GCSE Maths · Number

Estimation GCSE Questions and Worked Answers

An estimate is a nearby, easier answer used to judge size or make a practical decision. Round the inputs sensibly, keep the original operations and show that your answer is approximate.

Foundation & Higher6 worked examples10 original questions

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Start with the meaning

What you need to know about estimation

If 19 items each cost £4.90, you can quickly think ‘about 20 lots of £5’, or about £100. That is an estimate: a useful nearby value, not the exact total of £93.10. It helps you notice that a calculator answer such as £9.31 is the wrong size.

See the idea first

Make the numbers simpler, not the calculation different

A significant figure is a meaningful digit counted from the first non-zero digit. To round to one significant figure, keep that first digit, use the next digit to decide whether to round up, and preserve place value.

Original19 × £4.90a count multiplied by a price

Round inputs20 × £5the easier values stay close

Estimateabout £100keep pounds in the answer

Rounding to one significant figure
Original	Rounded	Why
368	400	First digit 3, next digit 6: round up
0.047	0.05	First non-zero digit 4, next digit 7
9.6	10	Rounding carries into the tens column

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Estimate a calculation

What the problem asks: Estimate 31.7 × 5.8, rounding both numbers to one significant figure.

How to solve it: Use 30 × 6 = 180. Follow the requested accuracy rather than calculating exactly first.

Check a calculator result

What the problem asks: A calculator gives 0.204 for 6.12 ÷ 0.3. Could it be correct?

How to solve it: About 6 ÷ 0.3 = 20. Dividing by 0.3 asks how many lots of 0.3 fit into 6: there are 20, so the answer must be larger than 6. The proposed value is far too small.

Decide whether an amount is definitely enough

What the problem asks: Show that £30 is enough for seven items each costing £3.95.

How to solve it: Use an upper estimate: each item costs less than £4, so the total is less than 7 × £4 = £28. This establishes the claim; an arbitrary rough estimate would not.

A reliable routine

Estimate a numerical calculation

Use rounded inputs when the task asks for an approximate answer or a reasonableness check. Rounding simplifies the arithmetic while keeping its scale. It does not guarantee an upper or lower bound unless you track the direction of the changes.

Follow any rounding instruction; otherwise choose nearby numbers that make mental arithmetic straightforward.
Rewrite the whole calculation with the rounded inputs, including brackets and the entire denominator.
Calculate in the normal order of operations.
Use ‘about’ or ≈, keep units, and judge what your estimate can actually show.

Check: Do not round small decimals to zero when that destroys the estimate. A rough estimate close to a budget cannot prove the exact bill fits.

Fully worked

Estimation GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

One significant figure

2 marks

Question

Estimate $48.2\times3.7$ using one significant figure for each number.

Round 48.2 to 50 and 3.7 to 4.

48.2\times3.7\approx50\times4

=200

Tip: show the rounded calculation, not only 200.

Example 2

A small decimal divisor

2 marks

Question

Estimate $6.3\div0.21$ using one significant figure.

The first non-zero digit in 0.21 is in the tenths column: 0.21 rounds to 0.2.

6.3\div0.21\approx6\div0.2

=60\div2

=30

Dividing by a number smaller than 1 increases a positive quantity.

Example 3

A multi-step fraction

3 marks

Question

Estimate $\dfrac{39.2\times5.8}{0.49}$ using one significant figure.

Keep the product above the fraction bar.

\frac{39.2\times5.8}{0.49}\approx\frac{40\times6}{0.5}

=\frac{240}{0.5}

=480

Dividing by one half doubles the result.

Example 4

Preserve a power

3 marks

Question

Estimate $\dfrac{9.7^2+20.8}{3.9}$ using one significant figure.

Round the inputs before calculating the square.

\frac{9.7^2+20.8}{3.9}\approx\frac{10^2+20}{4}

$=\frac{100+20}{4}$ $=\frac{120}{4}$ $=30$ Tip: 10² is 100, not 20.

Example 5

Prove a budget is sufficient

3 marks

Question

Show that £40 is enough to buy nine notebooks at £4.25 each without calculating the exact total.

We need a price above £4.25 whose total still fits £40. Using £4.50 would give £40.50, which is inconclusive, so choose the closer upper estimate £4.40. The actual cost is smaller than this estimate.

9\times4.40=39.60

The actual bill is less than £39.60, which is below £40. Therefore £40 is enough. Tip: ordinary nearest rounding is not automatically a safe upper estimate.

Example 6

Know the limit of an estimate

2 marks

Question

A journey is about 200 km and a car uses about 6 litres per 100 km. Estimate the fuel needed. Does this guarantee 12 litres is enough?

There are about two groups of 100 km.

2\times6=12

Estimated fuel: 12 litres. This does not guarantee sufficiency: both distance and fuel consumption are approximate and could be higher.

10 original questions · total 21 marks

Estimation GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 26 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

Round a whole number

1 mark

Round 684 to one significant figure.

Show worked answer

The first digit is 6; the next is 8, so round up.

684\approx700

Keep the hundreds place.

Round a small decimal

1 mark

Round 0.0368 to one significant figure.

Show worked answer

The first non-zero digit is 3 in the hundredths column. The next is 6.

0.0368\approx0.04

Estimate a product

2 marks

Estimate $29.1\times6.2$ using one significant figure.

Show worked answer

29.1\times6.2\approx30\times6

=180

Both inputs are close to the original values.

Estimate a quotient

2 marks

Estimate $83.6\div3.8$ using one significant figure.

Show worked answer

83.6\div3.8\approx80\div4

=20

Divide the rounded values.

A decimal denominator

2 marks

Estimate $4.9\div0.098$ using one significant figure.

Show worked answer

0.098 rounds to 0.1, not zero.

4.9\div0.098\approx5\div0.1

=50

Several operations

3 marks

Estimate $\dfrac{61\times2.9}{0.19}$ using one significant figure.

Show worked answer

\frac{61\times2.9}{0.19}\approx\frac{60\times3}{0.2}

=\frac{180}{0.2}

=900

Keep the denominator below the whole numerator.

An unreasonable display

2 marks

A student says 51.8 × 19.4 = 100.492. Use an estimate to explain why this is wrong.

Show worked answer

51.8\times19.4\approx50\times20

=1000

The proposed result is about ten times too small. An estimate detects the decimal-place error; it does not supply all exact digits.

An upper estimate

3 marks

Show that £25 is enough for six sandwiches costing £3.90 each.

Show worked answer

Each sandwich costs less than £4.

6\times4=24

The actual bill is less than £24, so £25 is enough.

When a rough answer is inconclusive

2 marks

You estimate a bill as £50 and have exactly £50. Can you conclude you can afford it? Explain.

Show worked answer

No. The exact bill could be above or below the estimate. Calculate more accurately or establish an upper estimate no greater than £50.

Estimate with brackets

3 marks

Estimate $2.8(19.6+31.2)$ using one significant figure.

Show worked answer

2.8(19.6+31.2)\approx3(20+30)

=3\times50

=150

Round the inputs, then preserve the brackets.

Examiner-style feedback

Common estimation mistakes

Rounding only the final exact answer

In an estimation question, show simpler input values and the resulting easier calculation.

Losing place value

0.047 rounds to 0.05 at one significant figure, not 5 or 0.

Treating ≈ as =

An estimate is not an exact equality. Track upper or lower rounding if you need a definite conclusion.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Choose nearby easy numbers.
Preserve the operations and units.
Show the rounded calculation.
Distinguish a rough check from a guaranteed bound.

Quick answers

Estimation FAQ

Must I always use one significant figure?

Use it when instructed. Otherwise suitable nearby values may give more useful mental arithmetic.

Can different estimates be valid?

Yes, if no exact rounding rule is prescribed. Show the values you used so the reasoning can be checked.

Build connected skills

What to revise next

Order matters

BIDMAS

Keep the right order after rounding.

Revise bidmas

Use powers of ten

Standard form

Compare very large and small quantities.

Revise standard form

Content standards

Curriculum and rights review

N14–N15: numerical estimation and rounding across tiers; this is not a sampling-estimation guide. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content