GCSE Maths Practice: integers-and-directed-numbers

Question 1 of 10

This question uses direction to show how dividing a positive by a negative produces a negative result. It connects number rules to real-world movement.

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\( \begin{array}{l}\text{A car reverses 4 miles in 2 hours. What is its average velocity?}\end{array} \)

Choose one option:

When one number is negative and the other positive, the result is negative. Interpret the sign as direction, not size.

Using Negative Division in Context

In GCSE Maths, negative numbers often represent direction, loss, or decrease. Division with negative values can be used to calculate rates of change in real-world scenarios, such as velocity, finance, or temperature. Here, we use a motion problem to understand division of a positive and a negative number.

Scenario

A car travels backwards 4 miles in 2 hours. The distance covered (−4 miles) is negative because the car moves in the reverse direction. The time is positive because time always moves forward. To find the average speed:

  1. Distance = −4 miles
  2. Time = 2 hours
  3. Speed = Distance ÷ Time = −4 ÷ 2 = −2 miles per hour

The negative sign in the result shows that the movement is backwards, not that speed is less in value.

Understanding the Rule

When dividing numbers with different signs (positive ÷ negative or negative ÷ positive), the answer is always negative. This is consistent with the rule of multiplication and division for directed numbers.

Worked Examples

  • 12 ÷ (−3) = −4
  • (−9) ÷ 3 = −3
  • 4 ÷ (−2) = −2
  • (−8) ÷ (−4) = 2 (because same signs give a positive)

Common Misunderstandings

  • Forgetting that dividing by a negative reverses the sign.
  • Thinking negative results mean smaller quantities rather than opposite direction.
  • Dropping the negative sign in intermediate steps.

Real-Life Applications

Directed numbers appear in science and finance. For instance, in physics, a negative speed or acceleration may indicate motion in the opposite direction. In banking, dividing a negative profit (loss) by time gives a rate of loss per month. Understanding how negatives interact through division ensures correct interpretation of data across different contexts.

FAQs

  • Q: Why is the answer negative?
    A: Because one number is positive and the other is negative — different signs always produce a negative result.
  • Q: What happens if both are negative?
    A: The negatives cancel, giving a positive result.
  • Q: Is speed ever negative in real life?
    A: No, but velocity can be negative because it shows direction.

Study Tip

Remember this quick rule: Same signs → positive, different signs → negative. When applying division in context, always decide whether the sign represents direction or gain/loss before finalising your answer.