The same number of pieces
Two cakes are the same size. One is cut into 3 equal slices and the other into 5 equal slices. Take 2 slices from each cake. Both plates hold 2 slices, but they do not hold the same amount of cake, because the slices are not the same size.
Fewer cuts, bigger pieces
Cutting a whole into fewer pieces makes each piece bigger. A third of a cake is bigger than a fifth, so 2 thirds are more cake than 2 fifths: is bigger than .
When two fractions have the same top number, the one with the smaller bottom number is bigger. This is the other way round from fractions with the same bottom number. 5 is bigger than 3, but is smaller than , because the bottom number counts the pieces the whole was cut into, and more pieces are smaller pieces.
Both bars have 2 pieces shaded. Thirds are bigger pieces than fifths, so the 2 shaded thirds cover more of the bar.
2/5 = 0.4
Make one half
Keep 2 pieces shaded and change the number of pieces. The fewer pieces the bar is cut into, the more the 2 shaded pieces cover: is less than a half, is exactly a half and is more.
Worked example: Three Pieces of Two Ribbons Cut Differently
Question Mei and Lin each have a ribbon of the same length. Mei cuts her ribbon into 5 equal pieces and uses 3 of them, so she uses 35 of her ribbon. Lin cuts her ribbon into 8 equal pieces and uses 3 of them, so she uses 38 of her ribbon. (a) Whose pieces are longer? (b) Who uses more ribbon?
1.The two ribbons are the same length. Cutting a ribbon into more pieces makes each piece shorter.
The ribbons are the same length. More pieces means shorter pieces. 2.(a) 5 pieces are fewer than 8 pieces, so Mei's pieces are longer. 15 is greater than 18.
(a) Mei's pieces are longer: 15 is greater than 18. 3.Both girls use 3 pieces. Shade 3 pieces on each strip.
Both girls use 3 pieces. 4.(b) Three longer pieces are more than three shorter pieces, so 35 is greater than 38. Mei uses more ribbon.
(b) Three longer pieces are more than three shorter pieces: 35 is greater than 38. Mei uses more.
Answer: (a) Mei's pieces; (b) Mei, because 35 is greater than 38
Common mistakes
- Thinking that 38 is greater because 8 is greater than 5. A bigger bottom number means the ribbon is cut into more pieces, so each piece is smaller.
- Saying the two girls use the same amount because both use 3 pieces. The pieces are not the same size: Mei's pieces are longer than Lin's pieces.
Comparing what is missing
The same rule compares two fractions that are each one piece short of a whole. is missing and is missing . and have the same top number, and an eighth is smaller than a sixth, so is missing less. It is closer to a whole, and is bigger than .
Worked example: Two Jugs Each One Part Short of Full
Question Jug A and jug B are the same size. Jug A is 56 full of water. Jug B is 78 full of water. (a) What fraction of each jug is empty? (b) Which jug has more water?
1.Jug A has 6 equal parts and 5 are full, so 6 − 5 = 1 part is empty. Jug B has 8 equal parts and 7 are full, so 8 − 7 = 1 part is empty.
Each jug has one empty part: 6 − 5 = 1 and 8 − 7 = 1. 2.(a) 16 of jug A is empty and 18 of jug B is empty.
(a) 16 of jug A is empty and 18 of jug B is empty. 3.Compare the empty parts. The top numbers are the same, and eighths are smaller than sixths, so 18 is less than 16.
Eighths are smaller than sixths, so 18 is less than 16. 4.(b) Jug B has the smaller empty part, so it is closer to full. Jug B has more water: 78 is greater than 56.
(b) Jug B has the smaller empty part, so it has more water: 78 is greater than 56.
Answer: (a) 16 of jug A and 18 of jug B; (b) Jug B
Common mistakes
- Saying the jugs hold the same amount because each jug is 1 part short of full. The missing parts are not the same size: 18 is smaller than 16.
- Choosing jug A because its empty part, 16, is the bigger fraction. A bigger empty part means less water, not more.