OCR GCSE Maths
Topic questions Ratio, proportion and rates of change Direct & inverse proportion
65 questions · 299 marks · about 4.5 hours
Also involves Converting between units
It takes 36 hours for 7 people to paint a fence. Assume that all people paint at the same rate.
How long would it take 15 people to paint the same fence? Give your answer in hours and minutes.
y is inversely proportional to the square of x. y = 5 when x = 6.
Find a formula linking x and y.
p is directly proportional to the square root of m. p = 15 when m = 25.
Find the value of p when m = 81.
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.
Also involves Substitution into formulae · Simplifying and manipulating expressions
y is directly proportional to the square of t. y = 14 when t = 2.
t is directly proportional to x. t = 12 when x = 3.
Find a formula for y in terms of x. Give your answer in its simplest form. You must show your working.
Also involves Standard units and compound measures
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.
A machine makes 15 boxes in 12 minutes. The machine works continuously.
Work out how many boxes are made by this machine in 7 hours.
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.
| Ingredient | Weight (kg) | Weight (kg) |
| Cement | 1 | 8 |
| Gravel | 3 | |
| Sand | 1.5 | |
| Total weight | 5.5 | 44 |
Also involves Using a calculator
A watch costs £207.50. The exchange rate is 1 dollar ($) = £ 0.83.
Calculate the price of the watch in dollars ($).
Also involves Fractions · Standard units and compound measures
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.
Here is a question and an incorrect solution.
Describe the error made and write out a correct solution.
A correct solution is........
Also involves Percentages
y is directly proportional to x.
Write down the percentage increase in y when x is increased by 100%.
z is inversely proportional to x.
Write down the percentage decrease in z when x is increased by 100%.
y is inversely proportional to x3. y = 4 when x = 12.
Find a formula linking x and y.
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.
The diagram shows a Penny Farthing bicycle.
The diameter of the large wheel is 130 cm.
The diameter of the small wheel is 46 cm.
On a short journey, the large wheel makes exactly 69 rotations. The small wheel also makes an exact number of rotations.
Work out the number of rotations made by the small wheel.
y is directly proportional to the cube of x.
Complete the table.
| x | 1 | 2 | |
| y | 7 | 875 |
x and y are related by the equation xy = 36.
Tick the correct statement.
y is inversely proportional to x4. y = 2.5 when x = 2.
Find a formula linking x and y.
Also involves Converting between units
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.
A volunteer packs boxes for a charity. They can pack 5 boxes in 45 seconds.
Use this information to show that they can pack 55 boxes in less than 9 minutes.
What assumption did you make in part (a) ?
Also involves Substitution into formulae
y is inversely proportional to the square root of x. y = 7 when x = 144.
Find the value of y when x = 16.
Also involves The four operations
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 volunteer packs boxes for a charity. They can pack 5 boxes in 45 seconds.
Use this information to show that they can pack 55 boxes in less than 9 minutes.
What assumption did you make in part (a) ?
y is directly proportional to the square of x.
Find the percentage decrease in y when x is decreased by 30%.
The time taken to complete a journey halves as the speed doubles.
On the axes below, sketch a graph to show this relationship.
It takes 40 minutes to fill a garden pond using water from 5 identical hose pipes.
Assuming the rate of flow of water from each hose pipe is the same, work out how many minutes it would take to fill the same garden pond using 2 of these hose pipes.
All of the loaves in a baker’s shop cost the same price. Rowan buys 3 loaves and pays £3.78. Azmi buys 5 loaves.
Work out how much Azmi pays.
Alex and Ling travel the same distance to school.
Alex walks to school in 20 minutes. Ling runs to school at twice the speed that Alex walks.
Find how many minutes it takes Ling to run to school.
Find a formula for y in terms of x.
Also involves Forming and solving equations
The time taken to paint a wall is inversely proportional to the number of people painting. It takes 40 minutes for 3 people to paint the wall if nobody stops painting.
Layla, Mia and Nina start painting the wall. After 10 minutes Layla stops painting. She leaves Mia and Nina to finish painting the wall.
Assume that Layla, Mia and Nina paint at the same rate.
Work out the total time taken to paint the wall.
y is inversely proportional to x3. y = 16 when x = 2.
Find the value of y when x = 8.
y is inversely proportional to x. y = 20 when x = 3.
Find the value of y when x = 12.
y is inversely proportional to x2. y = 9 when x = 2.
Find the value of y when x = 10.
Also involves The four operations
A conversion graph between pounds (£) and dollars ($) is shown below.
Explain fully how the graph shows that the number of dollars is directly proportional to the number of pounds.
Use the conversion graph to change £20 into dollars.
Some trainers cost £170 in the UK. The same trainers cost $195 in the USA.
Show that the trainers cost less in the USA.
If the trainers are brought from the USA there is an extra charge for tax and delivery.
Alex wants to pay the lowest total amount for the trainers.
Write down the maximum extra charge for tax and delivery that Alex should be willing to pay. Give your answer in dollars.
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.
Rowan’s bath has a hot tap and a cold tap. When turned on full, each tap on its own will fill the bath in 6 minutes.
Rowan turns both taps on full.
How long will it take to fill the bath?
Also involves Compound measures
Amit says
My normal typing speed is 40 words per minute. Therefore, I estimate that my normal typing speed is about 210 characters per minute.
Each letter, space and piece of punctuation counts as a character.
How many letters per word is Amit most likely to have used in making the estimate? Show how you decide.
Amit starts some homework at their normal typing speed. Amit types 52 words in 1 minute 12 seconds.
What may be true about the length of the words that Amit has just typed? Show how you decide.
Also involves Rearranging formulae
y is inversely proportional to the square root of x. y = 5 when x = 36.
Find a formula linking x and y.
Find the value of x when y = 20.
Also involves Using a calculator
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.
y is inversely proportional to the square of x. y = 2 when x = 5.
Find a formula linking x and y.
Also involves Forming and solving equations
x is directly proportional to y. y is directly proportional to z.
When x = 10, y = 60. When y = 8, z = 1.6.
Find a formula for z in terms of x.
Also involves The four operations · Converting between units
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.
y is inversely proportional to the square root of x. y = 7 when x = 25.
Find the value of y when x = 100.
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.
50 sweets weigh 200 g.
If each sweet weighs the same, work out the weight of 7 sweets.
b is directly proportional to a. b is 10 when a is 8.
Work out b when a is 9.
A graph is drawn below.
Explain how you know that y is not directly proportional to x.
Also involves Rearranging formulae
The table shows values of x and y.
| x | 4 | 16 | 36 |
| y | 6 | 3 | 2 |
a is inversely proportional to b2 and a = 3.75 when b = 4.
Find a formula linking a and b.
Also involves Using a calculator
Louiza changes £320 into euros. £1 is worth 1.14 euros.
How many euros does she receive?
This graph shows the cost of buying potatoes from a farm.
How much does it cost to buy 70 kg of potatoes?
What weight of potatoes can be bought for £38?
The cost per kilogram of potatoes is the same for any weight of potatoes.
How much will it cost to buy 180 kg of potatoes from the farm?
This graph shows the annual profits, in thousands of pounds, of the farm between 2013 and 2017.
Describe one misleading feature of the graph.
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.
y is inversely proportional to the square root of x. y is 40 when x is 9.
Find a formula linking x and y.
Also involves Standard units and compound measures
4 people take 3 hours to paint a fence.
Assume that all people paint at the same rate.
How long would it take one of these people to paint the same fence?
How long would it take 5 people to paint the same fence? Give your answer in hours and minutes.
(b) ……..….… hours ……..……. minutes
y is directly proportional to the square of x.
Find the percentage increase in y when x is increased by 15%.
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) ?
y is inversely proportional to x. y = 0.04 when x = 80.
Find the value of y when x = 32.
Also involves The four operations · Rounding
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.
Also involves Using a calculator
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.
A graph is drawn below.
Explain how you know that y is not directly proportional to x.
q is directly proportional to r. q is 68 when r is 20.
Work out q when r is 25.
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.
y is inversely proportional to x2 and y = 5 when x = 4.
Find a formula linking x and y.
Also involves Standard units and compound measures
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.
Also involves The four operations
Lillian works 7 hours each day for 5 days a week. She earns £420 each week.
How much does she earn per hour?
Lillian decides that she is going to work 7 hours each day for only 4 days a week. Her earnings are to be reduced by 20%. Lillian thinks that this reduction is reasonable.
Explain why a reduction of 20% is reasonable.
How much will Lillian earn working 4 days a week?
Also involves The four operations · Using a calculator
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.
| Note | Number of notes |
| €50 | 2 |
| €20 | 4 |
| €10 | 9 |
| €5 | 12 |
The exchange rate is £1 = €1.17.
Work out how much he will get in total when he changes these notes.
Also involves Motion graphs · Converting between units
The graph shows the speed, v metres per second (m/s), of a car at time t seconds.
Find the speed of the car at t = 7.
It is claimed that the car has accelerated from 0 to 60 miles per hour in the first 10 seconds.
Does the graph support this claim? Show your reasoning.
Use 1 mile = 1.6 kilometres.
Use the graph to estimate the acceleration at t = 7.
The speed of this car is directly proportional to the square of the time.
Find a formula linking v and t.
Georgina says that the graph shows that the speed of the car will continue to increase after 10 seconds.
Make one comment to show that this statement is incorrect.
Also involves Substitution into formulae · Rearranging formulae
y is inversely proportional to the square of x.
Complete the table.
| x | 10 | 6 | |
| y | 9 | 4 |
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