Bigger bottom, smaller piece
Why Is a Third Bigger Than a Quarter?
The bottom number counts how many pieces the whole was cut into, so a bigger one means smaller pieces.
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Step 1 of 5
The bottom number counts the cuts, not the amount.
In a third, the 3 does not mean you have three of anything. It says the whole was divided into three equal pieces, and you took one.
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Step 2 of 5
More pieces from the same cake means smaller pieces.
The cake never changes size. Cutting it into more parts can only make each part smaller, because they still have to add back up to one cake.
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Step 3 of 5
So a bigger bottom number is a smaller share.
This is the bit that feels backwards. Four is larger than three, yet a quarter is smaller than a third, because you are counting cuts.
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Step 4 of 5
With matching bottoms, just compare the tops.
Try it on three fifths against two thirds: the bigger top loses. Once the pieces are all the same size the hard part is over. Three fifths beats two fifths for the plain reason that three pieces beat two.
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Step 5 of 5
With different bottoms, recut into the same size.
To compare a third with a quarter properly, cut both into twelfths. A third becomes four twelfths and a quarter becomes three.
Twelve is the smallest number that both 3 and 4 divide into, which is why twelfths are the natural common size here.
The short version
The bottom number counts cuts, not amounts, so a bigger bottom is a smaller share. Once the bottoms match, the bigger top simply wins; when they do not, recut both — a third and a quarter become four twelfths and three.
Try it yourself
Cut one sandwich into 3 and an identical one into 4. Put a piece of each side by side; the third is visibly the bigger piece.