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Problem Solving Introduction

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Problem Solving Introduction

Use the scenario and data below to practise understanding, planning, solving and checking. Show enough working for someone else to follow your reasoning. If a question has more than one valid answer, give one and explain why it works.

Scenario: a low-waste community event

A group is planning food for an event in Aotearoa New Zealand. They have a budget of $180. Pies cost $6 each and serve one person; sandwich packs cost $9 each and serve two people. They need at least 24 servings. Let p be the number of pies and s the number of sandwich packs. Assume p and s are whole numbers greater than or equal to zero.

Item

Cost

Servings

Pie

Sandwich pack

Part A: Understand and plan

Start by translating the situation into mathematical conditions before trying combinations.

1.Which pair of inequalities correctly represents the budget and minimum-servings conditions? Choose one.

6p + 9s ≤ 180 and p + 2s ≥ 24
6p + 9s ≥ 180 and p + 2s ≤ 24
9p + 6s ≤ 180 and 2p + s ≥ 24
6p + 9s = 180 and p + 2s = 24

2.In your own words, explain what the variables p and s represent, and why they must be whole numbers.

3.A volunteer suggests buying 12 pies and 6 sandwich packs. Test this proposal against both conditions. State whether it works and show the cost and servings.

4.Determine whether a combination can provide exactly 24 servings and cost exactly $180. Show how you set up and solve the equations.

Part B: Solve, compare and check

There may be several workable options. A good solution also considers what the organisers value most.

5.Is buying 18 pies and 3 sandwich packs a valid plan? Select true or false, then justify your choice using both conditions.

TrueFalse

6.The group wants to minimise spending while providing at least 24 servings. What is the minimum possible cost? Give a combination that achieves it and explain why the cost cannot be lower.

7.Another organiser proposes 6 pies and 9 sandwich packs. Check the proposal. If it is valid, calculate its cost and servings; if not, identify which condition it fails.

8.A teammate says, “Because 12 sandwich packs cost $108, that must be the best plan in every way.” Explain one sense in which this plan is best, and one reason the group might still choose a different valid plan.

9.Fill in the missing words: A mathematical model is useful because it simplifies a real situation, but its results should be checked against the original and any assumptions made.

10.Imagine the budget changes to $120, but the group still needs at least 24 servings. Give one valid combination and describe a systematic way to search for other combinations without testing randomly.