C1.9 – Estimation

Table of Contents

IGCSE Mathematics  |  Core Topic

1. Rounding to Decimal Places #

Decimal places (d.p.) are the digits that come after the decimal point. Rounding to a certain number of decimal places means keeping only that many digits after the point.

Rounding Rule Look at the digit one place to the right of where you want to round:
  • If it is 5 or more → round up (increase the last kept digit by 1)
  • If it is 4 or less → round down (keep the last kept digit the same)
Worked Example — Round 3.746 to 2 decimal places
  1. We want 2 d.p., so we keep two digits after the point: 7 and 4
  2. Look at the next digit: 6  →  6 is 5 or more, so round up
  3. The 4 becomes 5. Answer: $3.746 \approx 3.75$
Worked Example — Round 12.995 to 2 decimal places
  1. We keep two digits after the point: 9 and 9
  2. Look at the next digit: 5  →  5 or more, so round up
  3. The second 9 rounds up to 10, which carries over: $12.99 \rightarrow 13.00$
  4. Answer: $12.995 \approx 13.00$
Important: Always write trailing zeros to show the degree of accuracy. For example, $13.00$ (2 d.p.) is not the same as just writing $13$ — the zeros show exactly how precise the answer is.

2. Rounding to Significant Figures #

Significant figures (s.f.) count from the first non-zero digit of the number. This method works for both large and small numbers.

How to count significant figures
  • Start counting from the first non-zero digit
  • Every digit after that counts — including zeros that sit between non-zero digits
  • Leading zeros (zeros before the first non-zero digit) do not count

IMAGE NEEDED: Diagram labelling significant figures in a number like 0.00304 — showing which zeros count and which do not

Google Images Search: “significant figures counting rules diagram IGCSE maths educational”

Same rounding rule applies: Look at the digit after the last significant figure you want to keep. If it is 5 or more, round up. If it is 4 or less, round down.

Rounding large numbers #

Worked Example — Round 5764 to 1 significant figure
  1. The first significant figure is 5 (the thousands digit)
  2. Look at the next digit: 7  →  7 is 5 or more, so round up
  3. Replace the remaining digits with zeros:   $5764 \approx 6000$
Worked Example — Round 5764 to 2 significant figures
  1. The first two significant figures are 5 and 7
  2. Look at the next digit: 6  →  6 is 5 or more, so round up
  3. The 7 becomes 8. Replace the rest with zeros:   $5764 \approx 5800$
Important: When rounding large numbers, you must replace removed digits with zeros as placeholders. Without the zeros, the place value would be wrong. For example, $5764$ rounded to 1 s.f. is $6000$, not $6$.

Rounding small decimal numbers #

Worked Example — Round 0.004 362 to 2 significant figures
  1. The leading zeros do not count. The first significant figure is 4
  2. The second significant figure is 3
  3. Look at the next digit: 6  →  6 is 5 or more, so round up
  4. The 3 becomes 4:   $0.004\,362 \approx 0.0044$

3. Estimation #

Estimation means finding an approximate answer quickly, without a calculator. It is very useful for checking whether your calculator answer looks reasonable.

Method for Estimating a Calculation
  1. Round each number in the calculation to 1 significant figure
  2. Carry out the simplified calculation by hand
  3. Write the answer using the $\approx$ symbol (means “approximately equal to”)
Worked Example — Estimate the value of $\dfrac{41.3}{9.79 \times 0.765}$
  1. Round each number to 1 s.f.:
    • $41.3 \approx 40$
    • $9.79 \approx 10$
    • $0.765 \approx 0.8$
  2. Substitute into the calculation: $$\frac{40}{10 \times 0.8} = \frac{40}{8} = 5$$
  3. The estimated answer is $\approx 5$
Worked Example — Estimate $\dfrac{318 \times 5.12}{0.491}$
  1. Round each number to 1 s.f.:
    • $318 \approx 300$
    • $5.12 \approx 5$
    • $0.491 \approx 0.5$
  2. Substitute and calculate: $$\frac{300 \times 5}{0.5} = \frac{1500}{0.5} = 3000$$
  3. The estimated answer is $\approx 3000$
Use estimation as a check: If your calculator answer is very different from your estimate, you have probably made an error. Go back and check your working.

4. Rounding to a Reasonable Degree of Accuracy #

In real problems, you must choose a sensible level of accuracy. The context of the problem tells you how precise your answer should be.

Context Reasonable accuracy Example
Money 2 decimal places \$14.57
Long distances Nearest km or whole number 240 km
Short lengths Nearest mm or cm 3.4 cm
Time Nearest minute or second 45 minutes
Large populations or counts Nearest thousand or 3 s.f. 2 400 000
General rule: Your answer should not be more precise than the data you were given. If the measurements in a question are given to the nearest whole number, do not give your final answer to 4 decimal places.
Worked Example — Choosing the right accuracy

A rectangle has length $8.3\text{ cm}$ and width $4.7\text{ cm}$ (both given to 1 d.p.).

Area $= 8.3 \times 4.7 = 39.01\text{ cm}^2$

Since the original measurements were given to 1 d.p., a reasonable answer is:

Area $\approx 39.0\text{ cm}^2$   (1 d.p.)

Giving the answer as $39.01\text{ cm}^2$ would suggest more precision than the measurements actually have.

Syllabus Reference — C1.9 Estimation #

  1. Round values to a specified degree of accuracy. Includes decimal places and significant figures.
  2. Make estimates for calculations involving numbers, quantities and measurements.
  3. Round answers to a reasonable degree of accuracy in the context of a given problem.
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