Comparing fractions · applications

Applications: Comparing Fractions

10 question types · Primary 3 · each worked step by step with a figure that follows the steps

PSLE

01

Two Pizzas Cut into the Same Number of Slices

strategySame Denominator / Count the Parts

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?

Ali1/81/81/81/81/81/81/81/8Ben1/81/81/81/81/81/81/81/8
Both pizzas are cut into 8 equal slices, so every slice is 18 of a pizza.
Both pizzas are cut into 8 equal slices. Every slice is the same size: 18 of a pizza.
step 1 of 4

Draw one strip for each pizza and cut each strip into 8 equal parts. The denominators (the bottom numbers) are the same, so all the slices are the same size. The boy with more slices eats more pizza.

  1. Both pizzas are cut into 8 equal slices. Every slice is the same size: 18 of a pizza.
  2. Ali eats 3 slices and Ben eats 5 slices. Shade them on the strips.
  3. (a) 5 slices are more than 3 slices of the same size, so 58 is greater than 38. Ben eats more pizza.
  4. (b) Ben eats 5 − 3 = 2 more slices. Check: 3 + 2 = 5.

answer(a) Ben; (b) 2 slices

techniqueSame Bottom Number

examsPSLE

Common pitfalls

  • 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.
02

Three Pieces of Two Ribbons Cut Differently

strategySame Numerator / Bigger Parts Make the Bigger Fraction

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?

Mei5 piecesLin8 pieces
The ribbons are the same length. More pieces means shorter pieces.
The two ribbons are the same length. Cutting a ribbon into more pieces makes each piece shorter.
step 1 of 4

Draw two strips of the same length. Cut one into 5 equal pieces and the other into 8 equal pieces. The numerators (the top numbers) are the same, so both girls use 3 pieces. The girl with the longer pieces uses more ribbon.

  1. The two ribbons are the same length. Cutting a ribbon into more pieces makes each piece shorter.
  2. (a) 5 pieces are fewer than 8 pieces, so Mei's pieces are longer. 15 is greater than 18.
  3. Both girls use 3 pieces. Shade 3 pieces on each strip.
  4. (b) Three longer pieces are more than three shorter pieces, so 35 is greater than 38. Mei uses more ribbon.

answer(a) Mei's pieces; (b) Mei, because 35 is greater than 38

techniqueSame Top Number

examsPSLE

Common pitfalls

  • 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.
03

Three Jars Sorted as Less Than, Equal to or More Than a Half

strategyCompare with a Half / Half of the Denominator

Three jars are the same size. Jar A is 310 full, jar B is 48 full and jar C is 712 full. (a) Which jar is exactly half full? (b) Which jar is less than half full, and which jar is more than half full?

A: 3/10lessB: 4/8C: 7/121/2
Half of 10 is 5. Jar A has only 3 parts, so it is less than half full.
Jar A is cut into 10 parts. Half of 10 is 5, so 510 is a half. Jar A has only 3 parts, so it is less than half full.
step 1 of 5

For each jar, find half of the denominator (the bottom number). That many parts is exactly a half of the jar. Then compare the numerator (the top number) with it.

  1. Jar A is cut into 10 parts. Half of 10 is 5, so 510 is a half. Jar A has only 3 parts, so it is less than half full.
  2. Jar B is cut into 8 parts. Half of 8 is 4, so 48 is exactly a half.
  3. Jar C is cut into 12 parts. Half of 12 is 6, so 612 is a half. Jar C has 7 parts, so it is more than half full.
  4. (a) Jar B is exactly half full.
  5. (b) Jar A is less than half full and jar C is more than half full.

answer(a) Jar B; (b) jar A is less than half full and jar C is more than half full

techniqueMore Than a Half

examsPSLE

Common pitfalls

  • Comparing the top number with the wrong half. For jar C the half is 6 parts out of 12, not 5 parts or 4 parts. Each jar has its own half, found from its own bottom number.
  • Saying 48 is not a half because it does not look like 12. 4 is half of 8, so 48 and 12 are the same amount.
04

Three Runners on One Track Put in Order

strategyUse a Half as the Benchmark / Then Compare the Same Denominator

Three friends run on the same track. Ana has run 78 of the track, Ben has run 512 of the track and Cal has run 58 of the track. (a) Who has run less than half of the track? (b) Write the three fractions in order from least to greatest.

Ana: 7/8Ben: 5/12lessCal: 5/81/2
Half of 12 is 6, so 512 is less than a half.
Compare each fraction with a half. Half of 12 is 6, so 512 is less than a half.
step 1 of 5

First compare each fraction with a half. A fraction that is less than a half is smaller than a fraction that is more than a half. Two fractions on the same side of a half are compared in another way.

  1. Compare each fraction with a half. Half of 12 is 6, so 512 is less than a half.
  2. Half of 8 is 4, so 58 and 78 are both more than a half.
  3. (a) Ben has run less than half of the track, so 512 is the least.
  4. 58 and 78 have the same denominator. 5 parts are fewer than 7 parts of the same size, so 58 is less than 78.
  5. (b) From least to greatest: 512, 58, 78.

answer(a) Ben; (b) 512, 58, 78

techniqueOrdering Fractions · More Than a Half · Same Bottom Number

examsPSLE · GCSE Higher

Common pitfalls

  • Putting 512 last because 12 is the biggest bottom number. A bigger bottom number means smaller parts, and 5 parts out of 12 is not even a half.
  • Writing the order from greatest to least. The question asks for least to greatest, so the smallest fraction comes first.
05

Two Jugs Each One Part Short of Full

strategyCompare the Missing Parts / The Smaller Gap Is Closer to One Whole

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?

Jug A5/6Jug B7/8
Each jug has one empty part: 6 − 5 = 1 and 8 − 7 = 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.
step 1 of 4

Each jug is one part short of full. Find the empty part of each jug. The jug with the smaller empty part is closer to full, so it 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.
  2. (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.
  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.

answer(a) 16 of jug A and 18 of jug B; (b) Jug B

techniqueSame Top Number · Ordering Fractions

examsPSLE

Common pitfalls

  • 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.
06

One Piece of Two against Two Pieces of Four

strategyEquivalent Fractions / Cut Each Part in Two

Kai and Lea each have a chocolate bar of the same size. Kai breaks his bar into 2 equal pieces and eats 1 piece. Lea breaks her bar into 4 equal pieces and eats 2 pieces. (a) What fraction of a bar does each child eat? (b) Who eats more chocolate?

Kai1/21/2Lea1/41/42/4
(a) Kai eats 12 of his bar and Lea eats 24 of her bar.
(a) Kai eats 1 of 2 equal pieces, which is 12 of his bar. Lea eats 2 of 4 equal pieces, which is 24 of her bar.
step 1 of 4

Draw the two bars one above the other, the same length. Cut each of Kai's pieces in two so that both bars have 4 pieces. Then the pieces are the same size and can be counted.

  1. (a) Kai eats 1 of 2 equal pieces, which is 12 of his bar. Lea eats 2 of 4 equal pieces, which is 24 of her bar.
  2. Cut each of Kai's halves into 2 smaller pieces. Now his bar has 4 pieces too, and his 1 half is 2 quarters.
  3. So 12 = 24. The two shaded parts are the same length.
  4. (b) Neither child eats more. 12 and 24 of the same bar are the same amount.

answer(a) Kai eats 12 and Lea eats 24; (b) neither, because 12 and 24 are the same amount

techniqueMore Than a Half · Halves, Quarters, Thirds

examsPSLE

Common pitfalls

  • Saying Lea eats more because she eats 2 pieces and Kai eats only 1 piece. Lea's pieces are half the size of Kai's pieces, so 2 of hers make 1 of his.
  • Saying 24 is greater because its numbers are bigger. Doubling both the top number and the bottom number gives a new name for the same amount.
07

Half of a Small Bottle and Half of a Large Bottle

strategyThe Same Fraction of Different Wholes / Find Each Amount First

A small bottle holds 300 ml when it is full. A large bottle holds 800 ml when it is full. Each bottle is half full of water. (a) How much water is in each bottle? (b) How much more water is in the large bottle than in the small bottle?

Small150 ml300 mlLarge?800 ml
Half of the small bottle is 300 ÷ 2 = 150 ml.
Half of the small bottle is 300 ÷ 2 = 150 ml.
step 1 of 4

Draw one bar for each bottle. The large bottle gets the longer bar because it holds more. Half of each bar is water, and the two halves are not the same length.

  1. Half of the small bottle is 300 ÷ 2 = 150 ml.
  2. Half of the large bottle is 800 ÷ 2 = 400 ml.
  3. (a) The small bottle has 150 ml of water and the large bottle has 400 ml of water.
  4. (b) The large bottle has 400 − 150 = 250 ml more water. Check: 150 + 250 = 400.

answer(a) 150 ml and 400 ml; (b) 250 ml

techniqueMore Than a Half · A Fraction of a Group

examsPSLE

Common pitfalls

  • Saying the bottles have the same amount of water because both are 12 full. A half of a bigger bottle is more than a half of a smaller bottle. Two fractions can be compared directly only when the wholes are the same size.
  • Subtracting the full sizes, 800 − 300 = 500 ml. The bottles are only half full, so the two halves must be found first.
08

A Mark between Two Places on a Number Line in Eighths

strategyNumber Line in Eighths / Change a Quarter to Eighths First

A stick is marked in eighths from 0 to 1. A leaf is at 14 and a stone is at 58. A snail stops at an eighth mark that is after the leaf and before the stone. (a) How many eighths is 14? (b) At which marks can the snail be?

01/82/83/84/85/86/87/81leaf 1/4stone 5/8
The stick has 8 equal parts. Each quarter is 2 eighths.
The stick is cut into 8 equal parts. 8 is double 4, so each quarter is 2 eighths.
step 1 of 4

Draw the number line from 0 to 1 with a mark at every eighth. Change 14 to eighths so that both places are counted in the same parts. Then read the marks between them.

  1. The stick is cut into 8 equal parts. 8 is double 4, so each quarter is 2 eighths.
  2. (a) 14 = 28, so the leaf is at the second mark.
  3. The stone is at 58, the fifth mark. The marks after the second and before the fifth are the third and the fourth.
  4. (b) The snail can be at 38 or at 48. 48 is the same as 12.

answer(a) 28; (b) 38 or 48

techniqueOrdering Fractions · Same Bottom Number

examsPSLE · GCSE Higher

Common pitfalls

  • Comparing 14 and 58 without changing 14 to eighths first. On a line marked in eighths, every place is counted in eighths.
  • Including 28 or 58 in the answer. Those marks are the leaf and the stone, and the snail is between them.
09

Who Has More Juice Left after Drinking

strategyDrinking Less Leaves More / The Comparison Turns Around

Amy and Raj each have a bottle of juice of the same size. Amy drinks 23 of her juice. Raj drinks 25 of his juice. (a) Who drinks less juice? (b) Who has more juice left, and what fraction of the bottle is it?

Amydrankdrank2/3Rajdrankdrank2/5
Fifths are smaller parts than thirds, so 25 is less than 23.
The top numbers are the same. Fifths are smaller parts than thirds, so 25 is less than 23.
step 1 of 5

Draw one strip for each bottle, the same length. Shade the part each child drinks. The bottles are the same size, so the child who drinks less has more left.

  1. The top numbers are the same. Fifths are smaller parts than thirds, so 25 is less than 23.
  2. (a) Raj drinks less juice.
  3. The bottles are the same size, so the child who drinks less has more left. That child is Raj.
  4. Raj's bottle has 5 equal parts and he drinks 2 of them, so 5 − 2 = 3 parts are left.
  5. (b) Raj has more juice left: 35 of his bottle. Check: Amy has 3 − 2 = 1 part left, which is 13. That is less than a half, and 35 is more than a half.

answer(a) Raj; (b) Raj, with 35 of his bottle left

techniqueSame Top Number · More Than a Half

examsPSLE

Common pitfalls

  • Answering Amy for part (b) because 23 is the greater fraction. 23 is what Amy drinks. The more a child drinks, the less is left.
  • Thinking 25 is greater than 23 because 5 is greater than 3. A bigger bottom number means smaller parts.
10

Every Numerator That Fits between a Half and One Whole

strategyList the Whole Numbers That Fit / Find the Two Ends First

Nora folds a paper strip into 8 equal parts. She colors more than 12 of the strip, but she does not color the whole strip. (a) How many parts make exactly 12 of the strip? (b) How many parts could Nora have colored? Give every possible number.

Strip4/8 = 1/21/2
(a) Half of 8 is 4, so 4 parts make exactly 12 of the strip.
(a) Half of 8 is 4, so 4 parts make exactly 12 of the strip: 48 = 12.
step 1 of 4

Draw the strip with its 8 equal parts. Find the number of parts that is exactly a half, and the number of parts that is the whole strip. Nora's number is between these two.

  1. (a) Half of 8 is 4, so 4 parts make exactly 12 of the strip: 48 = 12.
  2. More than a half means more than 4 parts, so Nora colors at least 5 parts.
  3. The whole strip is 8 parts, and 88 = 1. Less than the whole strip means fewer than 8 parts, so Nora colors at most 7 parts.
  4. (b) Nora could have colored 5, 6 or 7 parts. The fractions are 58, 68 and 78.

answer(a) 4 parts; (b) 5, 6 or 7 parts

techniqueMore Than a Half · Same Bottom Number

examsPSLE

Common pitfalls

  • Including 4 parts. 48 is exactly a half, and the question says more than a half.
  • Including 8 parts. 88 is the whole strip, and Nora does not color the whole strip.
Mr. Chalk Drill fraction arithmetic