GCSE Maths Practice: estimation

Question 10 of 10

Practise estimating multi-step calculations by rounding values before carrying out multiplication and division.

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\( \begin{array}{l} \textbf{Estimate:} \\ (498.7 \times 47.6) \div 112.4 \end{array} \)

Choose one option:

Round each number sensibly, multiply the numerator, then divide. Always check your result is of the right order of magnitude.

Understanding Multi-Step Estimation with Division

In GCSE Maths, multi-step estimation problems test your ability to simplify complex expressions using rounding and order of operations. The aim is not precision but reasoning — producing a value that is close enough to the real answer while demonstrating logical working. Here we estimate (498.7 × 47.6) ÷ 112.4 by rounding each number to one or two significant figures before performing the calculation.

Step-by-Step Method

  1. Identify all values: Look at each number in the expression to decide sensible rounding levels.
  2. Round appropriately: 498.7 → 500, 47.6 → 50, 112.4 → 110. Avoid rounding too far; keep proportions realistic.
  3. Multiply the rounded values in the numerator: 500 × 50 = 25 000.
  4. Divide by the denominator: 25 000 ÷ 110 ≈ 227. State your final estimate as 220 for simplicity.

Worked Examples

Example 1: (498.7 × 47.6) ÷ 112.4 → rounded ≈ 220.

Example 2: (256.3 × 39.8) ÷ 45.2 → round to 250, 40, 50 ⇒ (10 000 ÷ 50) = 200.

Example 3: (812.9 × 28.4) ÷ 95.6 → round to 800, 30, 100 ⇒ (24 000 ÷ 100) = 240.

Common Mistakes

  • Rounding inconsistently — e.g., rounding one value to 1 s.f. and another to 3 s.f.
  • Ignoring brackets or order of operations.
  • Forgetting that dividing by a slightly larger number decreases the result.
  • Writing a final answer without units when the context involves quantities.

Real-Life Applications

Estimation of combined multiplication and division is widely used in real life. Economists estimate average cost per unit by dividing total cost by approximate production volume. Engineers estimate load capacities or energy outputs using rounded inputs before precise testing. Shoppers mentally calculate discounts or price-per-item ratios by rounding values to easy multiples. These skills show how estimation simplifies decision-making under time pressure.

FAQs

Q1: Why round 112.4 to 110 instead of 100?
A: Rounding to 110 keeps the ratio realistic. Rounding too far could distort the estimate and make it less credible.

Q2: What happens if I round all to 1 s.f.?
A: You might get (500 × 50) ÷ 100 = 250, which overestimates the true value slightly — showing why thoughtful rounding matters.

Q3: How can I check that my estimate is reasonable?
A: Use magnitude sense: multiplying numbers near 500 and 50 gives roughly 25 000, and dividing by about 100 gives a result around 250, so 220 is perfectly sensible.

Study Tip

Show every rounded number clearly in your working. Even if your final answer differs from the model answer, examiners award marks for correct reasoning and order of operations. Always perform multiplication first when brackets indicate it, and finish with division.

Summary

Multi-step estimation problems strengthen mental arithmetic and number sense. By rounding each number sensibly, performing multiplication first, and dividing accurately, you can produce a quick, clear approximation that demonstrates full understanding of estimation techniques.