Multiplying and dividing fractions · applications

Applications: Multiplying and Dividing Fractions

12 problems

PSLE · GCSE Higher

01

A Fraction of One Part of a Group

heuristicFraction of a Fraction / Cut the Part, Then Count Pieces of the Whole

In a school choir, 35 of the members are girls. 14 of the girls wear glasses. (a) What fraction of the choir members are girls who wear glasses? (b) The choir has 120 members. How many girls in the choir do not wear glasses?

Choir1/51/51/51/51/5girls, 3/5 of the choir
The girls are 3 of the 5 equal fifths of the choir.
Draw the choir as one bar of 5 equal fifths and mark 3 of them as the girls.
step 1 of 5

Draw the choir as one bar and mark the girls' part. The second fraction is a fraction of that part only, so cut the part into quarters and then count the pieces against the whole bar.

All steps
  1. Draw the choir as one bar of 5 equal fifths and mark 3 of them as the girls.
  2. The girls who wear glasses are 14 of the girls' part, not of the whole bar. Cut every fifth into 4 equal pieces. The whole bar now has 5 × 4 = 20 pieces and the girls have 3 × 4 = 12 of them.
  3. 14 of the girls' 12 pieces is 3 pieces. (a) 14 × 35 = 320 of the choir members are girls who wear glasses.
  4. The girls who do not wear glasses have 12 − 3 = 9 pieces, which is 920 of the choir.
  5. (b) 920 × 120 = 54 girls do not wear glasses. Check: there are 35 × 120 = 72 girls, 14 × 72 = 18 of them wear glasses, and 72 − 18 = 54.

answer(a) 320; (b) 54 girls

techniqueMultiplying Fractions · A Fraction of an Amount

examsPSLE · GCSE Higher

Common pitfalls

  • Subtracting the fractions (35 − 14 = 720) to find the girls who do not wear glasses. The 14 is a fraction of the girls and the 35 is a fraction of the choir, so they cannot be subtracted until both are fractions of the choir.
  • Finding 14 of all 120 members, which is 30. Only the girls are counted in the 14, so it must be taken from the 72 girls.
02

Area of a Rectangle with Fractional Sides

heuristicArea Model / Columns Times Rows Inside a Unit Square

A square board has sides of 1 m. A rectangular poster 56 m long and 34 m wide is pasted on the board. (a) What is the area of the poster? (b) 25 of the poster is coloured blue. What is the area of the blue part?

5/6 m3/4 m1 m6 × 4 = 24 small rectangles
The board is 1 m2, cut into 6 columns and 4 rows.
Draw the board as a square with an area of 1 m2. Cut its length into 6 equal columns and its width into 4 equal rows. This makes 6 × 4 = 24 equal small rectangles.
step 1 of 5

Draw the board as a square of area 1 m2 and cut it into columns and rows that match the two denominators. The poster is then a whole number of small rectangles.

All steps
  1. Draw the board as a square with an area of 1 m2. Cut its length into 6 equal columns and its width into 4 equal rows. This makes 6 × 4 = 24 equal small rectangles.
  2. The poster is 5 columns long and 3 rows wide, so it covers 5 × 3 = 15 of the small rectangles.
  3. (a) The area of the poster is 56 × 34 = 1524 = 58 m2.
  4. The poster has 5 columns, so 25 of the poster is 2 of its columns. They hold 2 × 3 = 6 small rectangles.
  5. (b) The area of the blue part is 624 = 14 m2. Check: 25 × 58 = 1040 = 14.

answer(a) 58 m2; (b) 14 m2

techniqueMultiplying Fractions · Area and Perimeter · Simplest Form

examsPSLE · GCSE Higher

Common pitfalls

  • Changing both fractions to twelfths and multiplying only the numerators: 1012 × 912 = 9012. When fractions are multiplied, the denominators are multiplied as well, and no common denominator is needed.
  • Taking 25 of the whole board and giving 25 m2. The blue part is 25 of the poster, which covers only 58 of the board.
03

Fractional Pieces Cut from a Whole Length

heuristicDivide by a Fraction / The Fraction in the Quotient Counts Pieces, Not Metres

A carpenter cuts a plank 5 m long into pieces that are each 34 m long. (a) What is the greatest number of complete pieces he can cut? (b) What length of plank is left over, and what fraction of one piece is it?

Plank1 m1 m1 m1 m1 mOne piece3/4 mThe plank is 20 quarters of a metre. One piece is 3 quarters.
Measure both in quarters of a metre: 5 × 4 = 20 quarters, and 3 quarters in a piece.
Cut each metre of the plank into quarters. The plank is 5 × 4 = 20 quarters long and one piece is 3 quarters long.
step 1 of 5

Measure the plank and the piece in the same part, quarters of a metre. The division then counts groups of 3 quarters, and the quarters that do not make a full group are the leftover.

All steps
  1. Cut each metre of the plank into quarters. The plank is 5 × 4 = 20 quarters long and one piece is 3 quarters long.
  2. Put the quarters into groups of 3: 20 ÷ 3 = 6 remainder 2. This is the division 5 ÷ 34 = 5 × 43 = 203 = 623.
  3. (a) The greatest number of complete pieces is 6.
  4. The 6 pieces use 6 × 34 = 184 = 412 m of the plank, so 2 quarters are left over.
  5. (b) The length left over is 5 − 412 = 12 m. It is 2 of the 3 quarters in a piece, so it is 23 of a piece.

answer(a) 6 pieces; (b) 12 m, which is 23 of a piece

techniqueDividing by a Fraction · Multiplying Fractions

examsPSLE

Common pitfalls

  • Reading the 23 in 623 as 23 m. The quotient counts pieces, so 23 means two thirds of a piece, and two thirds of 34 m is 12 m.
  • Multiplying instead of dividing: 5 × 34 = 334. The pieces are shorter than 1 m, so there must be more than 5 of them.
04

A Fraction Shared Equally Among Several People

heuristicDivide a Fraction by a Whole Number / Cut the Parts Smaller Until They Share Exactly

Mr Lee pours 910 litre of juice equally into 6 cups. (a) How much juice is in each cup? (b) His children drink 4 of the cups. How much juice do they drink altogether?

1 litre1/101/101/101/101/101/101/101/101/109/10 litre of juice
One litre is 10 tenths, and the juice fills 9 of them.
Draw 1 litre as 10 equal tenths. The juice fills 9 of them.
step 1 of 5

Draw the juice as tenths of a litre. Nine tenths do not share among six cups as whole tenths, so cut every tenth in half and share the smaller parts.

All steps
  1. Draw 1 litre as 10 equal tenths. The juice fills 9 of them.
  2. Nine tenths cannot be shared among 6 cups as whole tenths, so cut every tenth into 2 equal parts. Each part is 120 litre, and the juice is 18 of these parts.
  3. Share the 18 parts among the 6 cups: 18 ÷ 6 = 3 parts in each cup.
  4. (a) Each cup holds 320 litre. This is the division 910 ÷ 6 = 910 × 16 = 960 = 320.
  5. (b) Four cups hold 4 × 320 = 1220 = 35 litre. Check: the other 2 cups hold 620 litre, and 1220 + 620 = 1820 = 910.

answer(a) 320 litre; (b) 35 litre

techniqueDividing a Fraction by a Whole Number · Multiplying Fractions

examsPSLE

Common pitfalls

  • Multiplying by 6 instead of by 16: 910 × 6 = 525 litres. Sharing makes each share smaller than the amount shared, so one cup must hold less than 910 litre.
  • Inverting the fraction instead of the whole number and working out 109 × 6. Only the divisor is inverted, so 6 becomes 16 and 910 stays as it is.
05

Equal Servings from a Mixed-Number Quantity

heuristicDivide a Fraction by a Fraction / Same-Sized Parts, Then Divide the Numerators

A pot holds 214 litres of soup. The soup is served in bowls that each hold 38 litre. (a) How many bowls can be filled? (b) After 4 bowls have been served, how much soup is left in the pot?

The pot holds 2 1/4 litres, which is 18 eighths of a litre.Pot1 litre1 litre1/4
In eighths of a litre, the pot holds 188 and one bowl holds 38.
Write both amounts in eighths. 214 = 94 = 188, so the pot holds 18 eighths of a litre and a bowl holds 3 eighths.
step 1 of 5

Write both amounts in eighths of a litre so that the parts are the same size. The division is then a division of whole numbers of eighths.

All steps
  1. Write both amounts in eighths. 214 = 94 = 188, so the pot holds 18 eighths of a litre and a bowl holds 3 eighths.
  2. Count the groups of 3 eighths in 18 eighths: 18 ÷ 3 = 6.
  3. (a) 6 bowls can be filled. This is the division 94 ÷ 38 = 94 × 83 = 6.
  4. After 4 bowls have been served, 6 − 4 = 2 bowls of soup are left in the pot.
  5. (b) The soup left is 2 × 38 = 68 = 34 litre.

answer(a) 6 bowls; (b) 34 litre

techniqueDividing a Fraction by a Fraction · Equivalent Fractions

examsPSLE · GCSE Higher

Common pitfalls

  • Inverting the first fraction instead of the divisor: 49 × 38 = 16. The divisor is the size of one bowl, 38, and it is the divisor that is inverted.
  • Dividing only the whole number, 2 ÷ 38, and leaving out the 14 litre. Change 214 into the improper fraction 94 before dividing.
06

A Recipe Made for Fewer Servings

heuristicFractional Scale Factor / Multiply Every Ingredient by the Same Fraction

A recipe for 10 servings of pancakes uses 56 kg of flour and 34 litre of milk. Mdm Wong makes only 4 servings. (a) How much flour does she need? (b) How much milk does she need?

Servings4 of 10 servings = 2/5
She makes 410 = 25 of the recipe, so she needs 25 of every ingredient.
Compare the servings. She makes 4 of the 10 servings, which is 410 = 25 of the recipe, so she needs 25 of every ingredient.
step 1 of 5

Find what fraction of the recipe she is making, in its simplest form. She needs that same fraction of every ingredient.

All steps
  1. Compare the servings. She makes 4 of the 10 servings, which is 410 = 25 of the recipe, so she needs 25 of every ingredient.
  2. The flour is 56 kg, which is 5 parts of 16 kg each. 25 of the 5 parts is 2 parts.
  3. (a) She needs 26 = 13 kg of flour. Check: 25 × 56 = 1030 = 13.
  4. For the milk, multiply the numerators and multiply the denominators: 25 × 34 = 620.
  5. (b) She needs 620 = 310 litre of milk.

answer(a) 13 kg; (b) 310 litre

techniqueMultiplying Fractions · Simplest Form

examsPSLE · GCSE Higher

Common pitfalls

  • Multiplying each ingredient by 4 because she makes 4 servings. The amounts in the recipe are for 10 servings, so the factor is 410, and she needs less than the recipe says.
  • Simplifying 620 by halving only the numerator and writing 320. The numerator and the denominator must both be divided by the same number.
07

A Rate Given with Two Fractions

heuristicUnitary Method with Fractions / Find the Amount for One Fifth of an Hour, Then for One Hour

A painter paints 34 of a wall in 25 of an hour, working at a steady rate. (a) How many walls of this size can he paint in 1 hour? (b) How many minutes does he take to paint one whole wall?

2/5 hour3/83/83/4 wall
In each 15 of an hour he paints 34 ÷ 2 = 38 of a wall.
In 25 of an hour he paints 34 of a wall. Halve both to find what he paints in 15 of an hour: 34 ÷ 2 = 38 of a wall.
step 1 of 5

Use the unitary method. Find what he paints in one fifth of an hour, then in five fifths. For the time, find how long one quarter of the wall takes, then four quarters.

All steps
  1. In 25 of an hour he paints 34 of a wall. Halve both to find what he paints in 15 of an hour: 34 ÷ 2 = 38 of a wall.
  2. One hour is 5 fifths of an hour, so in 1 hour he paints 5 × 38 = 158 walls.
  3. (a) He can paint 158 = 178 walls in 1 hour. This is the division 34 ÷ 25 = 34 × 52.
  4. For part (b), change the time to minutes: 25 × 60 = 24 minutes for 3 quarters of the wall. One quarter of the wall takes 24 ÷ 3 = 8 minutes.
  5. (b) The whole wall is 4 quarters, so it takes 4 × 8 = 32 minutes.

answer(a) 178 walls; (b) 32 minutes

techniqueDividing a Fraction by a Fraction · Dividing a Fraction by a Whole Number

examsPSLE · GCSE Higher

Common pitfalls

  • Multiplying the two fractions: 34 × 25 = 310. A rate is the amount of work divided by the time, so the fractions are divided.
  • Dividing the time by the work for part (a): 25 ÷ 34 = 815. That is the number of hours for one wall, not the number of walls in one hour.
08

One Quantity Given as a Fraction of Another

heuristicDivide to Find the Whole / The Numerator Counts the Known Units

A kitten has a mass of 45 kg. This is 23 of the mass of a rabbit. (a) What is the mass of the rabbit? (b) What is the total mass of the two animals?

Kitten1 unit1 unit4/5 kgRabbit1 unit1 unit1 unit?
The rabbit is 3 equal units, and the kitten is 2 of them.
Draw the rabbit as 3 equal units. The kitten is 23 of the rabbit, so the kitten is 2 of those units.
step 1 of 4

The rabbit is the whole, so draw it as 3 equal units. The kitten is 23 of it, which is 2 of those units, and the kitten's mass is known.

All steps
  1. Draw the rabbit as 3 equal units. The kitten is 23 of the rabbit, so the kitten is 2 of those units.
  2. 2 units = 45 kg, so 1 unit = 45 ÷ 2 = 25 kg.
  3. (a) The rabbit is 3 units: 3 × 25 = 65 = 115 kg. This is the division 45 ÷ 23 = 45 × 32.
  4. (b) The two animals are 2 + 3 = 5 units altogether: 5 × 25 = 2 kg. Check: 45 + 115 = 2.

answer(a) 115 kg; (b) 2 kg

techniqueDividing a Fraction by a Fraction · Dividing a Fraction by a Whole Number

examsPSLE · GCSE Higher

Common pitfalls

  • Multiplying: 23 × 45 = 815 kg. That is 23 of the kitten. The rabbit is the whole, so it is heavier than the kitten, and it is found by dividing.
  • Inverting the kitten's mass instead of the divisor: 54 × 23. The divisor is 23, so it is 23 that becomes 32.
09

Successive Fractions of a Tank of Water

heuristicProduct of Several Fractions / Cancel Common Factors Before Multiplying

A water tank was full on Monday morning. At the end of Monday, 56 of the water was still in the tank. At the end of Tuesday, the tank held 910 of the water it held at the end of Monday. At the end of Wednesday, it held 23 of the water it held at the end of Tuesday. (a) What fraction of the full tank was left at the end of Wednesday? (b) The full tank holds 480 litres. How many litres of water were used on Tuesday?

Full12 partsMonday5/6
Of the 12 parts of the full tank, 56 are left on Monday: 10 parts.
Each day keeps a fraction of the day before, so the fractions are multiplied. Draw the full tank as 12 equal parts. At the end of Monday 56 of them are left, which is 10 parts.
step 1 of 6

Draw the full tank as 12 equal parts, a number that 6 divides exactly. Each day's fraction is taken from the parts that the day before left.

All steps
  1. Each day keeps a fraction of the day before, so the fractions are multiplied. Draw the full tank as 12 equal parts. At the end of Monday 56 of them are left, which is 10 parts.
  2. At the end of Tuesday the tank holds 910 of the 10 parts, which is 9 parts: 56 × 910 = 912 = 34 of the tank.
  3. At the end of Wednesday it holds 23 of the 9 parts, which is 6 parts.
  4. (a) 612 = 12 of the full tank was left. With canceling, 56 × 910 × 23 = 12.
  5. On Tuesday the water went from 10 parts to 9 parts, so 1 part was used. One part is 112 of the tank.
  6. (b) 112 × 480 = 40 litres were used on Tuesday.

answer(a) 12; (b) 40 litres

techniqueMultiplying Fractions · A Fraction of an Amount · Simplest Form

examsPSLE · GCSE Higher

Common pitfalls

  • Taking 110 of the full tank for Tuesday: 110 × 480 = 48 litres. The 910 is a fraction of the water left at the end of Monday, which was 400 litres, so Tuesday used 110 × 400 = 40 litres.
  • Adding the three fractions instead of multiplying them. Each fraction is a fraction of a different amount, the amount the day before left, so the fractions are multiplied.
10

A Mixed Number of Equal Packets

heuristicMixed Number Times a Fraction / Change to an Improper Fraction First

Each packet of flour has a mass of 38 kg. Mrs Goh buys 6 packets and uses 212 packets to bake bread. (a) What mass of flour does she use? (b) What mass of flour does she have left?

6 packets3/8 kg3/8 kg3/8 kg3/8 kg3/8 kg3/8 kgIn halvesShe uses 2 1/2 packets, which is 5 half packets.
212 = 52, so she uses 5 half packets.
Write the mixed number as an improper fraction: 212 = 52. She uses 5 half packets.
step 1 of 5

Count in half packets. The mixed number 212 is 5 halves, so find the mass of half a packet and multiply.

All steps
  1. Write the mixed number as an improper fraction: 212 = 52. She uses 5 half packets.
  2. Half a packet has a mass of 38 ÷ 2 = 316 kg.
  3. (a) She uses 5 × 316 = 1516 kg of flour. This is the product 52 × 38 = 1516.
  4. She has 6 − 212 = 312 packets left, which is 7 half packets.
  5. (b) She has 7 × 316 = 2116 = 1516 kg of flour left. Check: 1516 + 2116 = 3616 = 214 kg, which is 6 × 38.

answer(a) 1516 kg; (b) 1516 kg

techniqueMultiplying Fractions · Dividing a Fraction by a Whole Number

examsPSLE · GCSE Higher

Common pitfalls

  • Multiplying only the whole number, 2 × 38 = 68, and leaving out the half packet. The half packet adds another 316 kg.
  • Working out 2 × 38 and then adding 12, which treats the half packet as 12 kg. Half a packet is half of 38 kg, which is 316 kg.
11

Reagent Shared into Vials, Then from Vials onto Plates

heuristicDivide a Fraction by a Whole Number Twice / Then Count the Plates a New Supply Fills

A laboratory dispenser holds 78 litre of a liquid reagent. It empties all of it equally into 14 identical test vials. A robotic arm then takes 9 of the vials and shares their reagent equally among 6 culture plates. (a) How much reagent is in each vial? (b) How much reagent does each culture plate receive? (c) A fresh container holds 214 litres of the reagent. How many culture plates can it fill if each plate receives the same amount as before? Give the answers to (a) and (b) as fractions of a litre in their simplest form.

Dispenser7/8 L = 14/16 L
Cut each of the 7 eighths of a litre in two: 14 parts of 116 litre, one for each vial.
The dispenser holds 7 eighths of a litre. Seven eighths cannot be shared among 14 vials as whole eighths, so cut every eighth into 2 equal parts: the reagent is 14 parts of 116 litre.
step 1 of 6

Draw one litre as a bar and follow the reagent through each share. Where the parts do not share out exactly, cut every part in two until they do. The last share is the amount for one plate, and the new supply is counted in parts of that size.

All steps
  1. The dispenser holds 7 eighths of a litre. Seven eighths cannot be shared among 14 vials as whole eighths, so cut every eighth into 2 equal parts: the reagent is 14 parts of 116 litre.
  2. (a) Each vial gets one part, 116 litre. This is the division 78 ÷ 14 = 78 × 114 = 116.
  3. The 9 vials hold 916 litre. Nine sixteenths do not share among 6 plates as whole sixteenths, so cut every sixteenth into 2: that makes 18 parts of 132 litre, and 18 ÷ 6 = 3 parts for each plate.
  4. (b) Each plate receives 332 litre. This is the division 916 ÷ 6 = 996 = 332.
  5. Count the fresh supply in thirty-seconds of a litre: 214 = 94 = 7232 litres, which is 72 parts.
  6. (c) Each plate takes 3 parts, so the container fills 72 ÷ 3 = 24 plates. Check: 24 × 332 = 7232 = 214 litres.

answer(a) 116 litre; (b) 332 litre; (c) 24 plates

techniqueDividing a Fraction by a Whole Number · Dividing a Fraction by a Fraction · Multiplying Fractions

examsPSLE

Common pitfalls

  • Sharing the whole 78 litre among the 6 plates. Only 9 of the 14 vials go to the plates, so the plates share 916 litre, and each receives 332 litre rather than 748 litre.
  • Working out 332 ÷ 94 = 124 in (c) and answering 124. That is the fraction of the new container that one plate uses. The number of plates is the supply divided by the amount for one plate, 94 ÷ 332 = 24.
12

A Bag of Birdseed in Quarter-Cup Scoops and Three-Quarter-Cup Mornings

heuristicDividing by a Fraction Counts How Many Fit / Count the Quarters, Then Group Them in Threes

Hana's class has a bag with 6 cups of birdseed for the class bird feeder. (a) A small scoop holds 14 cup. How many small scoops of seed are in the bag? (b) The class puts 34 cup of seed in the feeder every morning. For how many mornings does the bag last?

Bag1 cup1 cup1 cup1 cup1 cup1 cup6 cups
Cut each of the 6 cups into quarters: each cup holds 4 quarter-cups.
Cut each of the 6 cups into quarters. Each cup holds 4 quarter-cups.
step 1 of 4

Draw the 6 cups as 6 bars and cut each one into quarters. Then count how many quarter-cups fit, and how many groups of 3 quarter-cups fit.

All steps
  1. Cut each of the 6 cups into quarters. Each cup holds 4 quarter-cups.
  2. (a) The 6 cups hold 6 × 4 = 24 quarter-cups, so there are 24 small scoops of seed: 6 ÷ 14 = 24.
  3. One morning's seed, 34 cup, is 3 quarter-cups. Put the 24 quarter-cups into groups of 3: 24 ÷ 3 = 8 groups.
  4. (b) The bag lasts 8 mornings: 6 ÷ 34 = 8. Check: 8 × 34 = 244 = 6 cups.

answer(a) 24 small scoops; (b) 8 mornings

techniqueDividing by a Fraction

examsPSLE

Common pitfalls

  • Multiplying by the fraction: 6 × 34 = 412 mornings. Each morning uses less than a whole cup, so 6 cups must last more than 6 mornings. The question asks how many 34 cups fit into 6 cups, which is a division.
  • Multiplying the 24 small scoops by 3 and answering 72 mornings. One morning's seed is 3 small scoops, so there are fewer mornings than scoops: 24 ÷ 3 = 8.