MISSION · Number sense
Equivalent Fractions
Why one half and four eighths are the same piece of pizza
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Cutting the same amount into more pieces makes more pieces, but it doesn’t make more amount.
The same bar, cut differently
Shade half a chocolate bar. Now score a line down the middle of every piece. Nothing was eaten and nothing was added, but the shading you already had is now 2 pieces out of 4. Score them again: 4 out of 8.
\[\frac{1}{2} = \frac{2}{4} = \frac{4}{8} = \frac{50}{100}\]Every one of those is the same amount of chocolate. They differ only in how finely the bar was cut, which is a fact about the knife and not about the chocolate.
The rule, and why it works
To make an equivalent fraction, multiply the top and the bottom by the same number.
\[\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\]Multiplying the bottom by 4 cuts each piece into 4 smaller ones, so there are four times as many pieces in the whole. Multiplying the top by 4 says you’re holding four times as many of those smaller pieces. Four times as many pieces, each a quarter the size, is the same amount — the two changes cancel exactly.
This is also why multiplying only the top is not allowed. That would hand you more pieces without making them smaller, which really is more chocolate.
Simplifying is the same move backwards
Going from \(\frac{8}{12}\) to \(\frac{2}{3}\) means dividing top and bottom by 4 — gluing pieces back together into bigger ones. A fraction is in lowest terms when no number above 1 divides both parts, which is the same as saying the pieces can’t be glued any further.
To get there in one step, divide by the greatest common factor. For \(\frac{18}{24}\) that’s 6, giving \(\frac{3}{4}\) immediately. If you can’t spot it, halve repeatedly and you’ll arrive at the same place with more steps.
What it’s for
Equivalent fractions are how you compare. Is \(\frac{3}{5}\) bigger than \(\frac{5}{8}\)? The pieces are different sizes, so the numerators tell you nothing. Recut both into fortieths — \(\frac{24}{40}\) and \(\frac{25}{40}\) — and now the pieces match, so counting them settles it. \(\frac{5}{8}\) wins, barely.
Every fraction skill that comes later, adding included, runs on this one move.
Common mix-ups
- Adding instead of multiplying. \(\frac{1}{2}\) and \(\frac{2}{3}\) are not equivalent, even though you added 1 to both parts. Only multiplying preserves the amount.
- Changing only one part. Both the top and the bottom, by the same number, always.
- Assuming bigger numbers mean a bigger fraction. \(\frac{50}{100}\) is exactly the same size as \(\frac{1}{2}\).
- Stopping before lowest terms and thinking the answer is wrong. \(\frac{4}{8}\) is a correct answer; it’s just not the tidiest one.
Check yourself
Write three fractions equal to \(\frac{3}{4}\). Then simplify \(\frac{20}{30}\) to lowest terms.
\(\frac{6}{8}\), \(\frac{9}{12}\), and \(\frac{30}{40}\) all work — any multiplier does. \(\frac{20}{30}\) divides top and bottom by 10 to give \(\frac{2}{3}\).
LEVEL 3 · REFLECT
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