GCSE Maths Practice: integers-and-directed-numbers

Applying BIDMAS in Real-Life Contexts

Word problems help you apply mathematical operations to real-world scenarios. This question blends negative numbers and order of operations in a financial context—a situation you might meet in business, economics, or data analysis. Understanding how to manage operations involving negatives is key to making sense of profit and loss, temperature changes, and movement above or below zero.

Interpreting the Problem

Imagine a diver descending 16 metres below the surface of the sea. Each metre of depth is represented by a negative number because it is below the reference point (sea level). The diver then ascends in four equal stages. To find how much depth is reduced per stage, you divide −16 by −4. Because a negative divided by a negative gives a positive result, each stage represents a positive change upwards. After completing the four stages, the diver returns towards the surface and then swims another 6 metres upward above the previous level. The combination of these movements describes the expression used in this problem.

Step-by-Step Approach

1. Identify operations: There is a division followed by an addition.
2. Apply BIDMAS: Division must come before addition.
3. Handle signs correctly: Negative ÷ Negative = Positive.
4. Add remaining values: Once the division is done, combine it with the positive increase.

This structure ensures you reach the correct result regardless of context.

Worked Examples (Different from the Question)

• Example 1: A hiker climbs 12 metres up a hill after dividing her previous 8-metre descent into equal recovery steps. The operation could resemble (−8) ÷ (−2) + 12.
• Example 2: A company reports a loss of £20,000 spread evenly over five months, then gains £4,000 in the next quarter. The calculation is similar in structure, using negatives for losses and positives for gains.
• Example 3: The temperature drops 15 °C over 3 hours, then rises another 5 °C. That operation mirrors the same order of logic.

Common Errors

• Performing addition before division, which violates BIDMAS.
• Forgetting that two negatives divided make a positive.
• Mixing up the physical meaning of a sign in the context—for instance, mistaking a depth increase for another decrease.

Real-Life Relevance

Negative numbers appear in many daily measurements: bank balances, altitude, temperature, and changes in data trends. In finance, dividing a total loss among several periods and then adding subsequent gains follows exactly the same rules. Understanding this connection between arithmetic and interpretation allows you to solve applied problems more confidently.

FAQs

Q1: Why do two negatives divide to make a positive?
A: Because the direction of change reverses twice—one negative indicates a downward trend, and dividing by another negative reverses it again, making it upward.

Q2: What if I add before dividing?
A: You would get the wrong result because you would be combining quantities in the wrong order, just like mixing totals before calculating averages.

Q3: How can I recognise BIDMAS quickly in word problems?
A: Look for key verbs: shared equally or divided indicates division; increased by or added indicates addition. Always handle the sharing or dividing first.

Study Tip

When facing a word problem, rewrite the text as a clean arithmetic expression. Then apply BIDMAS step by step. Highlight negative signs and check if the problem involves a reversal (e.g., a loss turning into a gain). This habit improves accuracy and helps you link mathematical logic with real-world understanding—an essential skill for higher-level GCSE Maths and beyond.