💡 Chapter 2: Process Information – Turning Data into Solutions
Welcome to the "Process Information" chapter! In the previous section, "Organise Information," you learned how to gather and structure data. Now, we take the crucial next step: actually doing the work with that data to find the answer. This is where the real problem-solving begins!
Don't worry if problems look daunting at first. Problem solving is like following a complex recipe: you must perform the right operations in the right order. This chapter gives you the toolkit to carry out those operations effectively.
2.1 Perform Appropriate Operations with Information
This skill is all about knowing which mathematical calculation or logical step to perform using the information you have extracted.
Use One or More Items of Information Appropriately
In Thinking Skills problems, "operations" usually means simple arithmetic (addition, subtraction, multiplication, division, percentages, ratio), but the difficulty lies in identifying which operation is needed and when.
- Obvious Operations: These are straightforward. For example, calculating the total cost of groceries by adding individual prices, or finding the percentage discount.
- Deductive Operations: These require a logical step first. For example, if you are planning a schedule, you need to calculate not just the duration of tasks, but also the waiting time or transition time between them to determine the total time required.
🔥 Simple Analogy: The Journey Calculator
Imagine you have three pieces of information:
- Distance: 100 km
- Average Speed: 50 km/h
- Break Time: 30 minutes
The operation needed to find the driving time is 100/50 = 2 hours. The operation to find the total journey duration is adding the driving time and the break time (2 hours + 0.5 hours = 2.5 hours). You must use both pieces of information appropriately.
Apply a Model to a Given Situation
A Model in problem solving is essentially a set of fixed rules, instructions, or a formula that describes how a system works. Your task is to input the specific data from the scenario into this pre-defined system.
How Models Work: Threshold Values and Rules
Models often include threshold values, where the calculation changes once a certain limit is passed. Think of it like a tiered pricing structure:
- Example 1: Taxi Fares. The charge is $5 for the first 5 km (the threshold), and $2 for every kilometer after that. If you travel 12 km, you must apply two different operations: \(5\) km at \(\$5\) + \(7\) km at \(\$2\) = \(\$19\).
- Example 2: Traffic Rules. A set of rules defining how traffic moves through a specific junction (e.g., "Right turns are only allowed between 10:00 and 14:00"). You apply these rules (the model) to a specific event (a car attempting a right turn at 15:00).
Key Takeaway 2.1: Processing information means performing the necessary logical or mathematical steps. Pay close attention to rules and thresholds—they dictate how the calculations must change.
2.2 Identify Cases that Satisfy Given Criteria
Often, a problem won't have just one obvious solution, but a list of possible outcomes, or a situation with many rules (criteria) that must all be met simultaneously. This section focuses on systematic checking and searching.
Search Through All Possible Solutions
This is often referred to as "search space reduction" or Systematic Listing.
🔎 Step-by-Step Search Strategy
- List or Identify Possibilities: Determine the range of potential answers (the 'search space'). Example: If the solution must be a whole number between 1 and 10, that’s your search space.
- Apply Criteria Systematically: Take the first criterion (rule) and eliminate all possibilities that fail it.
- Apply Subsequent Criteria: Use the next rule to filter the remaining possibilities further.
- Identify Final Case(s): The solutions left standing are the ones that satisfy all criteria.
Did you know? Sometimes the question will ask for the number of solutions, not the solutions themselves. Make sure you read the question carefully!
Identify Criteria Not Been Met in a Proposed Solution
This skill is the reverse of searching: you are given a proposed answer and must act as the quality checker, ensuring it hasn't broken any rules.
❌ Common Mistake to Avoid
Students often check a proposed solution against only one or two of the rules. For a solution to be valid, it must meet every single criterion defined in the problem scenario.
Example: A proposed team selection must satisfy these criteria: A) At least 5 members; B) More men than women; C) Total age less than 150. If the team has 6 members and a total age of 100, but has equal men and women, it fails criterion B, and is therefore an invalid solution.
Quick Review: The Criteria Check
When checking a solution against criteria (rules):
- List every rule clearly.
- Check the solution against each rule individually.
- If any one rule is broken, the solution is invalid.
Key Takeaway 2.2: When dealing with criteria, be a detective! List all the rules and check systematically, ensuring no rule is ignored when validating a potential answer.
2.3 Make Appropriate Deductions
Deduction is the ability to use the facts you have to logically infer new facts that were not explicitly stated. This is often the hardest, but most rewarding, part of problem solving.
Draw Conclusions Based on the Information Available
This involves forming logical chains using the relationships between different pieces of data.
How to Deduce New Information
- Using Relationships: If you know 'A is taller than B' and 'B is taller than C', you can deduce 'A is taller than C'. This simple logical relationship allows you to determine a new piece of information (A's relationship to C) that wasn't given directly.
- Considering Rules and Patterns: If a pattern is defined (e.g., numbers placed in a grid must add up to 10 in every row), and you have some numbers, you can deduce the value of the missing numbers required to complete the pattern.
- Evaluating Consequences (Optimisation): This is crucial when choosing the "best" outcome. Given a range of options, you must deduce the consequence of each action and compare the results to find the most favourable one. Example: Which delivery route is fastest? Deduce the total time for Route X, Route Y, and Route Z, then choose the minimum time.
🧠 Analogy: The Domino Effect
Deduction is like setting up dominoes. Fact A knocks over Fact B, which knocks over Fact C, leading you to the final Conclusion D. You must clearly establish the strength of the links between the facts.
Make Inferences from Numerical Patterns
If you are given a set of numerical data (a table of sales figures, travel times, or production numbers), you may be asked to infer why the data looks the way it does.
- Example: A bus company's data shows ticket sales are high on weekdays, but spike significantly every Friday afternoon.
- Deduction (Fact): Sales spike on Friday afternoon.
- Inference (Reason/Conclusion): The reason for this pattern is likely people leaving for weekend trips, or students travelling home from school/university.
Making a strong inference means providing a plausible reason that logically accounts for the observed numerical behaviour.
Key Takeaway 2.3: Deductions are logical jumps. Use all the given facts and relationships to establish new, unstated information, and always consider the consequences of different choices.