Problem-solving Pack

Mathematics

Year 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

d = \ldots

linking distance

d

(m) and time

t

(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.

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

k

.
(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?

Working space

5Problem 5 of 12

Investigation. Decide which of these is a proportional relationship and justify by sketching the graph.

(a) Earnings

E

at a flat hourly rate (no fee):

E = 12h

.
(b) Taxi fare:

F = 4 + 1.5d

(flat fee + per km).
(c) Area of a square:

A = s^2

.
(d) Cost of pencils at 30p each:

C = 0.30n

Working space

6Problem 6 of 12

Volume vs surface area. A cube of side

s

has volume

V = s^3

and surface area

S = 6s^2

.

(a) Are

V

and

s

directly proportional? Justify.
(b) Are

S

and

s

directly proportional? Justify.
(c) Find the ratio

V/S

in terms of

s

. What happens to this ratio as

s

grows?

Working space

7Problem 7 of 12

Currency conversion. On a particular day, 1 USD = 0.92 CHF.

(a) Write the conversion equation

C = kU

.
(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

d

, speed

v

and time

t

are linked by

d = vt

.

(a) For a fixed distance of 60 km, complete the table:

|

v

(km/h) | 60 | 30 | 20 | 15 |
|------------|----|----|----|----|
|

t

(h) | ? | ? | ? | ? |

(b) Is

t

directly proportional to

v

? Justify.
(c) Define inverse proportion and write

t

in the form

t = k/v

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

P = P_0 + kt

and identify

P_0

and

k

.
(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.

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

C

(chf) as a function of distance

d

(km).
(b) Find

C

for

d = 320

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?

Working space