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Advanced Proportional Reasoning Challenges

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Advanced Proportional Reasoning Challenges

Show your working for calculation questions. Include units where they are given.

Number Machine Patterns

Use the input-output information to identify each rule.

1.A sorting machine turns 3 into 9, 5 into 15 and 7 into 21. Ruby-Mae says an input of 9 will produce 24. Explain the mistake in her prediction and find the correct output.

2.A machine gives these results: input 3 produces 15, input 5 produces 35, and input 7 produces 63. The output is the input multiplied by the next odd number after it. What input produces an output of 143?

3.A machine has been tested with inputs 2, 4 and 6, producing outputs 6, 14 and 22. One proposed rule is “multiply the input by 4, then subtract 2.” Another is “add 6 to the input, then multiply by 2.” Which rule fits all three tests? Show how you know.

Multi-Step Proportional Reasoning

Track each change in order. Take care when a percentage is increased or reduced.

4.A processing line starts with 160 cans. Its first stage increases the count by 25%, the next stage doubles it, and a final inspection removes 20% of the cans. How many cans leave the line? A worker got 200 cans. Describe a likely error and calculate the correct result.

Algebra and Number Machines

Use the machine rule to evaluate an input and solve for an unknown input.

5.A machine’s processing time is given by T = 3w + w² ÷ 4, where T is time in minutes and w is the batch mass in kilograms. Check the rule for a 4 kg batch, then find the positive batch mass that takes exactly 40 minutes.

Apply Your Reasoning

Use rates and proportional changes to solve the real-world problem.

6.A plant sorts 240 tonnes of material in a 4-hour morning shift. In the afternoon its hourly rate is 25% lower than in the morning. If the plant must sort 510 tonnes altogether, how many hours must the afternoon shift run? Assume the morning rate stays constant for all 4 hours.

7.A delivery service charges $12 per package. During a busy period, its rate increases by 25%. If a customer sends 8 packages at the increased rate, how much will they pay? Show how you calculate the new rate and total cost.