GCSE Maths Practice: estimation

Question 3 of 10

This question practises estimating division by rounding one number down and the other up, giving a safe lower-bound estimate.

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\( \begin{array}{l}\textbf{Estimate: }1075 \div 42.8\end{array} \)

Choose one option:

When estimating a division, round the top down and the bottom up to make the answer slightly smaller and realistic.

Understanding Estimation by Rounding

Estimation is a quick way to find an approximate answer without using a calculator. In this question we use rounding in opposite directions: we round the numerator (top number) down and the denominator (bottom number) up. This gives a safe, slightly smaller result and prevents overestimation. Estimation like this is vital in real-life tasks such as budgeting, shopping, or checking if an answer is sensible.

Why Round in Opposite Directions?

When dividing, if you make the top number smaller and the bottom number larger, your result becomes smaller. This creates a lower bound—an estimate that you can trust will not exceed the true value. It’s especially useful when exact calculation is unnecessary or when you need a conservative answer quickly.

Step-by-Step Method

  1. Look at both numbers. Identify their first significant figures.
  2. Round the numerator down to a nearby convenient value.
  3. Round the denominator up to simplify the division.
  4. Perform the division mentally or using simple multiples of ten.
  5. State your estimate clearly with an appropriate degree of accuracy (no units needed unless specified).

Worked Examples

Example 1: \(1075 \div 42.8\)
Top rounded down: \(1075 \to 1000\).
Bottom rounded up: \(42.8 \to 50\).
\(1000 \div 50 = 20\). Estimate ≈ 20.

Example 2: \(872 \div 29\)
Top down: \(872 \to 800\). Bottom up: \(29 \to 30\).
\(800 \div 30 ≈ 26.7\). Estimate ≈ 27.

Example 3: \(4580 \div 612\)
Round down/up: \(4580 \to 4000\), \(612 \to 600\).
\(4000 \div 600 ≈ 6.7\). Estimate ≈ 7.

Common Mistakes

  • Rounding both numbers in the same direction (which can distort the estimate).
  • Forgetting to keep the ratio realistic—if the divisor becomes too large, the result may be underestimated.
  • Confusing significant figures with decimal places.
  • Writing the exact answer instead of an estimate.

When to Use This Method

This form of estimation is especially useful when you need to check whether a calculator result is reasonable. For example, if your calculator shows 23.8 for \(1075 \div 42.8\), your estimate of 20 confirms it’s plausible because it’s in the same range.

Real-Life Applications

People use estimation constantly—when dividing a restaurant bill, planning fuel use, or comparing costs. Professionals in construction, science, and finance use conservative estimates to avoid risk. By rounding down the top and up the bottom, they make sure they never overspend or overpredict results.

FAQ

Q1: Why not just round both numbers normally?
A: That can give an overestimate. For division, lowering the top and raising the bottom gives a safer, smaller estimate.

Q2: How many significant figures should I round to?
A: One significant figure is usually enough for estimation unless stated otherwise.

Q3: Does it matter if I get 19 or 21 instead of 20?
A: No. Any reasonable approximation close to 20 shows correct understanding of estimation.

Study Tip

Write rounding arrows above numbers to remind yourself which way each moves—↓ for numerator, ↑ for denominator. Check that your final result makes sense by comparing it to nearby multiples you know.

Summary

Rounding in opposite directions during division is a powerful estimation skill. It gives a quick, conservative result that ensures your answer is realistic. Always round to convenient figures, divide mentally, and double-check that the estimate lies in a sensible range compared with the original numbers.