Proportion

Sundip has some £5, £10, £20 and £50 bank notes. She buys 30 litres of fuel for her car. One litre of fuel costs £1.50.

Sundip pays the exact amount of money for the fuel using three bank notes.

Work out the three bank notes that she uses to pay for the fuel.

(a) £........, £........ and £........

On one journey, Sundip’s car uses fuel at a rate of one litre every 16 kilometres. At the start of the journey, her car has 50 litres of fuel. At the end of the journey, her car has used of this fuel. Use the conversion 5 miles = 8 kilometres to work out the number of miles Sundip travels on this journey. You must show your working.

Three lengths of wood are in the ratio 2: 5: 7. The difference in size between two of the lengths of wood is 30 cm.

Kofi says,

The sum of the three lengths of wood must be at least 120 cm.

Is Kofi correct? Show how you decide.

Kofi is........ because........

The diagram shows a sector of a circle AOB and a circle with radius r cm.

The sector has radius 12 cm and angle AOB = 120°. The ratio of the area of the sector AOB to the area of the circle is 3: 4.

Calculate the value of r. You must show your working.

Sam sees a computer tablet for sale on the internet for 126 euros (€). Sam has £105 to spend.

The exchange rate is £1 = €1.15.

Can Sam buy the tablet? Show how you decide.

The diagram shows a large rectangle made from 10 smaller rectangles of equal area. The large rectangle is divided into two unequal parts, A and B.

Tick (✓) all the correct statements.

The area of Part A is 18 k.

Write an expression, in terms of k, for the area of Part B. Give your answer in its simplest form.

Sasha makes some cupcakes. Each cupcake is topped with red or white or yellow icing. of the cupcakes have red icing. of the cupcakes have white icing.

Work out the fraction of the cupcakes that have yellow icing.

Sasha makes all of the icing using butter and sugar in the ratio 1: 2. Each cupcake has 60 g of icing.

Sasha uses 0.7 kg of butter to make the yellow icing.

Find the total number of cupcakes Sasha makes. You must show your working.

Each member of a tennis club is classed as a junior, an adult or a senior.

• 16% of the members are seniors.

• The ratio of the number of juniors to the number of adults is 9: 5.

• There are 95 more juniors than seniors.

Work out the total number of members of the tennis club. You must show your working.

A bag contains only round tiles and square tiles. Each tile is either black or white.

The ratio of round tiles to square tiles is 3: 1. 45% of all the tiles are round black tiles. 9% of all the tiles are square white tiles.

Show that the proportion of round tiles that are black is less than the proportion of square tiles

that are black.

A recipe for cookies uses only butter, flour and sugar in the ratio 2: 4: 1. Using this recipe, 160 g of butter is needed to make 10 cookies.

Using this recipe, work out how much flour is needed to make 25 cookies.

Leo pays £44.98 for 26 litres of diesel. Mia puts 60 litres of the same diesel into her car. Mia has £103 with which to pay for her diesel.

Does Mia have enough money to pay for her diesel? Show how you decide.

These are the ingredients for making some scones.

10 ml of milk weighs 10.4 g.

Work out the total weight of all the ingredients.

100 students vote in a school election.

1 vote is spoiled and is not counted. The remaining votes are for Jamal or Layla and are in the ratio 3: 8.

How many more votes does Layla have than Jamal?

Machine A makes enough lollipops to fill 300 packs.

There are 8 lollipops in each pack.

Show that 2400 lollipops are made by machine A.

Machine B makes 3600 lollipops in the same time it took by machine A to make 2400 lollipops.

Machine B makes lollipops one at a time and at a constant rate.

What fraction of the time needed to fill 300 packs is saved if machine B is used rather than machine A? Give your answer in its simplest form.

Shop A and shop B have special offers on the same cupcakes.

Show that the special offer cost of 6 cupcakes from Shop A is £6.25.

Gabi wants 25 cupcakes for a party.

Which shop will be cheapest and by how much? Show how you decide.

(b) Shop........ by........ p

Felix makes craft figures at a constant rate. He can make 5 craft figures in 40 minutes.

Find the number of craft figures Felix can make in 4 hours.

Darcie makes craft figures 10% quicker than Felix.

Work out how long Darcie takes to make 15 craft figures.

Sam eats 30% of a 600 g pack of cereal every day.

Sam has no cereal left. The cereal is no longer sold in 600 g packs. Instead, the cereal is now sold in 400 g packs.

Sam wants to continue to eat the same amount of cereal every day.

Work out the minimum number of 400 g packs of cereal that Sam must buy to have enough for 7 days. You must show your working.

A bathroom shower uses water at a rate of 15 litres per minute.

How much water is used in a bathroom shower that lasts 12 minutes?

This table shows the weight of each ingredient used to make 5.5 kg of concrete.

Complete this table to show the weight of each ingredient used to make 44 kg of the same concrete.

A watch costs £207.50. The exchange rate is 1 dollar ($) = £ 0.83.

Calculate the price of the watch in dollars ($).

When Layla walks to school she always takes 28 minutes. When Layla cycles to school her speed is double her walking speed.

One day Layla leaves home at 08 20 and cycles to school.

Find the time that Layla arrives at school.

On another day Layla cycles half-way to school. She walks the rest of the way at her normal walking speed.

Find how many minutes Layla takes to get to school on this day.

The same dog food is sold in three different sized packs. The diagram shows the price of each pack.

Which pack is the best value for money? Show how you decide.

The........ pack because........

Two bags of fruit contain only apples and bananas. In bag X, the ratio of apples to bananas is 5: 7. In bag Y, of the fruit are apples.

Finley says

Bag X and bag Y contain the same number of apples.

Tick the correct statement.

Show how you decided.

Finley adds 4 apples to bag X. The ratio of apples to bananas is now 11: 14.

How many bananas are in bag X?

Alex and Emma use this recipe to make scones. The recipe shows the ingredients needed to make 16 scones.

Alex makes 24 scones.

He uses a tablespoon to measure the amount of milk he needs. The size of a tablespoon is 15 ml.

How many tablespoons of milk should Alex use to make 24 scones?

Emma has 1 kg of flour and 240 g of sultanas. She has lots of all of the other ingredients.

Work out the maximum number of scones that Emma can make. You must show your working.

Four apples cost £1.80.

Find the cost of five of these apples.

By rounding each value to one significant figure, estimate the cost of 8.2 kg of bananas at 73p per kg. Give your answer in pounds.

In a computer game the player can either win or lose. A student thinks the ratio of the probability of winning to the probability of losing is 2: 3.

The student plays two games.

Use the information to complete the tree diagram.

Find the probability that the student wins at least one of the two games.

The student now thinks the ratio of the probability of winning to the probability of losing has changed to 2: 5.

Explain the effect this change will have on your answer to part (b).

A box contains only red, green and black pens. The ratio of red pens to green pens to black pens is 1: 4: 11.

Work out the percentage of the pens that are green.

There are 24 more green pens than red pens.

Work out the total number of pens in the box.

A large box of chocolates contains dark, milk and white chocolates. When Riley opens the box, the ratio of dark to milk to white chocolates is 3: 2: 4. Riley’s family eat 6 of the dark chocolates, none of the milk chocolates and all of the white chocolates. The ratio of dark to milk chocolates is now 9: 8.

How many white chocolates did Riley’s family eat?

450 g of flour costs £1.44.

Work out the cost of 1 kg of this flour.

Cookies are made using these ingredients.

How many eggs are needed to make 48 cookies?

How much sugar is needed to make 6 cookies?

Ashley has 520 g of cranberries and plenty of the other ingredients. Ashley thinks this is enough to make at least 80 cookies.

Is Ashley correct? Show working to support your answer.

Darcie makes 100 cookies. They are put into packets, each holding 6 cookies. Each packet of 6 cookies is sold for £1.35. Darcie sells all of these packets.

Work out how much money Darcie receives.

Taylor has a full bottle of medicine. The bottle holds 20 doses of medicine.

Each day Taylor takes one dose of medicine out of the bottle. After 8 days, there are 180 millilitres of medicine left in the bottle.

Work out how many millilitres of medicine the bottle holds when full.

A zoo counts its animals. The ratio of antelope to zebra is 3: 2. The ratio of meerkats to zebra is 7: 3.

Write the number of antelope as a percentage of the number of zebra.

There are 15 more meerkats than antelope.

Work out the number of zebra in the zoo.

On Heidi’s bookcase, the ratio of fiction to non-fiction books is 2: 3. Heidi removes 2 fiction books from the bookcase. The ratio of fiction to non-fiction books is then 5: 8.

How many books are left on the bookcase in total?

Morgan makes 15 cakes. Each cake needs a piece of ribbon 18 cm long for decoration.

Ribbon is only sold in rolls of 1.2 metres, which can then be cut into 18 cm pieces. One roll of ribbon costs 92p.

Calculate the cost of the ribbon that Morgan must buy to decorate the 15 cakes. You must show your working.

Blake changes £450 into dollars. £1 is worth 1.34 dollars.

Blake says

450 ÷ 1.34 = 335.82 Therefore, £450 is worth 335.82 dollars.

Is Blake correct or incorrect? Give a reason for your decision.

Blake is........ because........

A bag contains 35 balls. Each ball is either red or green. The ratio of red balls to green balls is 3: 2.

Work out the smallest number of balls of each colour that have to be added to the bag so that the ratio of red balls to green balls becomes 7: 3. You must show your working.

Ashley has £7 to spend on fruit. The table shows the prices.

Ashley buys 2 pineapples and 3 kilograms of bananas. Ashley spends the remaining money on strawberries.

Work out the mass, in grams, of strawberries that Ashley buys. You must show your working.

Two supermarkets, A and B, have special offers on the same packet of biscuits.

Dan buys one packet of these biscuits.

Which supermarket is best value for Dan? Show how you decide.

Supermarket........ because........

Darcy buys three packets of these biscuits.

Which supermarket is best value for Darcy? Show how you decide.

Supermarket........ because........

A recipe for a batch of jam needs 3 oranges, 5 lemons and 1.5 kg of sugar. A cook uses the recipe to make lots of batches of jam. They use 16 more lemons than oranges in total.

F ind how much sugar the cook should use.

A bag contains 35 balls. Each ball is either red or green. The ratio of red balls to green balls is 3: 2.

Work out the smallest number of balls of each colour that have to be added to the bag so that the ratio of red balls to green balls becomes 7: 3. You must show your working.

The parallelogram and the trapezium have the same area.

The ratio of x: y is 3: 5.

Find the value of x and the value of y. You must show your working.

A recipe for a batch of jam needs 3 oranges, 5 lemons and 1.5 kg of sugar. A cook uses the recipe to make lots of batches of jam. They use 16 more lemons than oranges in total.

Find how much sugar the cook should use.

A machine can dig, on average, 2 cm of tunnel each minute. It operates 24 hours each day.

Work out how many days it should take to dig a tunnel of length 3.5 km. Give your answer to the nearest day.

The machine actually digs an average of 2.5 cm of tunnel each minute for most of the time and an average of 1.5 cm each minute for the rest of the time.

How would this affect your answer to part (a) ?

A local theatre is putting on a show. 50 child tickets are sold. The ratio of the number of child tickets sold to the number of adult tickets sold is 5: 2.

The cost of a child ticket is £2.50. The cost of an adult ticket is £5.00.

Work out the total amount paid for the tickets.

Frankie goes on holiday. They change £375 into euros (€) at a rate of £1 = €1.15. They spend €217.49 of this money. After the holiday, Frankie changes the remaining euros back into pounds at a rate of £1 = €1.28.

Work out how many pounds Frankie gets back.

A machine can dig, on average, 2 cm of tunnel each minute. It operates 24 hours each day.

Work out how many days it should take to dig a tunnel of length 3.5 km. Give your answer to the nearest day.

The machine actually digs an average of 2.5 cm of tunnel each minute for most of the time and an average of 1.5 cm each minute for the rest of the time.

How would this affect your answer to part (a) ?

In a survey about music, some students were asked whether they like pop (P) and whether they like rap (R).

The Venn diagram shows some of the results. x students liked both types of music.

The ratio of the number of students who liked pop to the number who liked rap was 5: 2.

Work out the total number of students in the survey.

One of the students is selected at random.

Find the probability that this student does not like rap given that they like pop.

A biased five-sided spinner is numbered 1, 2, 3, 4 and 5.

The table shows the probability of the spinner landing on 1, 2 and 4.

The spinner is four times more likely to land on 5 than on 3.

Complete the table.

Ben and Sundip are making pancakes using the ingredients below.

The ratio of the amount of water to the amount of milk needed is 75: 200.

Write this ratio in its simplest form.

Ben makes 18 pancakes.

Work out how much flour he needs.

Sundip has 225 g of butter and 10 eggs. She has plenty of the other ingredients.

Work out the maximum number of pancakes that she can make.

The shape below is formed by a rectangle of width 4 cm and a regular pentagon. For the rectangle, the ratio of the width to the length is 2: 5.

Work out the perimeter of the shape.

A bag of sweets contains jellies, mints and toffees.

The ratio of jellies to mints is n: 2. The ratio of mints to toffees is 5: 3 n.

Work out the ratio of jellies to toffees. Give your answer in its simplest form.

A shop sells the same milk in three different sized cartons. The diagram shows the price of each carton.

Which carton is the best value for money? Show how you decide.

A student only buys milk on a Saturday morning. They use 120 ml of milk each day. Any unused milk has to be thrown away at the end of the following Friday.

Show that it is cheaper for the student to buy the milk they need in 300 ml cartons than in

500 ml cartons.

Lily buys and sells microwaves.

She buys each one for £32 and sells it for £60. She also pays £7 for the delivery of each microwave she sells.

If she sells a microwave that is faulty then Lily must pay for its repair and redelivery. This costs her another £25 for each faulty microwave.

Last month, 6 out of the 80 microwaves Lily sold were faulty.

This month she has orders for 133 microwaves.

Calculate her expected percentage profit on this month’s order. Showing your working in the boxes below may help you present your work.

A bag only contains red, blue, yellow and white counters. A counter is taken at random from the bag. The table shows the probability it is red and the probability it is blue.

There are twice as many yellow counters as white counters in the bag.

Complete the table.

Sundip is going on holiday. She wants to change £400 into euros(€).

Bank A will change her £400 into €452.

Bank B changed £250 into €280 for Sundip’s friend. It will use the same rate to change Sundip’s £400 into euros.

At which bank will Sundip receive the most euros and by how many? Show your working.

At Bank........ Sundip will receive €........ more

The mass of flour used in a recipe doubles as the number of biscuits made doubles.

On the axes below, sketch a graph to show this relationship.

Here are some of the ingredients for a recipe to make 10 biscuits.

Jane followed the recipe and used 432 g of butter. All of the sugar used came from a new 2 kg bag.

Find the number of biscuits she made.

Find the mass of the sugar, in grams, that Jane has left in the bag.

Carol makes birthday cards. Each card takes the same amount of time to make.

She makes 3 cards in 48 minutes. She has an order for 80 cards.

Can she complete this order in 3 days if she works 8 hours each day? Show how you decide.

Carol makes birthday cards. Each card takes the same amount of time to make.

She makes 3 cards in 48 minutes. She has an order for 80 cards.

Can she complete this order in 3 days if she works 8 hours each day? Show how you decide.

Bob makes dry concrete by mixing cement, sand and stone in the ratio 1: 2: 3 by weight. He buys the cement, sand and stone in bags as shown in this table.

He packs the dry concrete into 30 kg bags.

Bob buys just enough cement, sand and stone to make 50 bags of dry concrete.

Show that Bob buys 500 kg of sand.

Bob sells the 50 bags of dry concrete for a total of £396.

Calculate Bob’s percentage profit.

Sundip and Emma have some money. The ratio of Sundip’s money to Emma’s money is 3: 5. Emma spends £450 of her money. The ratio of Sundip’s money to Emma’s money is now 2: 3.

Find how much money Sundip has.

Rashid is making cupcakes using these ingredients.

How many eggs does he need to make 60 cupcakes?

How much butter is needed to make 5 cupcakes?

Rashid has 210 g of cocoa powder and plenty of the other ingredients. He says that he can make at least 75 cupcakes.

Is he correct? Explain your reasoning.

A sunflower grows at a rate of 4 cm each day.

How many days does it take to grow from a height of 80 cm to more than 1.06 m?

If the sunflower grows at a faster rate, how would this affect your answer to part (a) ?

One day, a group of people had a driving test. 40 of this group were men and the rest were women. of the men and of the women passed the driving test. The number of men and women that passed the driving test was the same.

Work out the number of women that took the driving test that day.

Delroy drives 240 miles. His car averages 40 miles per gallon of petrol. Petrol costs £1.30 per litre.

1 gallon is 4.5 litres.

How much does Delroy spend on petrol for this journey?

A bag contains some counters.

• There are 300 counters in the bag.

• There are only red, white and blue counters in the bag.

• The probability of picking a blue counter is

• The ratio of red counters to white counters is 2: 1.

Calculate the number of red counters in the bag.

Louiza changes £320 into euros. £1 is worth 1.14 euros.

How many euros does she receive?

Liu wants to decorate some cakes with shapes.

She has 140 shapes. Each shape is a star or a heart. The ratio of the number of stars: number of hearts is 4: 3. She wants to put 5 stars and 4 hearts on each cake.

How many cakes can Liu decorate? Show full working to support your answer.

Tea bags of similar quality are sold in three different sized packs:

Which pack is the best value for money? Show how you decide.

Explain why someone may buy a pack which is not the best value for money.

Hector can run 400 metres in 66 seconds.

Use this information to show that he could run 5 kilometres in less than 14 minutes.

Hector tries to run 5 kilometres in less than 14 minutes.

Give one reason why he might not achieve this.

In a village the ratio of males to females is 2: 1.

40% of the people in the village are right-handed males. 25% of the people in the village are right-handed females.

Show that the proportion of females who are right-handed is greater than the proportion of males

who are right-handed.

Pippa owns a snack bar.

Spread costs £3.20 for a 1 kilogram tub and £6.15 for a 2 kilogram tub.

Pippa buys enough spread to last for 14 days.

What is the lowest price Pippa can buy this spread for? Show your working.

In 2016, Pippa paid £1650 rent. In 2017, the rent increased by 14%.

Calculate the amount of rent she paid in 2017.

Here is the floor plan of a rectangular room.

Tim buys carpet tiles for this room.

Each tile is a square measuring 50 cm by 50 cm. The tiles are only sold in packs of ten. Each pack costs £20. Tim pays for fitting at a rate of £7.50 per square metre, with any fraction of a square metre rounded up.

Work out the total cost of the tiles and fitting.

Ifsaw noticed this information on her car’s dashboard at the end of her journey. She started her journey with a full tank of fuel and her miles travelled set to zero.

Work out how far Ifsaw’s car can travel on a full tank of fuel.

What assumption have you made when answering part (a) ?

Tea Biscuits can be bought in packets of 20 or packets of 24. All biscuits are identical in size and quality.

Nada says

The packet of 24 biscuits is better value.

Is Nada correct? Show how you decide.

Nada is........ because........

Ifsaw noticed this information on her car’s dashboard at the end of her journey. She started her journey with a full tank of fuel and her miles travelled set to zero.

Work out how far Ifsaw’s car can travel on a full tank of fuel.

What assumption have you made when answering part (a) ?

Students deliver catalogues and leaflets to houses. One day, they have to deliver 360 catalogues and 1440 leaflets. Each student can deliver either 15 catalogues or 80 leaflets in 1 hour. Each student can only work for 8 hours.

Work out the minimum number of students needed.

A 100 g packet of tea costs £4.16. A 25 g packet of the same tea costs £1.05.

Which packet is better value for money? Show how you decide.

This pie chart shows how the employees of a business travel to work.

Find the ratio of the number of employees who cycle to work to the number of employees who walk to work. Give your answer in its simplest form.

80 employees travel to work by car.

Work out the number of employees who cycle to work and the number of employees who walk to work.

Elsie changes to a decimal.

This is her working.

Where a line of working is wrong, write the correct working beside it.

Ali has 1 litre of squash. He always mixes 0.05 litres of squash with 200 ml of water to make a glass of drink.

Find the total volume of the drink that Ali can make. Give your answer in litres.

A company makes sweets. The sweets are put into packets.

Here are some facts.

Calculate the mean number of sweets in one packet.

Sweets are made on 288 days each year.

Calculate the number of sweets made each year. Give your answer in standard form.

The company has 152 machines making the sweets. Each machine operates for 15 hours each day.

Calculate the number of sweets made by one machine each hour. Give your answer as an ordinary number correct to the nearest 10.

State one assumption you have made in part (c)(i).

Two bags each contain only red counters and yellow counters.

In Bag A, the ratio of red counters to yellow counters is 1: 4.

Sharon says

The proportion of the counters that are red is the same in both bags.

Explain why Sharon is not correct.

The number of counters in the two bags is the same.

Complete the table below to show how many counters of each colour could be in the bags.

In another bag, Bag C, the ratio of red counters to yellow counters is 3: 4. If 3 of the red counters are removed from Bag C, the ratio of red counters to yellow counters is 3: 5.

How many yellow counters are in Bag C?

Donald swims 3 lengths of a swimming pool in 93 seconds.

Use this information to show that he could swim 100 lengths in under 55 minutes.

What assumption did you make in part (a) ?

Donald tries to swim the 100 lengths in under 55 minutes.

Suggest one reason why he might not achieve this.

Maria mixes white paint and red paint in the ratio 2: 3. She makes a total of 15 litres of paint.

How much more red paint does she need to add to the mixture so that the ratio of white paint to red paint becomes 1: 5?

There is a total of 250 men, women and children on a train. The ratio of men to women is 4: 5. The ratio of women to children is 10: 7.

How many men are on the train?

A company makes sweets. The sweets are put into packets.

Here are some facts.

Calculate the mean number of sweets in one packet.

Sweets are made on 288 days each year.

Calculate the number of sweets made each year. Give your answer in standard form.

The company has 152 machines making the sweets. Each machine operates for 15 hours each day.

Calculate the number of sweets made by one machine each hour. Give your answer as an ordinary number correct to the nearest 10.

State one assumption you have made in part (c)(i).

Two bags each contain only red counters and yellow counters.

In Bag A, the ratio of red counters to yellow counters is 1: 4.

Sharon says

The proportion of the counters that are red is the same in both bags.

Explain why Sharon is not correct.

The number of counters in the two bags is the same.

Complete the table below to show how many counters of each colour could be in the bags.

In another bag, Bag C, the ratio of red counters to yellow counters is 3: 4. If 3 of the red counters are removed from Bag C, the ratio of red counters to yellow counters is 3: 5.

How many yellow counters are in Bag C?

Apple crumble is made using these ingredients.

Susumu makes apple crumble to serve 12 people.

How much flour should he use?

Natalie makes apple crumble for 2 people.

How much butter should she use?

Abena has 1.3 kg of apples and plenty of the other ingredients.

Can she make apple crumble for 15 people? Explain how you got your answer.

Nathan works out 23 × 12.4 without a calculator.

This is Nathan’s working.

Nathan’s working is incorrect.

Explain the error that Nathan has made and work out the correct answer.

Four friends buy cinema tickets using this offer.

Cinema tickets

Buy 3 tickets and get a ticket free

They each pay £6.45.

How much does a ticket cost?

The population of a village is in the following ratios.

• men: children = 11: 3

• women: children = 5: 2

Find the ratio men: women. Give your answer in its simplest form.

There are 36 children in the village.

Find the total population of the village.

Harry needs dollars to go on holiday. He can buy $50 for £40.

How much will $720 cost at the same rate?

Tony returns from holiday with these notes.

The exchange rate is £1 = €1.17.

Work out how much he will get in total when he changes these notes.

A village has a population of 4200 and a population density of 700 people per km². An estate is built next to the village. The estate has an area of 2 km² and a population density of 800 people per km².

Work out the population density for the village and the estate together.

........ people per km²