You can only add pieces that are the same size. But multiplying is taking a part of a part, and that works whatever the sizes are.
Adding and subtracting
Rewrite both with a common bottom, then add the tops and leave the bottom alone. Three fifths plus one fifth is four fifths, just as three apples plus one apple is four apples.
One half plus one half is not two quarters. That would say half a chapati plus half a chapati leaves you with half a chapati.
Multiplying
Multiply the tops and the bottoms; no common bottom is needed. Multiplying by a fraction smaller than 1 makes the answer smaller.
Six divided by a half asks how many halves fit into six, and the answer is twelve. Small pieces means many of them fit.
| Operation | What to do | Example |
|---|---|---|
| Add or subtract | Common bottom first, then the tops only | 2/3 + 1/4 = 8/12 + 3/12 = 11/12 |
| Multiply | Tops together, bottoms together | 2/3 × 3/4 = 6/12 = 1/2 |
| Divide | Turn the second one upside down, then multiply | 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9 |
Mixed numbers
Change a mixed number into an improper fraction first: two and a half becomes five halves. Do all the work in improper form and change back at the end.
Two and a half means two PLUS a half, which is five halves. Two TIMES a half is one. Everywhere else in mathematics, symbols side by side mean multiply.
For 8/15 × 5/12, cancel first: it becomes 2/3 × 1/3 = 2/9, all with small numbers.
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