A jar contains 8 red counters and 12 blue counters.
(a)Write the ratio of red counters to blue counters in its simplest form.(1)
(b)What fraction of the counters in the jar are red? Give your answer in its simplest form.(1)
3
Priya and Tom share £420 in the ratio 3 : 4. Work out how much each person receives.
(2)
4
Amara, Ben and Chidi share 720 kg of gravel in the ratio 2 : 3 : 4 for a building project. Work out the mass each person receives.
(3)
5
A recipe uses 300 ml of stock and 1.5 litres of water. Write the ratio of stock to water in its simplest form.
(2)
6
Shampoo A costs £2.75 for 250 ml. Shampoo B costs £4.20 for 400 ml. Which shampoo is better value for money? You must show your working.
(3)
7
A recipe for 6 people uses 450 g of rice and 3 eggs.
(a)Work out how much rice is needed to serve 8 people.(2)
(b)Work out how many eggs are needed to serve 15 people. Give a sensible whole number of eggs.(2)
8
It takes 5 builders 12 days to build a wall, all working at the same constant rate. Assuming the same rate of work, how many days would it take 8 builders to build an identical wall?
(3)
9
A train travels 210 km in 2 hours 30 minutes at a constant speed.
(a)Calculate the average speed of the train in km/h.(2)
(b)Assuming the same average speed, find the time taken to travel 350 km. Give your answer in hours and minutes.(2)
10
A metal block has a mass of 540 g and a volume of 60 cm3.
(a)Calculate the density of the metal in g/cm3.(1)
(b)A second block made from the same metal has a volume of 85 cm3. Calculate its mass.(2)
11
The exchange rate is £1 = $1.28.
(a)Convert £250 to dollars.(1)
(b)Convert $192 to pounds sterling. Give your answer to the nearest penny.(2)
12
A sofa has a price of £680 before VAT. VAT is charged at 20%.
(a)Calculate the total price of the sofa including VAT.(2)
(b)In a sale, the total price (including VAT) is reduced by 15%. Calculate the sale price.(2)
13
A map has a scale of 1 : 25000.
(a)The distance between two villages on the map is 8.4 cm. Calculate the real-life distance in kilometres.(2)
(b)A field has an area of 12 cm2 on the map. Calculate the real-life area of the field in km2.(3)
14
A water tank is filled at a constant rate. After 4 minutes it contains 60 litres of water. After 10 minutes it contains 150 litres of water.
(a)Find the rate at which the tank is being filled, in litres per minute.(2)
(b)Find how much water was in the tank at the start, when time = 0 minutes.(2)
15
A shade of paint is made by mixing blue and white paint. Batch 1 mixes blue and white in the ratio 3 : 7. Batch 2 mixes blue and white in the ratio 1 : 4. Grace mixes 400 g of Batch 1 with 500 g of Batch 2 to make a new batch of paint. Find the ratio of blue to white paint in the new mixture, in its simplest form.
(5)
16
A population of rabbits increases by 20% each year. The starting population is 150.
(a)Find the population after 2 years.(2)
(b)Find the single multiplier that could be used to calculate the population after 3 years directly from the starting population.(2)
17
The ratio of Ali's age to Beth's age is 3 : 5. In 4 years' time, the ratio of their ages will be 2 : 3. Find their current ages.
(4)
18
A car uses fuel at a rate of 1 litre per 14 km. Petrol costs £1.48 per litre.
(a)How many litres of fuel are needed for a 420 km journey?(1)
(b)Find the total cost of the fuel for this journey.(2)
19
The braking distance, d metres, of a car is directly proportional to the square of its speed, v mph. When v = 30 mph, d = 15 m. Find the braking distance when v = 60 mph.
(4)
20
Nadia is baking and uses flour, sugar and butter in the ratio 5 : 2 : 3 by mass. She has 600 g of flour, 300 g of sugar and 150 g of butter, and cannot buy any more of any ingredient. Find the maximum total mass of mixture she can make, and find how much of each ingredient will be left over.
(6)
Mark scheme · KS3.M-R1 Ratio, Proportion and Rates of Change
Question 1
B1 3 : 4 cao
Answer: 3 : 4
Question 2
(a) B1 2 : 3 cao
(a) Answer: 2 : 3
(b) B1 2/5 cao oe
(b) Answer: 2/5
Question 3
M1 420 / 7 = 60 (finds value of one part)
A1 Priya = £180 and Tom = £240 cao
Answer: Priya = £180, Tom = £240
Question 4
M1 720 / 9 = 80 (finds value of one part)
A1 Amara = 160 kg and Ben = 240 kg correct
A1 Chidi = 320 kg cao
Answer: Amara = 160 kg, Ben = 240 kg, Chidi = 320 kg
Question 5
M1 1.5 litres = 1500 ml (correct unit conversion)
A1 1 : 5 cao
Answer: 1 : 5
Question 6
M1 Shampoo A: £1.10 per 100 ml (or equivalent unit price)
M1 Shampoo B: £1.05 per 100 ml (or equivalent unit price)
A1 Shampoo B is better value, since it is cheaper per 100 ml, ft from both M marks
Answer: Shampoo B is better value (£1.05 per 100 ml vs £1.10 per 100 ml)
Question 7
(a) M1 450/6 x 8 (correct method, or equivalent scaling)
(a) A1 600 g cao
(a) Answer: 600 g
(b) M1 3/6 x 15 = 7.5 (correct method)
(b) A1 8 eggs, with reason that eggs must be whole, ft oe
(b) Answer: 8 eggs
Question 8
M1 5 x 12 = 60 (total builder-days for the job)
M1 60 / 8 (correct inverse proportion method)
A1 7.5 days cao
Answer: 7.5 days
Question 9
(a) M1 2 h 30 min = 2.5 h (correct time conversion)
(a) A1 84 km/h cao
(a) Answer: 84 km/h
(b) M1 350/84 (correct method, ft from part a)
(b) A1 4 hours 10 minutes cao
(b) Answer: 4 hours 10 minutes
Question 10
(a) B1 9 g/cm3 cao
(a) Answer: 9 g/cm3
(b) M1 9 x 85 (correct method, ft from part a)
(b) A1 765 g cao
(b) Answer: 765 g
Question 11
(a) B1 $320 cao
(a) Answer: $320
(b) M1 192 / 1.28 (correct method)
(b) A1 £150.00 cao
(b) Answer: £150.00
Question 12
(a) M1 680 x 1.2 (or finds 20% of 680 and adds to 680)
(a) A1 £816 cao
(a) Answer: £816
(b) M1 816 x 0.85 (or finds 15% of 816 and subtracts), ft from part a
(b) A1 £693.60 cao
(b) Answer: £693.60
Question 13
(a) M1 8.4 x 25000 = 210000 cm (correct method)
(a) A1 2.1 km cao
(a) Answer: 2.1 km
(b) M1 Area scale factor = 250002 = 625000000 (correct squaring of length scale)
(b) M1 12 x 625000000 = 7500000000 cm2 (correct application to map area)
(b) A1 0.75 km2 cao oe awrt
(b) Answer: 0.75 km2
Question 14
(a) M1 (150-60)/(10-4) (correct method for rate of change/gradient)
(a) A1 15 litres per minute cao
(a) Answer: 15 litres per minute
(b) M1 60 - 4 x 15 (correct method, ft from part a)
(b) A1 0 litres cao
(b) Answer: 0 litres
Question 15
M1 Batch 1: blue = 120 g, white = 280 g
M1 Batch 2: blue = 100 g, white = 400 g
M1 Total blue = 220 g, total white = 680 g
A1 Ratio 220 : 680 seen (or correct unsimplified equivalent)
A1 11 : 34 cao oe
Answer: 11 : 34
Question 16
(a) M1 150 x 1.2 x 1.2 (or 150 x 1.22), correct method
(a) A1 216 cao
(a) Answer: 216 rabbits
(b) M1 1.23 seen (correct method)
(b) A1 1.728 cao
(b) Answer: 1.728
Question 17
M1 Sets up ages as 3x and 5x and forms the equation (3x+4)/(5x+4) = 2/3
M1 Correctly cross-multiplies and simplifies to 9x+12 = 10x+8 or equivalent
A1 x = 4
A1 Ali = 12 and Beth = 20 cao
Answer: Ali = 12 years old, Beth = 20 years old
Question 18
(a) B1 30 litres cao
(a) Answer: 30 litres
(b) M1 30 x 1.48 (correct method, ft from part a)
(b) A1 £44.40 cao
(b) Answer: £44.40
Question 19
M1 Sets up d = k x v2
A1 k = 1/60 oe (e.g. 0.0167)
M1 Substitutes v = 60 into d = k x v2
A1 60 m cao
Answer: 60 m
Question 20
M1 Expresses amounts used as 5x (flour), 2x (sugar), 3x (butter) for some x
M1 Finds the maximum x allowed by each ingredient: x≤120 (flour), x≤150 (sugar), x≤50 (butter)
A1 Correctly identifies butter as the limiting ingredient, x = 50