Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Rounding, Estimation & Bounds

In one sentence. Significant figures, one-significant-figure estimates that sanity-check everything, and the upper and lower bounds of a rounded measurement.

Number — one of 8 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Eratosthenes. Learn Rounding, Estimation & Bounds, teach it, then defend it against Eratosthenes.

Where this sits in the specification

Specification 4MA1, 1.8 Degree of accuracy (bounds calculations: Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 1.8 — Degree of accuracy (bounds calculations: Higher)

1 · Understand it

What it is

Rounding keeps the significant part of a number (0.04962 → 0.050 to 2 s.f.). Estimation rounds every value to 1 s.f. so you can predict an answer's size in seconds. Bounds run rounding backwards: if a length is '5.6 cm to 1 d.p.', the true value lies anywhere from 5.55 (lower bound) up to but not including 5.65 (upper bound).

Why it works

Every measurement is really an interval — a ruler marked in millimetres cannot distinguish 5.598 from 5.612; both READ as 5.6. The bounds are the edges of everything that rounds to the stated value: half a rounding-unit either side. That's why the upper bound of 5.6 (to 0.1) is 5.65, not 5.7.

When a question is asking for it

'Estimate' in a question ALWAYS means round to 1 s.f. first — doing the exact calculation scores nothing. Bounds appear as 'to the nearest…' or 'correct to…' wording; maximum/minimum questions about calculated quantities (speed, area, density) are bounds questions in disguise.

How to hold it in your head

For a maximum of a DIVISION, you want the biggest top over the smallest bottom — like sharing the most cake between the fewest people. Maximum of a product: both bounds up. Minimum: flip every choice.

2 · The mistakes that cost marks

  • "The upper bound of 5.6 (to 1 d.p.) is 5.7." It's 5.65 — half the rounding unit, not the whole unit.
  • Leading zeros counted as significant: in 0.04962, the first significant figure is the 4. To 2 s.f.: 0.050 (and the trailing zero SHOWS the accuracy).
  • Using upper bounds for everything when maximising. For a quotient, the maximum needs the LOWER bound of the denominator; for a difference a − b, the maximum needs the LOWER bound of b.
  • 'Estimate' answered with a calculator-exact value. The examiner wants 1-s.f. rounding and visible working — the estimate IS the answer.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Estimate (29.6 × 6.84) ÷ 0.47.

  1. Round each to 1 s.f.: 30, 7, 0.5.
  2. (30 × 7) ÷ 0.5 = 210 ÷ 0.5.
  3. Dividing by a half doubles: 420.

Answer: ≈ 420

A rectangle has area 50 cm², to the nearest cm², and length 8.4 cm, to 1 d.p. Work out the lower bound of its width.

  1. Width = area ÷ length.
  2. Bounds: area 49.5 to 50.5 cm²; length 8.35 to 8.45 cm.
  3. SMALLEST width = smallest area ÷ LARGEST length = 49.5 ÷ 8.45 (minimum of a division: least on top, most underneath).
  4. = 5.8579… cm.

Answer: 5.86 cm (3 s.f.)

a = 7.4 and b = 2.6, each correct to 1 decimal place. Work out the upper and lower bounds of a − b, and give a − b to a suitable degree of accuracy.

  1. Maximum of a SUBTRACTION: biggest first number minus SMALLEST second: 7.45 − 2.55 = 4.9.
  2. Minimum: smallest first minus biggest second: 7.35 − 2.65 = 4.7.
  3. Suitable accuracy: round both bounds to fewer and fewer figures until they agree. To 2 s.f. they are 4.9 and 4.7 (disagree); to 1 s.f. both are 5.

Answer: Upper bound 4.9, lower bound 4.7; a − b = 5 (1 s.f.)

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Round 0.04962 to 2 significant figures.
  2. Grade 7 — By rounding each number to 1 significant figure, estimate the value of (48.7 × 3.12) ÷ 0.53.
  3. Grade 8 — A length is measured as 5.6 cm, correct to 1 decimal place. Write down its upper bound. (answer in cm)
  4. Grade 9 — x = 4.5 and y = 2.8, each correct to 1 decimal place. Work out the maximum possible value of x ÷ y, giving your answer to 3 significant figures.
  5. 9+ — A sprinter runs 100 m, measured to the nearest metre, in 12.3 seconds, measured to 1 decimal place. Work out the upper bound of the average speed in m/s, to 3 significant figures. (answer in m/s)

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Rounding, Estimation & Bounds, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Rounding, Estimation & Bounds works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Rounding, Estimation & Bounds against Eratosthenes — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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