Printable · GCSE Foundation · ages 14-16
Ratio, proportion and rates of change worksheet — GCSE Foundation
Fifteen questions across the ratio, proportion and rates of change statements at Foundation tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
Non-calculator
Ratio, proportion and rates of change worksheet — GCSE Foundation
MathsUKwww.geekhero.co.uk
- 1.A taxi journey of m miles costs C pounds, where C = 2.5m + 3. A driver says the ratio C : m is the same for every journey. Is the driver correct? Give a reason for your answer.
- 2.Work out the time taken to travel 120 km at an average speed of 60 km/h.
- 3.£120 is shared between three cousins in the ratio 3:4:5. Work out the largest share.
- 4.Jamal is paid £54 for working 6 hours. Work out his rate of pay, in £ per hour.
- 5.£2000 is invested in a savings account that pays 5% compound interest each year. Work out the value of the investment at the end of 2 years.
- 6.Square P has sides of length 3 cm. Square Q has sides of length 12 cm. Write the ratio of the area of square P to the area of square Q in its simplest form.
- 7.The ratio of the amount Noah has saved to the amount Grace has saved is 4:5. Noah has saved £200. Work out how much they have saved altogether.
- 8.Divide 84 in the ratio 3:4. Work out the smaller share.
- 9.A scale drawing of a garden uses a scale of 1 : 20. Write down a formula for the real length, L metres, in terms of the length on the drawing, d centimetres, given that 1 metre = 100 centimetres.
- 10.A garden plan is enlarged by a scale factor of 3. A flower bed on the original plan is 5 cm long. Work out the length of the flower bed on the enlarged plan, in centimetres.
- 11.Freya buys three items whose prices are in the ratio 2:3:5. Altogether she pays £400. Work out the price of the most expensive item.
- 12.Write 1/5 as a percentage.
- 13.A force of 250 N acts on an area of 0.5 m². Using pressure = force ÷ area, work out the pressure, in pascals.
- 14.In a bag the ratio of red counters to blue counters is 7 : 4. Write the number of red counters as a fraction of the number of blue counters.
- 15.Order these three values from smallest to largest: 3/8, 0.43, 41%.
Answer key
- (b) The ratio C : m is not constant because the formula includes a fixed charge of £3 as well as the charge per mile. — For the ratio C : m to stay constant, C must be directly proportional to m, i.e. C = km with no constant term. Because of the +3 fixed charge, C is not directly proportional to m: for example m = 1 gives C = 5.5 (ratio 5.5 : 1), while m = 10 gives C = 28 (ratio 2.8 : 1) — the ratio has changed.
- (c) 2 hours — Method: time = distance ÷ speed, and the units of the speed fix the unit of the time. Working: 120 ÷ 60 = 2, and because the speed is in kilometres per hour the time is in hours. Answer: 2 hours. The distractors: 0.5 hours comes from dividing the speed by the distance, 60 ÷ 120, the division the wrong way round; 2 minutes comes from the correct division but the wrong unit, reading kilometres per hour as kilometres per minute; 180 minutes comes from adding 120 and 60 instead of dividing.
- (b) £50 — Method: add the parts of the ratio, divide the amount by the number of parts to find the value of one part, then multiply by the parts in the largest share. Working: 3 + 4 + 5 = 12 parts, £120 ÷ 12 = £10 for one part, and the largest share is 5 parts, so 5 × £10 = £50. Answer: £50. The distractors: £10 is the value of one part only; £30 is the 3-part share, which is the smallest one; £40 is the 4-part share, the middle one.
- (a) £9.00 — Rate of pay = total pay ÷ number of hours. £54 ÷ 6 = £9.00 per hour. £324.00 comes from multiplying the pay by the hours instead of dividing (£54 × 6). £0.11 comes from dividing the hours by the pay instead of the pay by the hours (6 ÷ 54). £48.00 comes from subtracting the hours from the pay (£54 − 6) instead of dividing.
- (b) £2205.00 — With compound interest each year's interest is worked out on the value at the start of that year, so a 5% rise is a multiplier of 1.05 applied once per year. After the first year: 2000 × 1.05 = 2100. After the second year: 2100 × 1.05 = 2205. The question asks for the value of the investment, not for the interest earned, so the answer is £2205.00.
- (c) 1 : 16 — Work out each area before comparing. Square P has area 3 × 3 = 9 cm². Square Q has area 12 × 12 = 144 cm². The ratio of the areas is 9 : 144, and both parts divide by 9: 9 ÷ 9 = 1 and 144 ÷ 9 = 16, giving 1 : 16. Notice that the sides are in the ratio 1 : 4, and the areas are in the ratio of the squares of those parts, which is what always happens when a length is scaled. Stopping at 1 : 4 would compare the sides and never the areas, and cubing the parts to reach 1 : 64 is the rule for volumes rather than for areas. The order matters too: the question names square P first, so its area must be the first part of the ratio.
- (c) £450 — Method: use the equal ratios 4:5 = 200:x to find Grace's savings, then add the two amounts. Working: Noah's £200 is 4 parts, so one part is £200 ÷ 4 = £50; Grace has 5 parts, so 5 × £50 = £250; altogether £200 + £250 = £450. Answer: £450. The distractors: £250 is Grace's savings on their own, which is the middle step rather than the total the question asks for; £360 comes from reading £200 as the 5 parts instead of the 4, giving one part of £40 and a total of 9 × £40; £400 comes from doubling £200, which treats the two savings as equal and ignores the ratio altogether.
- (c) 36 — Add the parts of the ratio: 3 + 4 = 7. Divide the total by the number of parts: 84 ÷ 7 = 12, so one part is worth 12. The smaller share has 3 parts: 3 × 12 = 36. (48 is the larger share, using 4 parts instead of 3. 42 comes from splitting 84 in half, treating the ratio as if it were 1:1. 28 comes from dividing 84 by 3 — one of the ratio numbers — instead of dividing by the total number of parts, 7.)
- (c) L = d/5 — The scale 1 : 20 means each cm on the drawing represents 20 cm in real life, so the real length in cm is 20d. Converting to metres by dividing by 100: L = 20d/100 = d/5.
- (d) 15 cm — An enlargement by scale factor 3 means every length is multiplied by 3: 5 × 3 = 15 cm. Dividing by the scale factor instead, 5 ÷ 3 = 1.7 (to 1 d.p.), gives 1.7 cm — dividing shrinks the plan rather than enlarging it. Adding the scale factor instead of multiplying, 5 + 3 = 8, gives 8 cm. Squaring the scale factor, as you would for an area, 5 × 3² = 45, gives 45 cm — length scales by the factor itself, not its square.
- (d) £200 — Method: add the parts of the ratio, divide the total paid by the number of parts to find the value of one part, then multiply by the parts in the most expensive item. Working: 2 + 3 + 5 = 10 parts, £400 ÷ 10 = £40 for one part, and the most expensive item is 5 parts, so 5 × £40 = £200. Answer: £200. The distractors: £40 is the value of one part; £80 is the 2-part item, the cheapest of the three; £120 is the 3-part item.
- (a) 20% — Method: a percentage is a number of parts per hundred, so divide the numerator by the denominator and multiply by 100. Working: 1 ÷ 5 = 0.2, and 0.2 × 100 = 20; the same result comes from scaling 1/5 to 20/100. Answer: 20%. The distractors: 5% comes from reading the denominator 5 straight off as the percentage; 25% comes from confusing 1/5 with 1/4, the fraction met most often in percentage work; 50% comes from a place value slip in the division, writing 1 ÷ 5 as 0.5 instead of 0.2.
- (c) 500 — Pressure = force ÷ area = 250 ÷ 0.5 = 500 pascals. Getting 125 comes from multiplying the force by the area instead of dividing (250 × 0.5 = 125). Getting 249.5 comes from subtracting the area from the force instead of dividing. Getting 50 comes from misreading the area as 5 m² instead of 0.5 m² and dividing 250 by 5.
- (d) 7/4 — A part-to-part ratio a : b gives the fraction a/b when the first quantity is written as a fraction of the second, so 7 : 4 gives 7/4. Writing 4/7 puts the parts the wrong way round — blue as a fraction of red, not red as a fraction of blue. Writing 7/11 uses the total number of counters, 7 + 4 = 11, as the denominator instead of the number of blue counters — that is red as a fraction of the whole bag, not red as a fraction of blue. Writing 11/7 has both the wrong denominator and the parts inverted.
- (a) 3/8, 41%, 0.43 — Method: convert every value to a decimal so they can be compared on the same scale, then order them. Working: 3/8 = 0.375, 41% = 0.41, and 0.43 stays as 0.43, so from smallest to largest the decimals are 0.375, 0.41, 0.43, giving the order 3/8, 41%, 0.43. Answer: 3/8, 41%, 0.43. The order 3/8, 0.43, 41% comes from comparing 0.43 and 41% as raw digits (43 versus 41) without converting 41% into the decimal 0.41 first, wrongly placing 0.43 before 41%. The order 0.43, 41%, 3/8 comes from placing the values in completely reversed order, from largest to smallest instead of smallest to largest. The order 41%, 0.43, 3/8 comes from ordering the values by their TYPE (percentage, then decimal, then fraction) rather than by their actual size.
Build your own mix at the worksheet builder.