Divide a fraction by a fraction

Division of fractions is rare in everyday life because it’s too cumbersome for casual use. It is also rare in sciences, where people use decimals. We only want division of fractions to avoid a hole in our math theory. We have division, and we have fractions... Does our division work for fractions? If so, how?!

One-page wonders

These are books about fraction division you can fold out of single piece of paper.


By Carol Cross and Madison Cross Sugg:

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Prerequisite concepts for fraction division

- ratio
- proportion (equivalent ratios)
- common denominators
- units and changes in unit sizes (unitizing)

If you understand these ideas, you are ready to…

Learn To Divide Fractions In One Page!

Divide external image gif.latex?%5Cfrac%7B2%7D%7B5%7D by external image gif.latex?%5Cfrac%7B3%7D%7B4%7D. Rephrase the question as, “What is the ratio of external image gif.latex?%5Cfrac%7B2%7D%7B5%7D to external image gif.latex?%5Cfrac%7B3%7D%7B4%7D?”
Visual method: the rectangle model
Look at the picture until you see the answer!
  • Slice one side of the rectangle into 5 and the other into 4. There are 20 total units.
  • external image gif.latex?%5Cfrac%7B2%7D%7B5%7D means 2*4 units out of 20
  • external image gif.latex?%5Cfrac%7B3%7D%7B4%7D means 3*5 units out of 20
  • Their ratio is 2*4 to 3*5, or external image gif.latex?%5Cfrac%7B8%7D%7B15%7D
Numeric method 1: common denominator
  • Write the ratio external image gif.latex?%5Cfrac%7B2%7D%7B5%7D%3A%5Cfrac%7B3%7D%7B4%7D
  • Find a common denominator external image gif.latex?%5Cfrac%7B2*4%7D%7B5*4%7D%3A%5Cfrac%7B3*5%7D%7B4*5%7D or external image gif.latex?%5Cfrac%7B8%7D%7B20%7D%3A%5Cfrac%7B15%7D%7B20%7D
  • 20 times more on both sides makes the ratio equivalent to 8:15
  • Write as a fraction external image gif.latex?%5Cfrac%7B8%7D%7B15%7D
Numeric method 2: ratio to one
  • Rephrase the question as, "The ratio external image gif.latex?%5Cfrac%7B2%7D%7B5%7D%3A%5Cfrac%7B3%7D%7B4%7D is equivalent to the ratio of what to 1?"
  • 4 times more on both sides makes the ratio equivalent to external image gif.latex?%5Cfrac%7B2*4%7D%7B5%7D%3A3
  • 3 times less on both sides makes the ratio equivalent to external image gif.latex?%5Cfrac%7B2*4%7D%7B5*3%7D%3A1
  • Carry out the multiplications: external image gif.latex?%5Cfrac%7B8%7D%7B15%7D
Create your own examples and solve them visually and numerically. People who worked out many examples summarized this algorithm. If you work with enough examples, you will come up with this or your own algorithm, too.

The result of dividing a fraction by a fraction is, again, a fraction. To find its numerator and numerator, cross-multiply. To find the numerator, multiply the numerator of the dividend by the denominator of the divisor. To find its denominator, multiply the denominator of the dividend by the numerator of the divisor.

For dividing positive or negative fractions, use the same rules that apply to integers to determine the signs.
external image gif.latex?%5Cfrac%7B2%7D%7B5%7D%3A%28-%5Cfrac%7B3%7D%7B4%7D%29%3D-%5Cfrac%7B8%7D%7B15%7Dand external image gif.latex?%28-%5Cfrac%7B2%7D%7B5%7D%29%3A%28-%5Cfrac%7B3%7D%7B4%7D%29%3D%5Cfrac%7B8%7D%7B15%7D

Credits and links

Bonuses: don't let meaningless algorithms happen to your kids!

Madison's stop-motion proof that Ma and Pa Kettle were right!