Multiply fractions: what it means and how to use it
Multiply numerators together and denominators together.
Multiply fractions is one of 3 fractions formulas in the arithmetic section of this library, and it is used at middle school level.
Why multiply fractions works
Multiplying by 3/5 means taking 3 out of every 5 equal parts. If you slice a bar into 5 strips across and 9 strips down, you get 45 small squares, and the piece you keep is 3 strips by 10 strips, which is 30 squares. Counting rows and columns is why the tops multiply and the bottoms multiply.
What each symbol means
$a,c$ are numerators and $b,d$ are denominators.
Multiply fractions: when it holds
$b\ne0$ and $d\ne0$; cancel common factors before or after multiplying.
When it stops applying
It does not apply to mixed numbers until you convert them. Multiplying 1 1/2 by 2 is not 1 times 2 with the half tacked on; change it to 3/2 first and you get 3, which is the right answer.
Multiply fractions: a worked example
$\frac{3}{5}\cdot\frac{10}{9}=\frac{30}{45}=\frac{2}{3}$.
The mistake to avoid
What people do: Students hunt for a common denominator before multiplying, the way they were taught to do for addition.
Why it goes wrong: Nothing needs matching here. Multiplication never asks for pieces of the same size, so the extra step wastes time and gives more chances to slip up.
Do this instead: Multiply straight across: 3/5 times 10/9 gives 30/45, which simplifies to 2/3. Common denominators are only needed when you add or subtract.
Multiply fractions: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,c$ are numerators and $b,d$ are denominators.
- Check the conditions before substituting. $b\ne0$ and $d\ne0$; cancel common factors before or after multiplying.
- 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 multiply 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 lesson — 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 multiply fractions
Why does multiplying sometimes make the number smaller?
Because taking a fraction of something means taking part of it. Half of 8 is 4, so multiplying by any number below 1 shrinks what you started with, which feels backwards until you read the times sign as of.
Can I cancel before I multiply?
Yes, and it saves work. In 3/5 times 10/9 the 3 and the 9 share a factor of 3, and the 5 and the 10 share a factor of 5, so you can reduce to 1/1 times 2/3 and read off 2/3 with no simplifying afterwards.
How do I multiply a fraction by a whole number?
Put the whole number over 1. Then 4 times 2/7 is 4/1 times 2/7, which is 8/7. In practice that means the top gets multiplied and the bottom is left alone.
What happens when I square a fraction?
Both the top and the bottom get squared, so (2/3)² is 4/9. Since 4/9 is about 0.44 and 2/3 is about 0.67, squaring a proper fraction always pulls it closer to zero.