Why Is a Third Bigger Than a Quarter?

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.

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.

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.

Step 4 of 5

With matching bottoms, just compare the tops.

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.

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.