MISSION · Number sense
Multiplying and Dividing Fractions
Why multiplying can make things smaller, and why dividing flips
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Multiplying fractions is easier than adding them — straight across, no common denominator — and dividing is multiplying by the upside-down version.
Straight across
\[\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\]Tops multiply, bottoms multiply. No common denominator, no rearranging. It genuinely is simpler than addition, which surprises people who expect multiplication to be the harder operation.
The picture explains why. Take a square, shade \(\frac{2}{3}\) of it going across, then shade \(\frac{4}{5}\) of that going down. The overlap is 8 small pieces out of the 15 the square has been cut into. Multiplying the bottoms counts the pieces; multiplying the tops counts the shaded ones.
“Of” means times
\(\frac{1}{2}\) of 6 is 3, and that “of” is a multiplication: \(\frac{1}{2} \times 6 = 3\).
Recognising that connects a lot of separate-seeming things. Finding half of something, taking \(\frac{3}{4}\) of a recipe, working out \(25\%\) of a bill — all the same operation.
It also explains the result that bothers everyone: multiplying can make a number smaller. \(\frac{1}{2} \times 6\) is less than 6, because taking half of something obviously is. The rule “multiplying makes things bigger” was only ever true for whole numbers above 1, and fractions are where it stops applying.
Dividing: flip and multiply
\[\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}\]Turn the second fraction upside down — its reciprocal — and multiply. The rule works but looks like a trick, so here’s why it isn’t.
\(\frac{3}{4} \div \frac{1}{2}\) asks: how many halves fit into three quarters? A half is two quarters, and three quarters holds one whole half plus another half of a half. So \(1\frac{1}{2}\), which is what the arithmetic gave.
More formally: dividing by \(\frac{1}{2}\) is asking how many halves fit in, and twice as many halves fit as wholes — so dividing by a half is the same as multiplying by 2. Flipping the fraction is exactly that swap.
And this explains the other unsettling result: dividing can make a number bigger, when you divide by something less than 1. Six divided by a half is 12, because twelve halves fit into six.
Cancel before you multiply
\[\frac{3}{8} \times \frac{4}{9}\]Multiplying first gives \(\frac{12}{72}\), which then needs simplifying. Instead, cancel across the diagonals: 3 and 9 share a 3; 4 and 8 share a 4. That leaves \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\), with smaller numbers throughout.
Same answer, less arithmetic, fewer chances to slip.
Common mix-ups
- Finding a common denominator. That’s for adding. Multiplication doesn’t need one.
- Flipping the wrong fraction when dividing. The second one — the divisor — gets inverted.
- Assuming multiplying always makes things bigger. Multiplying by a proper fraction shrinks.
- Forgetting a whole number is a fraction. \(6 = \frac{6}{1}\), which is what makes the flip work.
Check yourself
Work out \(\frac{2}{5} \times \frac{3}{4}\) and \(\frac{2}{3} \div \frac{4}{9}\).
Straight across gives \(\frac{6}{20} = \frac{3}{10}\). For the division, flip: \(\frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}\).
LEVEL 3 · REFLECT
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