

10 ÷ 1/3   exactly:      10 × 3/1  =  30
           as decimals:  10 ÷ 0.333  =  30.030030...

3/4 ÷ 1/3  exactly:      3/4 × 3/1 =  9/4  =  2.25
           as decimals:  0.75 ÷ 0.333  =  2.252252...

Thirty batches or thirty and a bit: the difference was manufactured by writing 1/3 as 0.333. This is the same failure mode as 0.1 + 0.2 = 0.30000000000000004, where a calculator stores numbers in binary floating point instead of keeping the fraction whole. Stay in fractions until the last step, and if you want a decimal at the end, converting a fraction to a decimal is one division at the end rather than several in the middle.

Common mistakes, quickly

If dividing is still fiddly, it is usually because the fractions came in ugly. Adding fractions teaches the common denominator machinery that makes everything else easier to read.

Every number on this page was computed exactly and checked before publishing.

Or type 3/4 ÷ 2/5 and get 1 7/8 back

Prism has a fraction mode that divides exactly and reduces for you, so 10 ÷ 1/3 is 30 and not 30.03. Fractions are free forever, no subscription. See the app.