← Dossiers de révision
Problem-solving Pack
MathematicsYear 8 · 8.3 Proportions
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
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
2Problem 2 of 12
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
3Problem 3 of 12
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
4Problem 4 of 12
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
5Problem 5 of 12
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
6Problem 6 of 12
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
7Problem 7 of 12
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
8Problem 8 of 12
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
9Problem 9 of 12
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
10Problem 10 of 12
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
11Problem 11 of 12
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
12Problem 12 of 12
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