Estimating and Checking Calculations Revision Notes | GCSE AQA Higher Maths

1. Why Estimation Matters

You're in an exam. Your calculator shows 47.8 × 23.1 = 1104.18. But is this right? In mathematics, estimation is crucial for verifying results quickly, especially when working under pressure, as in exams. Being able to estimate can help you catch input errors on your calculator and ensures you are more confident in your answers. Without these skills, there’s a risk of relying too heavily on technology without verifying the results, which could lead to costly mistakes. By rounding numbers to one significant figure or using approximation methods, you are creating a solid foundation for evaluating whether your final answers are reasonable.

Without estimation skills, you'd never know if you'd accidentally pressed the wrong button. Estimation is your safety net - it catches errors before they cost you marks.

In real life, estimation helps you spot when something's gone wrong. If your phone bill calculator says £47,000 for one month, you know there's a problem. If a recipe calculator suggests 50 eggs for a cake, you'd question it.

Examiners love testing this skill because it shows mathematical thinking, not just button-pushing. You'll see questions asking you to estimate first, then calculate, then comment on whether your answer is reasonable.

2. The Estimation Toolkit

The key to good estimation is rounding to 1 significant figure. This means keeping only the first non-zero digit and replacing the rest with zeros.

Here's how it works: 47.8 becomes 50, and 23.1 becomes 20. Now our calculation becomes 50 × 20 = 1000. Much easier to work out in your head, and close to our calculator answer of 1104.18.

Sometimes you need to be clever about your rounding. If you're multiplying, round one number up and one down to balance out the error. For 47.8 × 23.1, you could use 50 × 20 or 45 × 25 - both give estimates in the range 1000–1125.

3. Step-by-Step Estimation Methods

Worked example: Estimating a Complex Calculation

Estimate the value of $\frac{297 \times 48.3}{19.7}$

1. $297 \rightarrow 300$ $48.3 \rightarrow 50$ $19.7 \rightarrow 20$ — Keep only the first non-zero digit in each number
2. $\frac{297 \times 48.3}{19.7} \approx \frac{300 \times 50}{20}$ — Replace original numbers with rounded versions
3. $300 \times 50 = 15000$ — Multiply the rounded values in the numerator
4. $\frac{15000}{20} = 750$ — Divide to get the final estimate

Answer: 750 (exact answer is 728.4, so our estimate is excellent)

Order of magnitude checks are your first line of defence against silly errors. If you're calculating the area of a football pitch and get 0.007 m², you know something's wrong - football pitches are thousands of square metres, not fractions.

Worked example: Using Order of Magnitude to Check an Answer

A student calculates that 4.7 × 10³ ÷ 2.3 × 10² = 2.04 × 10⁶. Check if this is reasonable.

1. $\frac{5 \times 10^3}{2 \times 10^2} \approx \frac{5 \times 10^3}{2 \times 10^2}$ — Round coefficients to simple numbers
2. $\frac{5}{2} = 2.5$ — Calculate the numerical part
3. $\frac{10^3}{10^2} = 10^{3-2} = 10^1 = 10$ — Subtract the indices when dividing powers
4. $2.5 \times 10 = 25 = 2.5 \times 10^1$ — The answer should be around 25, not 2 million

Answer: The student's answer of 2.04 × 10⁶ is wrong - it should be around 2 × 10¹

4. Checking Calculator Results

Calculators are brilliant, but they can't think. If you input the wrong numbers or press the wrong buttons, they'll give you the wrong answer with complete confidence.

The smart approach is to estimate first, calculate second, then compare. If your estimate was 750 and your calculator gives 728.4, you're probably right. If your calculator gives 72.84 or 7284, you know you've made an error.

Worked example: The Estimate-Calculate-Check Method

Find the value of $\sqrt{47.8 \times 23.1}$ and verify it's reasonable

1. $\sqrt{50 \times 20} = \sqrt{1000}$ — Round to 1 s.f. before calculating
2. $\sqrt{1000} \approx \sqrt{1024} = 32$ — 1024 = 32², so √1000 ≈ 32
3. $47.8 \times 23.1 = 1104.18$ $\sqrt{1104.18} = 33.23$ — Use calculator for exact calculation
4. Estimate: 32, Exact: 33.23 — Very close! The answer is reasonable

Answer: 33.23 (estimate of 32 confirms this is correct)

Common calculator errors include: pressing × instead of +, forgetting brackets, or entering numbers in the wrong order. A quick estimation catches these mistakes before they cost you marks.

6. Exam Techniques and Common Traps

Exam questions often ask you to estimate first, then calculate, then comment on your answer. This three-step process shows mathematical thinking and usually carries method marks even if your final answer is slightly wrong.

Watch out for questions that give you a 'wrong' calculator result and ask if it's reasonable. These test whether you can spot errors using estimation. Always do a quick order of magnitude check.

In reverse problems, you might be given an estimated answer and asked to find what calculation could have produced it. Work backwards from the estimate to see what the original numbers might have been.

Worked example: Exam-Style Estimation Question

Sarah calculates $\frac{4.8 \times 97.3}{0.52}$ and gets 897.7. Without using a calculator, explain whether this answer is reasonable. [3 marks]

1. $4.8 \rightarrow 5$, $97.3 \rightarrow 100$, $0.52 \rightarrow 0.5$ — [1 mark for appropriate rounding]
2. $\frac{5 \times 100}{0.5} = \frac{500}{0.5} = 1000$ — [1 mark for correct estimation calculation]
3. Estimate: 1000, Sarah's answer: 897.7 — [1 mark for reasonable conclusion about closeness]

Answer: Yes, the answer is reasonable because 897.7 is close to the estimate of 1000