Divide fractions is one of 3 fractions formulas in the arithmetic section of this library, and it is used at middle school level.
Why divide fractions works
Dividing asks how many copies of the second fraction fit inside the first. One whole holds 3/2 copies of 2/3, so 4/7 of a whole holds 4/7 of that many. Counting how many fit is the same as multiplying by the flipped fraction, which is why the divide sign turns into a times sign.
What each symbol means
$c/d$ is the divisor; $d/c$ is its reciprocal.
Divide fractions: when it holds
$b,c,d\ne0$ because neither a denominator nor the divisor may be zero.
When it stops applying
You cannot use it when the fraction you are dividing by is 0, because 0/5 has no reciprocal to flip to. Dividing by nothing is not a question with an answer, and the formula simply has no value there.
Divide fractions: a worked example
$\frac{4}{7}\div\frac{2}{3}=\frac{4}{7}\cdot\frac{3}{2}=\frac{6}{7}$.
The mistake to avoid
What people do: Students flip the first fraction instead of the second one.
Why it goes wrong: Only the number you are dividing by gets turned over. Flipping the wrong one changes the question completely.
Do this instead: Keep the first fraction exactly as it is and flip the one after the divide sign: 4/7 divided by 2/3 is 4/7 times 3/2, which is 6/7. Flipping the first instead would give 7/6, a number bigger than 1 rather than smaller.
Divide fractions: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $c/d$ is the divisor; $d/c$ is its reciprocal.
- Check the conditions before substituting. $b,c,d\ne0$ because neither a denominator nor the divisor may be zero.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most arithmetic slips.
Where this formula fits
- Subject
- Arithmetic formulas — 13 entries in this library
- Topic
- Fractions
- Level
- Middle school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where divide fractions comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- A Practical Guide to Fractions — the lesson behind this formula: compare, combine, multiply, and divide rational quantities.
- Arithmetic Calculator — check your substitution and the value it produces.
- Study arithmetic — the subject guide that explains the ideas these formulas compress.
- Arithmetic Practice — questions that make you retrieve the formula instead of recognising it.
- All 13 arithmetic formulas — the full grouped reference, or the complete formula library.
Questions about divide fractions
Which fraction do I flip?
The second one, the one right after the divide sign. A good memory line is keep, change, flip: keep the first, change divide to multiply, flip the last.
Why does dividing sometimes make the answer bigger?
Because small pieces fit into a whole many times over. Half a pizza holds two quarter slices, so 1/2 divided by 1/4 is 2, which is larger than either number in the question.
What is the reciprocal of a whole number?
Write it over 1 and turn it over, so the reciprocal of 5 is 1/5. That is why dividing by 5 and multiplying by 1/5 give the very same result.
Does a fraction bar mean the same thing as a divide sign?
Yes, they are two ways of writing one idea. A fraction stacked on top of another fraction is just this division written vertically, and you solve it with the same flip-and-multiply move.