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Problem-solving Pack
MathematicsYear 8 · Ratio and Proportion
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 36
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 36
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 36
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.
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4Problem 4 of 36
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?
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5Problem 5 of 36
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 36
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?
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7Problem 7 of 36
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 36
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 36
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 36
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 36
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 36
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
13Problem 13 of 36
Direct proportion + a graph. Distance is directly proportional to time for a rollercoaster: in 2 minutes it travels 36 m.
(a) Find the unit rate in metres per minute.
(b) Write an equation linking distance (m) and time (minutes).
(c) Use your equation to find how far the rollercoaster travels in 12 minutes.
(d) The rollercoaster's track is 540 m long. How long does it take to traverse the full track at constant speed?
(a) Find the unit rate in metres per minute.
(b) Write an equation linking distance (m) and time (minutes).
(c) Use your equation to find how far the rollercoaster travels in 12 minutes.
(d) The rollercoaster's track is 540 m long. How long does it take to traverse the full track at constant speed?
Working space
14Problem 14 of 36
Tour de Suisse training. A cyclist trains over a 4-week plan, increasing her weekly distance by 20% each week. In week 1 she covers 80 km.
(a) How far does she cycle in week 4?
(b) Find her total distance over the four weeks (to the nearest km).
(c) Express her week-4 distance as a percentage of her week-1 distance.
(d) Is the relationship between distance and week number proportional? Justify.
(a) How far does she cycle in week 4?
(b) Find her total distance over the four weeks (to the nearest km).
(c) Express her week-4 distance as a percentage of her week-1 distance.
(d) Is the relationship between distance and week number proportional? Justify.
Working space
15Problem 15 of 36
Testing proportionality. A student records distances driven by a car against fuel used:
| Fuel (L) | 5 | 10 | 15 | 20 |
|----------|---|----|----|----|
| Distance (km) | 60 | 120 | 180 | 240 |
(a) Show that distance is proportional to fuel and find .
(b) Write the equation.
(c) Predict the distance for 28 L of fuel.
(d) How much fuel is needed for 300 km?
| Fuel (L) | 5 | 10 | 15 | 20 |
|----------|---|----|----|----|
| Distance (km) | 60 | 120 | 180 | 240 |
(a) Show that distance is proportional to fuel and find .
(b) Write the equation.
(c) Predict the distance for 28 L of fuel.
(d) How much fuel is needed for 300 km?
Working space
16Problem 16 of 36
Recipe per person. A recipe to feed 8 people uses: 480 g flour, 320 g sugar, 6 eggs.
(a) Find the amount of each ingredient per person.
(b) How much of each is needed for 12 people?
(c) A chef has 1 kg flour, 800 g sugar, 10 eggs. What is the limiting ingredient if she wants to scale the recipe up exactly?
(a) Find the amount of each ingredient per person.
(b) How much of each is needed for 12 people?
(c) A chef has 1 kg flour, 800 g sugar, 10 eggs. What is the limiting ingredient if she wants to scale the recipe up exactly?
Working space
17Problem 17 of 36
Investigation. Decide which of these is a proportional relationship and justify by sketching the graph.
(a) Earnings at a flat hourly rate (no fee): .
(b) Taxi fare: (flat fee + per km).
(c) Area of a square: .
(d) Cost of pencils at 30p each: .
(a) Earnings at a flat hourly rate (no fee): .
(b) Taxi fare: (flat fee + per km).
(c) Area of a square: .
(d) Cost of pencils at 30p each: .
Working space
18Problem 18 of 36
Volume vs surface area. A cube of side has volume and surface area .
(a) Are and directly proportional? Justify.
(b) Are and directly proportional? Justify.
(c) Find the ratio in terms of . What happens to this ratio as grows?
(a) Are and directly proportional? Justify.
(b) Are and directly proportional? Justify.
(c) Find the ratio in terms of . What happens to this ratio as grows?
Working space
19Problem 19 of 36
Currency conversion. On a particular day, 1 USD = 0.92 CHF.
(a) Write the conversion equation .
(b) Convert 250 USD to CHF.
(c) Convert 460 CHF to USD.
(d) Is this a proportional relationship? Justify.
(a) Write the conversion equation .
(b) Convert 250 USD to CHF.
(c) Convert 460 CHF to USD.
(d) Is this a proportional relationship? Justify.
Working space
20Problem 20 of 36
Investigating inverse relationships. Distance , speed and time are linked by .
(a) For a fixed distance of 60 km, complete the table:
| (km/h) | 60 | 30 | 20 | 15 |
|------------|----|----|----|----|
| (h) | ? | ? | ? | ? |
(b) Is directly proportional to ? Justify.
(c) Define inverse proportion and write in the form .
(a) For a fixed distance of 60 km, complete the table:
| (km/h) | 60 | 30 | 20 | 15 |
|------------|----|----|----|----|
| (h) | ? | ? | ? | ? |
(b) Is directly proportional to ? Justify.
(c) Define inverse proportion and write in the form .
Working space
21Problem 21 of 36
Population growth. A town's population grows from 12 000 to 15 000 over 5 years. Assume linear (proportional) growth.
(a) Find the average annual growth rate (in absolute numbers).
(b) Write an equation and identify and .
(c) Estimate the population in 10 years.
(d) Is percentage growth proportional? Justify.
(a) Find the average annual growth rate (in absolute numbers).
(b) Write an equation and identify and .
(c) Estimate the population in 10 years.
(d) Is percentage growth proportional? Justify.
Working space
22Problem 22 of 36
Best buy investigation. A market sells potatoes:
- 1 kg sack: 2.20 chf
- 2.5 kg sack: 5.25 chf
- 5 kg sack: 9.50 chf
(a) Find the price per kg for each pack.
(b) Plot 'price' against 'mass'. Does the relationship form a straight line through the origin? Discuss.
(c) Which sack is best value per kg?
(d) Suggest a reason why larger sacks are cheaper per kg.
- 1 kg sack: 2.20 chf
- 2.5 kg sack: 5.25 chf
- 5 kg sack: 9.50 chf
(a) Find the price per kg for each pack.
(b) Plot 'price' against 'mass'. Does the relationship form a straight line through the origin? Discuss.
(c) Which sack is best value per kg?
(d) Suggest a reason why larger sacks are cheaper per kg.
Working space
23Problem 23 of 36
Petrol cost. Petrol costs 1.80 chf per litre. Your car uses 7 litres per 100 km.
(a) Write a formula for petrol cost (chf) as a function of distance (km).
(b) Find for km.
(c) If petrol rises to 2.00 chf/L, find the new cost for 320 km.
(d) For a 600-km trip, find the change in cost between the old and new petrol prices.
(a) Write a formula for petrol cost (chf) as a function of distance (km).
(b) Find for km.
(c) If petrol rises to 2.00 chf/L, find the new cost for 320 km.
(d) For a 600-km trip, find the change in cost between the old and new petrol prices.
Working space
24Problem 24 of 36
Hours and earnings. A part-time worker is paid 18 chf/h for the first 20 hours per week and 25 chf/h for any extra hours.
(a) Is total weekly pay directly proportional to hours worked? Justify.
(b) Plot weekly pay versus hours from 0 to 30 hours.
(c) The worker earns 510 chf one week. How many hours did she work?
(a) Is total weekly pay directly proportional to hours worked? Justify.
(b) Plot weekly pay versus hours from 0 to 30 hours.
(c) The worker earns 510 chf one week. How many hours did she work?
Working space
25Problem 25 of 36
Glacial scale (CdN November 2025). On a 1960 map with scale cm : m, the rectangle outlining a glacier measures 6 cm by 9 cm. On a 2010 map with scale cm : m, the rectangle outlining the same glacier measures 4 cm by 6 cm.
(a) Find the real dimensions of each rectangle in metres.
(b) Find the real area of each rectangle in square kilometres.
(c) Find the ratio Area(R1) : Area(R2) in simplest form.
(d) Find the linear scale factor between R1 and R2.
(e) Use a linear extrapolation to predict whether the glacier disappears by 2035.
(a) Find the real dimensions of each rectangle in metres.
(b) Find the real area of each rectangle in square kilometres.
(c) Find the ratio Area(R1) : Area(R2) in simplest form.
(d) Find the linear scale factor between R1 and R2.
(e) Use a linear extrapolation to predict whether the glacier disappears by 2035.
Working space
26Problem 26 of 36
Floor plan (1 : 50). A floor plan at 1 cm : 50 cm shows a room measuring 8 cm × 6 cm.
(a) Find the real dimensions.
(b) Find the real floor area in m².
(c) The plan also shows a circular table (radius 0.4 cm on the plan). Find its real radius and area.
(d) Show that the plan-area : real-area ratio is .
(a) Find the real dimensions.
(b) Find the real floor area in m².
(c) The plan also shows a circular table (radius 0.4 cm on the plan). Find its real radius and area.
(d) Show that the plan-area : real-area ratio is .
Working space
27Problem 27 of 36
Shadow stick. A 1.5 m vertical stick casts a 1 m shadow at the same time of day that a tree casts a 7 m shadow.
(a) Why are the shadow triangles similar?
(b) Find the tree height.
(c) The shadow of a flagpole is measured later in the day at 5 m, when the stick now casts a 2 m shadow. Find the flagpole height.
(a) Why are the shadow triangles similar?
(b) Find the tree height.
(c) The shadow of a flagpole is measured later in the day at 5 m, when the stick now casts a 2 m shadow. Find the flagpole height.
Working space
28Problem 28 of 36
Similar pictures. A photograph is 10 cm × 15 cm. A larger printed copy is similar and has area 600 cm².
(a) Find the linear scale factor.
(b) Find the larger picture's dimensions.
(c) If the smaller picture costs 4 chf and price is proportional to area, find the cost of the larger picture.
(a) Find the linear scale factor.
(b) Find the larger picture's dimensions.
(c) If the smaller picture costs 4 chf and price is proportional to area, find the cost of the larger picture.
Working space
29Problem 29 of 36
Scale-model car. A scale model of a car is 1 : 24. The real car is 4.8 m long, 1.8 m wide and 1.5 m tall.
(a) Find the model's dimensions in cm.
(b) Find the surface-area ratio (model : real) and the model's surface area if the real is 36 m².
(c) Find the volume ratio (model : real) and the model's volume if the real has interior volume 12 m³.
(a) Find the model's dimensions in cm.
(b) Find the surface-area ratio (model : real) and the model's surface area if the real is 36 m².
(c) Find the volume ratio (model : real) and the model's volume if the real has interior volume 12 m³.
Working space
30Problem 30 of 36
Designing a school garden. A scale drawing of a school garden uses 1 cm : 2 m.
(a) On the drawing, a flowerbed is a triangle with vertices (1 cm, 1 cm), (5 cm, 1 cm), (1 cm, 4 cm). Find the real vertices in metres (relative to the same origin).
(b) Find the real area of the flowerbed.
(c) A path of width 0.5 m surrounds the flowerbed on its three straight sides. Find the path width on the drawing.
(a) On the drawing, a flowerbed is a triangle with vertices (1 cm, 1 cm), (5 cm, 1 cm), (1 cm, 4 cm). Find the real vertices in metres (relative to the same origin).
(b) Find the real area of the flowerbed.
(c) A path of width 0.5 m surrounds the flowerbed on its three straight sides. Find the path width on the drawing.
Working space
31Problem 31 of 36
Similar shapes investigation. Two similar shapes have areas 50 cm² and 200 cm².
(a) Find the linear scale factor between them.
(b) The smaller has perimeter 30 cm. Find the larger perimeter.
(c) The smaller has 3 cm of trim around its border. Find the trim length on the larger shape.
(a) Find the linear scale factor between them.
(b) The smaller has perimeter 30 cm. Find the larger perimeter.
(c) The smaller has 3 cm of trim around its border. Find the trim length on the larger shape.
Working space
32Problem 32 of 36
Map of Geneva. On a 1 : 50 000 map, Cornavin station is 12 cm from UN headquarters along a straight line.
(a) Find the real distance in metres and kilometres.
(b) On a different map at 1 : 25 000, what would the same distance measure on the map?
(c) The area of Geneva inside the city limits is approximately 16 km². How much area would this occupy on the 1 : 50 000 map (in cm²)?
(a) Find the real distance in metres and kilometres.
(b) On a different map at 1 : 25 000, what would the same distance measure on the map?
(c) The area of Geneva inside the city limits is approximately 16 km². How much area would this occupy on the 1 : 50 000 map (in cm²)?
Working space
33Problem 33 of 36
Doll-house furniture. A doll house is 1 : 12. A real chair is 90 cm tall.
(a) Find the doll-house chair height in cm.
(b) A real living-room rug is 2.4 m × 1.8 m. Find the dimensions of the doll-house version.
(c) The real chair weighs 8 kg. Assuming the doll-house chair is made of similar material at the same density, estimate its mass.
(a) Find the doll-house chair height in cm.
(b) A real living-room rug is 2.4 m × 1.8 m. Find the dimensions of the doll-house version.
(c) The real chair weighs 8 kg. Assuming the doll-house chair is made of similar material at the same density, estimate its mass.
Working space
34Problem 34 of 36
Two similar boxes. Two cuboid boxes are similar. The smaller has dimensions and volume . The larger has linear scale factor relative to the smaller.
(a) Express the larger volume in terms of and .
(b) The smaller box has volume 250 cm³. The larger box has volume 2000 cm³. Find .
(c) The smaller has surface area 200 cm². Find the surface area of the larger.
(a) Express the larger volume in terms of and .
(b) The smaller box has volume 250 cm³. The larger box has volume 2000 cm³. Find .
(c) The smaller has surface area 200 cm². Find the surface area of the larger.
Working space
35Problem 35 of 36
Building's shadow. The Manhattan skyscraper "One World Trade Center" is 541 m tall. At noon, its shadow is 380 m long. A pedestrian 1.7 m tall is nearby.
(a) Find the angle of elevation of the sun to 1 d.p.
(b) Find the pedestrian's shadow length at the same time (assume parallel sunlight).
(c) Demonstrate that the shadow triangles are similar.
(a) Find the angle of elevation of the sun to 1 d.p.
(b) Find the pedestrian's shadow length at the same time (assume parallel sunlight).
(c) Demonstrate that the shadow triangles are similar.
Working space
36Problem 36 of 36
Cake tin scaling. A round cake tin has diameter 20 cm and depth 5 cm. A similar (linearly enlarged) cake tin is made with diameter 30 cm.
(a) Find the linear scale factor.
(b) Find the depth of the new tin.
(c) The volume of cake batter required to fill each tin is proportional to its volume. The small tin needs 1.5 litres. Find the volume for the larger tin.
(a) Find the linear scale factor.
(b) Find the depth of the new tin.
(c) The volume of cake batter required to fill each tin is proportional to its volume. The small tin needs 1.5 litres. Find the volume for the larger tin.
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