Adding Fractions When the Bottoms Match

Say you eat 1/5 of a chocolate bar in the morning and 2/5 in the afternoon. How much did you eat in total?

Picture the bar cut into 5 equal strips — fifths. In the morning you took 1 strip. In the afternoon you took 2 more. All the strips are the same size, so you just count them: 1 strip plus 2 strips is 3 strips. You ate 3/5 of the bar.

That's the whole method when the bottom numbers match: add the top numbers, and keep the bottom number exactly the same.

$\frac{1}{5} + \frac{2}{5} = \frac{3}{5}$

Simple — but here's the part worth slowing down for, because it's where people slip. Why doesn't the bottom change? Why isn't 1/5 + 2/5 equal to 3/10?

Because the bottom number isn't a quantity you're counting. It's the size of each piece. Fifths are fifths. When you gather more fifths together, you get more of them — but each one is still exactly one-fifth of the bar. The pieces don't shrink just because you have more of them. So the bottom stays 5, and only the count on top goes up.

Watch out for the classic mistake: 2/7 + 3/7 is 5/7, not 5/14. Adding the bottoms to get 14 would be claiming the pieces suddenly became half their size the moment you combined them — which is impossible. You had 2 sevenths and 3 sevenths; now you have 5 sevenths. Same size pieces, more of them.

Subtraction is the mirror image. If you have 5/8 of a tank of fuel and use 2/8, you're left with 3/8:

$\frac{5}{8} - \frac{2}{8} = \frac{3}{8}$

Take away 2 same-sized pieces from 5 same-sized pieces, and 3 remain. The bottom — the piece size — never moves.

So the rule is short: same bottom? Combine the tops, leave the bottom alone. And the reason behind it is even shorter: the bottom is the size of the pieces, and combining pieces doesn't change their size.

Which raises the obvious next question. What if the bottoms don't match — what if you want to add 1/2 and 1/3, halves and thirds, pieces of different sizes? You can't just count them; they don't line up. That's the puzzle the last article in this series solves.