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Problem-solving Pack
MathematicsYear 8 · 8.2 Ratio and Rates
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
World triathlon. Out of 2000 competitors at the World Triathlon, 1650 finished the race. The ratio of female to male finishers was .
(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?
(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?
Working space
2Problem 2 of 12
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.
(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.
Working space
3Problem 3 of 12
Carbon footprint (CdN 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.
- 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.
Working space
4Problem 4 of 12
Mixing paint. A shade of green is made by mixing blue and yellow paint in the ratio .
(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?
(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?
Working space
5Problem 5 of 12
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?
(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?
Working space
6Problem 6 of 12
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?
- 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?
Working space
7Problem 7 of 12
Combined ratio. In a school the ratio of boys to girls is . Of the boys, the ratio left-handed to right-handed is . Of the girls, the ratio is .
(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.
(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.
Working space
8Problem 8 of 12
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.
(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.
Working space
9Problem 9 of 12
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.
- 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.
Working space
10Problem 10 of 12
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?
(a) How many trees in 2027?
(b) In 2030?
(c) In which year does the forest first have more than 1500 trees?
Working space
11Problem 11 of 12
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?
- 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?
Working space
12Problem 12 of 12
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.
- 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.
Working space