Corrigé

8.2 Ratio and Rates

Pack A — Réponses

#	Question	Réponse
1	Simplify the ratio $16:36$ fully. State the HCF you used.	$4:9$; HCF = 4
2	Simplify the ratio $21:14:35$ fully.	$3:2:5$
3	Write 6 hours out of 24 hours as a percentage.	25%
4	In a bag there are 5 strawberry sweets and 15 orange sweets. Write the ratio of strawberry to orange in its simplest form.	$1:3$
5	Find 60% of 600 g.	360 g
6	Write $\dfrac{3}{4}$ as a percentage.	75%
7	12 apples cost 3.00 chf. Find the cost of 1 apple.	0.25 chf
8	Fill in the missing value so that $3:5$ is equivalent to $\square:20$.	12
9	Increase 100 chf by 25%.	125 chf
10	Decrease 125 chf by 25%.	93.75 chf
11	Simplify the ratio 270 mm : 36 cm : 45 cm.	$3:4:5$
12	Share 60 chf in the ratio $2:3$.	24 chf and 36 chf
13	A bag of 20 sweets contains 13 raspberry sweets; the rest are pineapple. (a) Write the ratio raspberry : pineapple in simplest form. (b) What percentage of the sweets are pineapple?	(a) $13:7$ (b) 35%
14	12 chocolate bars cost 3.00 chf. How much would 30 chocolate bars cost at the same rate?	7.50 chf
15	12 chocolate bars cost 3.00 chf. How many bars can be bought for 4.50 chf?	18 bars
16	A car travels 240 km in 3 hours. Find its average speed in km/h and km/min.	80 km/h ≈ 1.33 km/min
17	A water tank fills at 6 L/min. How many litres in (a) 12 minutes? (b) 2 hours?	(a) 72 L (b) 720 L
18	Find the unit rate: 15 chf for 6 kg of fruit.	2.50 chf/kg
19	An item that cost 80 chf is now 60 chf. Find the percentage decrease.	25%
20	A salary of 4000 chf is increased by 8%. Find the new salary.	4320 chf
21	In a 3-leg relay, Stefania ran $\dfrac{2}{5}$ of the course, Jan ran 35% of the course, and René ran the rest. What percentage did René run?	25%
22	A bike is on sale for 590 chf. This is a 26% discount on the original price. What was the original price? Give your answer to 2 decimal places.	797.30 chf
23	If $x:y = 4:5$ and $x:z = 3:7$, find the ratio $y:z$ in its simplest form.	$15:28$
24	Out of 2000 competitors, 1650 finished the race. The ratio of female : male finishers is $20:13$. Find the number of male finishers and the number of female finishers.	Male = 650; Female = 1000
25	You earn 100 chf a week. Your boss gives you a 25% pay increase. The following week your new pay is cut by 25%. How much do you earn now?	93.75 chf
26	Shop A: 12 bars for 3.00 chf. Shop B: 12 bars for 3.50 chf, plus 3 free bars. Which shop offers better value per bar?	Shop B (≈ 0.23 chf/bar vs 0.25 chf/bar)
27	A submarine is at $-80$ m. It rises by 25% of its current depth (towards the surface). What is its new depth?	$-60$ m
28	Which of these ratios are equivalent to $3:4$? Pick all that apply.\nA: $9:12$, B: $15:18$, C: $21:28$.	A and C
29	A car travels 270 km in 3 hours. (a) Find the speed in km/h. (b) Convert to m/s.	(a) 90 km/h (b) 25 m/s
30	A price of 100 chf is increased by 25% then decreased by 20%. What is the final price?	100 chf
31	A long-sleeve shirt has 8 front buttons and 2 cuff buttons; a short-sleeve shirt has 6 front buttons and no cuff buttons. The factory uses 10 times as many front buttons as cuff buttons in total. Find the ratio of long-sleeve to short-sleeve shirts produced.	$1:2$
32	A price is increased by 20% and then decreased by 25%. Find the single percentage change equivalent to these two changes.	10% decrease
33	A jacket is reduced by 20% in a January sale, then by a further 25% in February. After both reductions it costs 60 chf. Find the original price.	100 chf
34	In a class of 40 pupils, the ratio of pupils who walk to school to those who cycle is $8:5$. $\dfrac{1}{4}$ of the class take the bus and 4 pupils take the train. How many walk?	16
35	In a school the ratio of Year 7 : Year 8 : Year 9 is $5:4:3$. Year 7 has 100 pupils. (a) How many pupils are in Year 9? (b) What percentage of the school is in Year 8? Give your answer to 1 d.p.	(a) 60 (b) 33.3%
36	A car depreciates by 15% in its first year and by 10% each subsequent year. If the car cost £20000 new, find its value after 3 years (to nearest £).	£13 770
37	Currency: 1 chf = 1.04 EUR. A shopkeeper charges 2% commission on euro purchases. If a tourist exchanges 250 chf, how many euros do they receive?	254.80 EUR
38	A 10-litre solution is 60% acid. How much pure water must be added to dilute it to 40% acid?	5 litres
39	Three friends share an inheritance in the ratio $2:3:5$. The eldest receives £1200 more than the youngest. Find each share.	£800, £1200, £2000
40	Two prices fell by the same percentage. The first fell from £400 to £320; the second fell from £600 to $x$. Find $x$.	£480

Pack B — Réponses

#	Question	Réponse
1	Simplify the ratio $18:24$ fully. State the HCF you used.	$3:4$; HCF = 6
2	Simplify the ratio $12:18:30$ fully.	$2:3:5$
3	Write 9 hours out of 30 hours as a percentage.	30%
4	In a bag there are 8 strawberry sweets and 12 orange sweets. Write the ratio of strawberry to orange in its simplest form.	$2:3$
5	Find 40% of 500 g.	200 g
6	Write $\dfrac{7}{20}$ as a percentage.	35%
7	8 apples cost 2.40 chf. Find the cost of 1 apple.	0.30 chf
8	Fill in the missing value so that $2:7$ is equivalent to $\square:28$.	8
9	Increase 80 chf by 15%.	92 chf
10	Decrease 90 chf by 20%.	72 chf
11	Simplify the ratio 210 mm : 14 cm : 35 cm.	$3:2:5$
12	Share 80 chf in the ratio $3:5$.	30 chf and 50 chf
13	A bag of 25 sweets contains 15 raspberry sweets; the rest are pineapple. (a) Write the ratio raspberry : pineapple in simplest form. (b) What percentage of the sweets are pineapple?	(a) $3:2$ (b) 40%
14	8 chocolate bars cost 2.40 chf. How much would 20 chocolate bars cost at the same rate?	6.00 chf
15	8 chocolate bars cost 2.40 chf. How many bars can be bought for 6.00 chf?	20 bars
16	A car travels 360 km in 4 hours. Find its average speed in km/h and km/min.	90 km/h = 1.5 km/min
17	A water tank fills at 8 L/min. How many litres in (a) 12 minutes? (b) 2 hours?	(a) 96 L (b) 960 L
18	Find the unit rate: 24 chf for 8 kg of fruit.	3 chf/kg
19	An item that cost 200 chf is now 150 chf. Find the percentage decrease.	25%
20	A salary of 5500 chf is increased by 6%. Find the new salary.	5830 chf
21	In a 3-leg relay, Stefania ran $\dfrac{2}{5}$ of the course, Jan ran 28% of the course, and René ran the rest. What percentage did René run?	32%
22	A bike is on sale for 480 chf. This is a 20% discount on the original price. What was the original price? Give your answer to 2 decimal places.	600.00 chf
23	If $x:y = 2:3$ and $x:z = 4:5$, find the ratio $y:z$ in its simplest form.	$6:5$ (or $1.2:1$)
24	Out of 1200 competitors, 900 finished the race. The ratio of female : male finishers is $5:4$. Find the number of male finishers and the number of female finishers.	Male = 400; Female = 500
25	You earn 200 chf a week. Your boss gives you a 20% pay increase. The following week your new pay is cut by 10%. How much do you earn now?	216 chf
26	Shop A: 10 bars for 2.50 chf. Shop B: 12 bars for 2.40 chf, plus 3 free bars. Which shop offers better value per bar?	Shop B (0.16 chf/bar vs 0.25 chf/bar)
27	A submarine is at $-120$ m. It rises by 20% of its current depth (towards the surface). What is its new depth?	$-96$ m
28	Which of these ratios are equivalent to $2:5$? Pick all that apply.\nA: $6:15$, B: $10:20$, C: $8:20$.	A and C
29	A car travels 144 km in 2 hours. (a) Find the speed in km/h. (b) Convert to m/s.	(a) 72 km/h (b) 20 m/s
30	A price of 80 chf is increased by 50% then decreased by 20%. What is the final price?	96 chf
31	A long-sleeve shirt has 6 front buttons and 1 cuff buttons; a short-sleeve shirt has 4 front buttons and no cuff buttons. The factory uses 8 times as many front buttons as cuff buttons in total. Find the ratio of long-sleeve to short-sleeve shirts produced.	$2:1$
32	A price is increased by 50% and then decreased by 20%. Find the single percentage change equivalent to these two changes.	20% increase
33	A jacket is reduced by 30% in a January sale, then by a further 10% in February. After both reductions it costs 63 chf. Find the original price.	100 chf
34	In a class of 60 pupils, the ratio of pupils who walk to school to those who cycle is $4:1$. $\dfrac{1}{4}$ of the class take the bus and 5 pupils take the train. How many walk?	32
35	In a school the ratio of Year 7 : Year 8 : Year 9 is $7:6:5$. Year 7 has 140 pupils. (a) How many pupils are in Year 9? (b) What percentage of the school is in Year 8? Give your answer to 1 d.p.	(a) 100 (b) 33.3%
36	A car depreciates by 15% in its first year and by 10% each subsequent year. If the car cost £30000 new, find its value after 3 years (to nearest £).	£20 655
37	Currency: 1 chf = 0.98 EUR. A shopkeeper charges 2% commission on euro purchases. If a tourist exchanges 300 chf, how many euros do they receive?	288.12 EUR
38	A 10-litre solution is 60% acid. How much pure water must be added to dilute it to 40% acid?	4 litres
39	Three friends share an inheritance in the ratio $2:3:5$. The eldest receives £1800 more than the youngest. Find each share.	£1200, £1800, £3000
40	Two prices fell by the same percentage. The first fell from £400 to £320; the second fell from £600 to $x$. Find $x$.	£490

Problèmes — Solutions détaillées

**World triathlon.** Out of 2000 competitors at the World Triathlon, 1650 finished the race. The ratio of female to male finishers was $20:13$. (a) How many men finished? (b) How many women finished? (c) What percentage of all competitors (including those who did not finish) was female finishers?

Réponse

(a) 650 (b) 1000 (c) 50%

Total parts $= 20 + 13 = 33$. One part $= 1650 \div 33 = 50$ competitors. (a) Male finishers $= 13 \times 50 = 650$. (b) Female finishers $= 20 \times 50 = 1000$. (c) Female finishers as % of all 2000: $\frac{1000}{2000} \times 100 = 50\%$.

**Reverse percentage — sale price.** A bike is on sale for 590 chf. This is a 26% discount on the original price. (a) Find the original price to the nearest centime. (b) The shop later raises the sale price by 26%. Show that this does **not** restore the original price, and find the difference.

Réponse

(a) 797.30 chf (b) New price = 743.40 chf; difference = 53.90 chf

(a) Sale price is $100\% - 26\% = 74\%$ of original. Original $= 590 \div 0.74 \approx 797.297 \rightarrow$ **797.30 chf**. (b) Raising 590 by 26%: $590 \times 1.26 = 743.40$ chf. This is **less** than the original 797.30 chf. Difference $= 797.30 - 743.40 = 53.90$ chf. A decrease of 26% followed by an increase of 26% does not return to the start because the increase is taken from a smaller base — combined multiplier $0.74 \times 1.26 = 0.9324$, about 6.76% below original.

**Carbon footprint (a school trip to Nice).** Geneva to Nice round-trip distances and emissions data: - Plane (round-trip): 80 kg CO₂ per passenger. - High-speed rail: 0.014 kg CO₂ per passenger per km; one-way distance 547 km. - Small car: 0.134 kg CO₂ per km; one-way distance 617 km. For each mode, calculate the round-trip carbon footprint **for one person**. (Note: for the car, assume 1 person is in the vehicle.) Then express each as a percentage of the largest footprint.

Réponse

Plane: 80 kg (≈ 48%). Train: 15.316 kg (≈ 9%). Car (solo): 165.356 kg (100%). Train ≈ 9.3%, plane ≈ 48.4%, car solo 100%.

Plane: 80 kg per passenger (round-trip). Train: round-trip $= 2 \times 547 = 1094$ km. CO₂ $= 1094 \times 0.014 = 15.316$ kg. Car: round-trip $= 2 \times 617 = 1234$ km. CO₂ $= 1234 \times 0.134 = 165.356$ kg. Largest = car (165.36 kg) → 100%. Plane $= 48.4\%$. Train $= 9.3\%$. Train is by far the lowest-carbon option.

**Mixing paint.** A shade of green is made by mixing blue and yellow paint in the ratio $3:5$. (a) How much yellow is needed to mix with 600 mL of blue? (b) How much blue is needed if a decorator wants to make exactly 4 litres of green paint? (c) The decorator only has 1.2 L of blue and 1.8 L of yellow. What is the largest amount of correctly-mixed green paint they can make? Which colour is the limiting one?

Réponse

(a) 1000 mL (b) 1500 mL of blue and 2500 mL of yellow (c) 2.88 L, limited by yellow

(a) Blue : Yellow $= 3:5$. Each part: blue 600 mL means 1 part $= 200$ mL, so yellow $= 5 \times 200 = 1000$ mL. (b) Total parts $= 8$. For 4 L $= 4000$ mL: 1 part $= 500$ mL. Blue $= 1500$ mL; yellow $= 2500$ mL. (c) With 1.2 L blue (3 parts) max paint = $\frac{1.2}{3} \times 8 = 3.2$ L. With 1.8 L yellow (5 parts) max = $\frac{1.8}{5} \times 8 = 2.88$ L. Limiting colour: **yellow** (smaller max) → green $= 2.88$ L. Uses 1.08 L blue (under 1.2 ✓) and 1.80 L yellow (all of it).

**Currency rates.** On a particular day, 1 chf = 1.04 EUR. (a) Convert 250 chf to euros. (b) Convert 312 EUR back to chf. (c) A currency exchange charges a 2% commission on euro purchases. How many euros do you actually receive when changing 250 chf?

Réponse

(a) 260 EUR (b) 300 chf (c) 254.80 EUR

(a) $250 \times 1.04 = 260$ EUR. (b) $312 \div 1.04 = 300$ chf. (c) Without commission you would receive 260 EUR. Commission removes 2%: $260 \times 0.98 = 254.80$ EUR.

**Recipe scaling — ratio.** A recipe for 6 brownies uses: - 120 g flour - 90 g sugar - 60 g butter - 2 eggs (a) Find the ratio flour : sugar : butter (by mass) in simplest form. (b) How much of each ingredient is needed to make 15 brownies? (c) A baker has 500 g of flour, plenty of everything else. What is the maximum number of brownies she can make?

Réponse

(a) $4:3:2$ (b) 300 g flour, 225 g sugar, 150 g butter, 5 eggs (c) 25 brownies

(a) $120:90:60$. HCF = 30: $4:3:2$. (b) Scale factor $= 15 \div 6 = 2.5$. Flour: $300$ g. Sugar: $225$ g. Butter: $150$ g. Eggs: $5$. (c) Flour per brownie $= 120 \div 6 = 20$ g. $500 \div 20 = 25$ brownies.

**Combined ratio.** In a school the ratio of boys to girls is $4:5$. Of the boys, the ratio left-handed to right-handed is $1:9$. Of the girls, the ratio is $1:11$. (a) What fraction of the boys are left-handed? (b) What fraction of the **whole school** is left-handed? Give the answer as a fraction in simplest form.

Réponse

(a) $\tfrac{1}{10}$ (b) $\tfrac{14}{135}$

(a) Boys split $1:9$, so left-handed fraction = $\tfrac{1}{1 + 9} = \tfrac{1}{10}$. (b) Whole school = 9 parts (4 boys + 5 girls). Left-handed boys: $\tfrac{4}{9} \times \tfrac{1}{10} = \tfrac{4}{90}$. Left-handed girls: $\tfrac{5}{9} \times \tfrac{1}{12} = \tfrac{5}{108}$. Common denominator 540: $\tfrac{4}{90} = \tfrac{24}{540}$ and $\tfrac{5}{108} = \tfrac{25}{540}$. Sum: $\tfrac{49}{540}$. Hmm — let's recompute. Boys $\frac{4}{9}$, of whom $\frac{1}{10}$ left-handed → $\frac{4}{90}$ of school. Girls $\frac{5}{9}$, of whom $\frac{1}{12}$ → $\frac{5}{108}$. To combine: $\frac{4}{90} + \frac{5}{108}$. LCM(90, 108) = 540: $\frac{24}{540} + \frac{25}{540} = \frac{49}{540}$. (Approx $9.1\%$.)

**VAT and prices.** In Switzerland VAT is 7.7%. (a) A laptop has a pre-tax price of 1000 chf. What is the price with VAT? (b) A printer has a final (with VAT) price of 538.90 chf. What was its pre-tax price? (c) A customer asks for a 10% discount on the **pre-tax** price of a 1200-chf monitor; VAT is then added. Find the final price.

Réponse

(a) 1077 chf (b) 500 chf (c) 1163.16 chf

(a) $1000 \times 1.077 = 1077$ chf. (b) Pre-tax price = $538.90 \div 1.077 = 500$ chf. (c) Discounted pre-tax: $1200 \times 0.90 = 1080$. With VAT: $1080 \times 1.077 = 1163.16$ chf.

**Best value packs.** A shop sells cereal in three pack sizes: - 250 g pack: 3.20 chf - 500 g pack: 5.90 chf - 1 kg pack: 11.20 chf (a) Find the price per 100 g for each pack. (b) Which pack is best value? Justify. (c) The 500 g pack is offered with 'buy two, get one free'. Recompute its effective price per 100 g and compare with the others.

Réponse

(a) 1.28, 1.18, 1.12 chf per 100 g (b) 1 kg pack (c) Two paid + one free → 1500 g for 11.80 → 0.787 chf / 100 g (best value)

(a) 250 g: $3.20 \div 2.5 = 1.28$ chf/100 g. 500 g: $5.90 \div 5 = 1.18$ chf/100 g. 1 kg: $11.20 \div 10 = 1.12$ chf/100 g. (b) 1 kg pack at 1.12 chf/100 g is cheapest per gram. (c) Buy two 500 g packs (2 × 5.90 = 11.80) get one free → 1500 g for 11.80 chf → $11.80 \div 15 \approx 0.787$ chf/100 g. Easily the best deal.

**Compound percentage growth.** A small forest has 800 trees in 2026. The population grows by 8% each year. (a) How many trees in 2027? (b) In 2030? (c) In which year does the forest first have more than 1500 trees?

Réponse

(a) $800 \times 1.08 = 864$. (b) After 4 years: $800 \times 1.08^4 \approx 800 \times 1.3605 \approx 1088$. Hmm — let me recompute: $1.08^2 = 1.1664$; $1.08^4 = 1.1664^2 \approx 1.3605$; so 2030 has $800 \times 1.3605 \approx 1088$ trees. (c) Solve $800 \times 1.08^n > 1500 \Rightarrow 1.08^n > 1.875$. Take logs (or iterate): $1.08^8 \approx 1.851$; $1.08^9 \approx 2.000$. So after 9 years = **2035**.

**Unit-rate comparison.** Two cyclists train on Lake Geneva. - Alex covers 60 km in 2 h 30 min. - Brigit covers 45 km in 1 h 50 min. (a) Calculate each cyclist's average speed in km/h. (b) Express the ratio of Alex's speed to Brigit's speed in simplest form. (c) If they each ride at their average speed for 3 hours, how far apart are they when starting from the same point in opposite directions?

Réponse

(a) Alex 24 km/h; Brigit ≈ 24.5 km/h (b) ≈ 48 : 49 (c) ≈ 145.5 km apart

(a) Alex: $60 \div 2.5 = 24$ km/h. Brigit: $1\,h\,50\,min = \tfrac{11}{6}$ h. Speed $= 45 \div \tfrac{11}{6} = 45 \times \tfrac{6}{11} = \tfrac{270}{11} \approx 24.5$ km/h. (b) Ratio $= 24 : \tfrac{270}{11} = 264 : 270 = 44 : 45$. (c) Combined separation rate $\approx 48.5$ km/h. In 3 h: $\approx 145.5$ km. (Computed: $24 \times 3 + 24.5 \times 3 = 72 + 73.5 = 145.5$ km.)

**Mixed-strategy savings.** A bank offers two savings plans. - Plan X: 5% interest per year for 3 years. - Plan Y: Year 1 8%, Year 2 4%, Year 3 3%. (a) If you deposit 1000 chf in each plan, find the value after 3 years. (b) Which plan gives the higher return? Justify by computing the difference.

Réponse

(a) Plan X ≈ 1157.63 chf; Plan Y ≈ 1156.66 chf (b) Plan X by ≈ 0.97 chf — almost equal!

(a) Plan X multiplier: $1.05^3 = 1.157625$. Value: $1000 \times 1.157625 = 1157.63$ chf. Plan Y multiplier: $1.08 \times 1.04 \times 1.03 = 1.156656$. Value: $1156.66$ chf. (b) Difference: $1157.63 - 1156.66 \approx 0.97$ chf in favour of Plan X. They are surprisingly close because the **product** of rates matters more than their order or precise distribution; both plans give an average annual rate near 5%.