Year 9 Maths • AC9M9M05 • Authored homework
Direct proportion, scale and rates
Use direct proportion, scale factors and rates to model practical situations, compare options and interpret answers in context. Show working, use correct units or notation, and check reasonableness.
Part A
Short-answer questions
What does direct proportion mean?
If 5 kg of apples cost $18, find the cost per kilogram.
A map scale is 1:50 000. What real distance does 3 cm represent?
A car travels 180 km in 2.5 hours. Find its average speed.
Scale the ratio 2:5 so the first part is 14.
Explain why a graph of direct proportion passes through the origin.
Convert 72 km/h to metres per second.
Find the constant of proportionality if y = kx and y = 24 when x = 6.
Correct this error: doubling one quantity always means direct proportion.
Give one real situation modelled by a rate.
Part B
Long-answer questions
A recipe for 4 people uses 300 g of rice. Scale the recipe for 10 people and explain the proportional reasoning.
A model car has scale 1:24. If the model is 18 cm long, find the real car length in metres.
Compare two internet plans using cost per gigabyte and explain which is better value for different users.
A tap fills 18 L in 3 minutes. Model the amount of water after t minutes and use it to find the time for 75 L.
A cyclist travels 42 km in 1.75 hours. Calculate average speed and interpret the answer.
A map shows two towns 7.6 cm apart with scale 1:250 000. Find the real distance and express it in kilometres.
Explain the difference between direct proportion and a relationship with a fixed starting cost. Use examples.
Create a direct-proportion table and graph, then identify the constant of proportionality.
Solve a multi-step travel problem involving speed, time, distance and a rest break.
Write a full modelling response for a proportional situation, including assumptions, equation, calculation and interpretation.
Part C
Research and understanding task
Research a practical rate or scale context, such as fuel use, map scale, recipe scaling, exchange rates, speed, population density or printing scale. Build a proportional model and solve one realistic problem.