Finding a Common Denominator

Let's try to add 1/2 and 1/3.

The tempting move is to add straight across: 1 + 1 on top, 2 + 3 on the bottom, giving 2/5. It's tidy. It's also wrong — and there's a way to know it's wrong before doing any real maths.

You started with 1/2 and added something to it. So the answer has to be bigger than 1/2. But 2/5 is smaller than 1/2. An answer that's smaller than what you started with, after adding? Impossible. So 2/5 can't be right.

Here's what actually goes wrong. Back in the previous article, adding worked because the pieces were the same size — fifths and fifths, so you could just count them. But halves and thirds are different sizes. A half-piece and a third-piece don't line up. Counting "1 half plus 1 third" is like being asked how much money you have when someone hands you 1 coin and 1 banknote — you can't just say "2" until you express them in the same units.

So the strategy is: make the pieces the same size first. And you already know the tool for that — equivalent fractions.

We need a bottom number that both 2 and 3 fit into evenly. The multiples of 2 are 2, 4, 6, 8... The multiples of 3 are 3, 6, 9... The first number that shows up on both lists is 6. That's our common denominator. (Because 6 is the smallest one that works, it's called the least common denominator.)

Now rewrite each fraction over 6, using the multiply-top-and-bottom trick:

$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \qquad \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

Both are sixths now — same-sized pieces. So we're back to the easy case from before: add the tops, keep the bottom.

$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

And 5/6 passes the sanity check: it's bigger than 1/2, exactly as it should be. Drag the strips against a sixths-strip in the experiment and you'll see the shaded half cover 3 parts and the shaded third cover 2 — five parts of six, right there in paper.

The full recipe, then, for adding any two fractions:

1. If the bottoms already match, just add the tops (you learned this last time).
2. If they don't, find a common denominator — a number both bottoms divide into.
3. Rewrite each fraction over that denominator using equivalent fractions.
4. Now the bottoms match, so add the tops and keep the bottom.
5. Simplify if you can.

Subtraction follows the identical path — same-size the pieces first, then subtract. Every hard-looking fraction sum is really just this: get the pieces to the same size, then count. That single idea — you can only combine equal-sized pieces — is the thread running through this entire series.