Subtracting fractions

A fraction names pieces of a whole, and the denominator tells how big those pieces are. Three fourths are pieces of size 1/4, while one third is a piece of size 1/3. You cannot subtract 1/4-sized pieces from 1/3-sized pieces directly, any more than you can subtract cars from bikes and get a meaningful count. The sizes have to match first.

Leo's mistake was to subtract the tops (3 - 1 = 2) and add the bottoms (4 + 3 = 7), writing 2/7. He treated the fractions as if their pieces already matched. The operation never got a fair start because the pieces were different sizes, and mixing the two directions doubles the error: the top subtracts one way while the bottom adds the other way, so the answer means nothing at all.

The fix is renaming: cut every piece so the wholes use the same size. Fourths and thirds both divide into twelfths, so 3/4 becomes 9/12 (top and bottom times 3) and 1/3 becomes 4/12 (top and bottom times 4). Now the pieces are identical in size, and the two fractions describe the same two bars with the same grid lines.

With matching pieces, subtraction is simple: 9 twelfths minus 4 twelfths leaves 5 twelfths, or 5/12. The denominator stays 12 because the piece size did not change. Check the answer by adding back: 5/12 + 4/12 = 9/12 = 3/4, which is where you started, so the subtraction undoes perfectly. Addition is the reverse gear for subtraction, and it checks every time.