Introduction to Searching for Solutions

Welcome to one of the most practical parts of the Thinking Skills course! In the section "Searching, scheduling and optimising," we look at how to find answers when the solution isn't immediately obvious. In this chapter, Search for solutions in simple situations, we focus on scenarios where you have a limited amount of information and need to find a specific result that fits certain rules.

Think of this like a puzzle. You have all the pieces, but you need to figure out how they fit together to reach a goal. Whether it’s figuring out how many stamps you need for a letter or finding the right combination of coins to pay for a snack, you are "searching" for a solution. Don't worry if this seems tricky at first—once you have a systematic approach, it becomes much easier!

What Does "Searching for a Solution" Mean?

In Paper 1 (Problem Solving), you will often be given a set of rules (a model) and a goal. "Searching" means testing different possibilities until you find one that works. In simple situations, the number of possibilities is usually small, so you can check them one by one without getting overwhelmed.

Example: If a shop sells notebooks for \( \$3 \) and pens for \( \$2 \), and you spent exactly \( \$12 \), how many of each could you have bought? You "search" by trying different combinations of notebooks and pens until the total hits \( \$12 \).

Key Skill: Identifying Constraints

Before you start searching, you must identify the constraints (the "rules" of the problem). These are the boundaries you cannot cross. Common constraints include:

  • Fixed totals: You must spend exactly \( \$20 \).
  • Minimum/Maximum limits: You must buy at least \( 2 \) items.
  • Availability: There are only \( 5 \) tickets left.

Quick Review: A search is just a way of checking which possibilities fit the "rules" of the problem.

Systematic Searching: The Step-by-Step Method

The biggest mistake students make is "random guessing." Randomly picking numbers is slow and makes it easy to miss the correct answer. Instead, use a systematic search.

Step 1: Understand the Goal

What are you actually looking for? Is it a number of items, a specific time, or a total cost? Read the question carefully to ensure you know what "success" looks like.

Step 2: Start from an Extreme

Usually, it is easiest to start with the highest or lowest possible value for one of the variables. This gives your search a clear starting point.

Step 3: Test and Adjust (Trial and Improvement)

Try a value. If it’s too high, lower it. If it’s too low, increase it. Keep track of your results so you don't repeat the same mistake.

Step 4: Check All Conditions

In the 9694 syllabus, a solution often has to satisfy more than one rule. Even if your answer meets the "total cost" rule, does it also meet the "minimum quantity" rule?

Key Takeaway: Being systematic means being organized. If you test options in a logical order (like \( 1, 2, 3, 4... \)), you are much less likely to get confused.

Applying Simple Models

A "model" in Thinking Skills is just a description of how something works. For this chapter, you will apply a simple model (Syllabus point 2.2) to find your solution.

Analogy: Think of a simple model as a recipe. The "search" is the process of trying different amounts of ingredients to see which one makes the cake taste right according to the recipe's rules.

Example Scenario:

A car park charges \( \$5 \) for the first hour and \( \$2 \) for every hour after that. You have \( \$13 \). What is the maximum number of hours you can park?

  • The Model: \( \text{Cost} = \$5 + (\$2 \times (\text{Total Hours} - 1)) \)
  • The Search:
    Try \( 3 \) hours: \( \$5 + (\$2 \times 2) = \$9 \) (Too low, I have money left).
    Try \( 4 \) hours: \( \$5 + (\$2 \times 3) = \$11 \) (Still have money left).
    Try \( 5 \) hours: \( \$5 + (\$2 \times 4) = \$13 \) (Perfect!).

Did you know? In Paper 1, you can often earn marks for "correct steps" even if your final answer is wrong. This is why writing down your search process is so important!

Communicating Your Reasoning

The 2028 syllabus edition places a heavy emphasis on communicating reasoning. When you search for a solution, don't just write down the final number.

1. Label your values: Instead of just writing "\( 10 \)", write "\( 10 \text{ tickets} \)" or "\( \$10 \)".

2. Show your "failed" attempts: If you tested a value and it didn't work, show why. This justifies why your final answer is the correct one. For example: "If \( 6 \) tickets are bought, the cost is \( \$12 \), which exceeds the \( \$10 \) limit."

Common Mistakes to Avoid

  • Ignoring "Simple Deductions": Sometimes you can narrow your search before you even start. If you need a total of \( \$11 \) using only \( \$2 \) coins and \( \$5 \) notes, you know you must use at least one \( \$5 \) note because you can't make an odd number with only \( \$2 \) coins.
  • Stopping too early: Sometimes there might be more than one solution, or the question might ask for the maximum or minimum. Always check if a "better" solution exists.
  • Miscalculating Ratios: Many simple searches involve ratios or percentages. Double-check your arithmetic, as calculators are allowed and strongly recommended for Paper 1.

Chapter Summary

Search for solutions in simple situations is about finding the right answer among a small set of possibilities by following specific rules.
Key Points:

  • Always identify the constraints first.
  • Use a systematic approach rather than guessing.
  • Apply the model provided in the scenario.
  • Label your working with words and units to gain full credit.
  • If a situation gets more complex, you may need to look at the next chapter: "Search for solutions in more complicated situations".

Don't worry if you find the search slow at first. With practice, you'll start to see patterns that help you "narrow the search" much faster!