Same Bottom Number

Pieces of the same size

Two chocolate bars of the same size are each broken into 5 equal pieces. Mia eats 3 pieces of her bar and Leo eats 2 pieces of his. Every piece is one fifth of a bar, so all the pieces are the same size, and Mia, who ate more pieces, ate more chocolate. 3/5 is bigger than 2/5.

When two fractions have the same bottom number, their pieces are the same size, so compare the top numbers. The bigger top number is the bigger fraction: 5/8 is bigger than 3/8, and 7/10 is bigger than 4/10.

The bottom number stays as it is. Mia ate one piece more than Leo, and that piece is a fifth: 3/5 − 2/5 = 1/5. Only the number of pieces changed, so the bottom number is still 5.

Both bars are cut into 5 equal pieces, so every piece is the same size. 3 shaded pieces cover more of the bar than 2.

Worked example: Two Pizzas Cut into the Same Number of Slices

Question Ali and Ben each have a pizza of the same size. Each pizza is cut into 8 equal slices. Ali eats 38 of his pizza. Ben eats 58 of his pizza. (a) Who eats more pizza? (b) How many more slices does he eat?

1.Both pizzas are cut into 8 equal slices. Every slice is the same size: 18 of a pizza.

Both pizzas are cut into 8 equal slices, so every slice is 18 of a pizza.
2.Ali eats 3 slices and Ben eats 5 slices. Shade them on the strips.

Ali eats 3 slices and Ben eats 5 slices.
3.(a) 5 slices are more than 3 slices of the same size, so 58 is greater than 38. Ben eats more pizza.

(a) The slices are the same size, so 58 is greater than 38. Ben eats more.
4.(b) Ben eats 5 − 3 = 2 more slices. Check: 3 + 2 = 5.

(b) Ben eats 5 − 3 = 2 more slices.

Answer: (a) Ben; (b) 2 slices

Common mistakes

Saying the two boys eat the same amount because both fractions have an 8. The 8 only tells the size of one slice. The top numbers 3 and 5 tell how many slices each boy eats.
Subtracting the bottom numbers as well and writing 8 − 8 = 0. The slices stay eighths. Only the number of slices changes, so the difference is 2 slices, which is 28 of a pizza.

More comparing fractions problems, worked step by step →

When the pieces do not match

Counting works only once the pieces are the same size. 3/4 and 5/7 have different bottom numbers, and a quarter is bigger than a seventh, so 5 pieces against 3 pieces does not say which fraction is bigger.

Cut the pieces again until they match. Cut each quarter into 7 and each seventh into 4, and both bars are in 28ths, because 4 × 7 = 28. Then 3/4 = 21/28 and 5/7 = 20/28. The pieces are the same size, so count them: 21 is more than 20, and 3/4 is bigger than 5/7 by one piece, 1/28.

$\frac{3}{4} vs \frac{5}{7}$

3/4 and 5/7 are cut into pieces of different sizes, so 5 > 3 says nothing: the numerators cannot be compared until the pieces match

Cut both bars into 28ths and find the extra piece

Before the cut, the bars hold 3 quarters and 5 sevenths, pieces of two different sizes. Cut both into 28ths and the shaded parts are counted in pieces of one size: 21 against 20.