# Degree of Accuracy

> Edexcel International GCSE Mathematics A · 4MA1 (2016 Spec)
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-degree-of-accuracy/

This guide covers rounding to powers of 10, decimal places and significant figures, identifying upper/lower bounds, estimation, and Higher-only multi-step bound calculations for Edexcel IGCSE Math A.

**先修:** Understanding of place value for integers and decimals; Ability to perform basic arithmetic operations (multiplication, division, addition, subtraction)

## 学习目标

- Round integers to a given power of 10, specified number of decimal places or significant figures
- Identify upper and lower bounds for values given to a stated degree of accuracy
- Use estimation by rounding values to 1 significant figure to approximate calculation results
- (Higher only) Solve multi-step problems combining bounds for products, quotients and real-world contexts

## Rounding to Decimal Places, Powers of 10 and Significant Figures

Rounding reduces the precision of a value to match a required degree of accuracy. You will be asked to round to a given power of 10, number of decimal places (dp), or significant figures (sf).

**Significant Figures** — Digits that carry meaning contributing to a value's precision. Leading zeros are not significant; trailing zeros after a decimal point are significant.

*例:* 4.02 has 3 sf; 0.0051 has 2 sf; 1200 has 2 sf if rounded to the nearest 100.

**例题:** Round 2746 to the nearest 100, 3.14159 to 2 dp, and 0.00487 to 2 sf.

1. Round 2746 to nearest 100: Look at the tens digit (4), which is less than 5, so round down. Result = 2700.
2. Round 3.14159 to 2 dp: Look at the 3rd decimal digit (1), less than 5, so keep the first 2 decimals. Result = 3.14.
3. Round 0.00487 to 2 sf: First non-zero digit is 4 (1st sf), next digit 8 (2nd sf), next digit 7 ≥5 so round the 8 up to 9. Result = 0.0049.

> **tip**
>
> Only round your final answer in exam questions; keep full precision for all intermediate steps to avoid rounding errors.

> **考试提示:** If you are asked to round to a power of 10 (e.g. nearest 10, 100, 1000), circle the digit in that place value first before checking the next digit to decide if you round up or down.

*计算器:* allowed

## Estimation using 1 Significant Figure

Estimation is used to quickly check if a calculation result is reasonable. For Edexcel IGCSE, you will always round every value in the calculation to 1 significant figure first, then compute the result.

**例题:** Estimate the value of (392 × 0.71) / 19.8

1. Round each value to 1 sf: 392 → 400, 0.71 → 0.7, 19.8 → 20
2. $$Estimate = \frac{400 × 0.7}{20}$$
3. Calculate numerator first: 400 × 0.7 = 280
4. Divide by 20: 280 / 20 = 14. Final estimated value is 14.

**概念自测**

1. Estimate 512 × 2.9 / 10.3

   - 1.5
   - 15
   - 150

   *解析:* Round 512→500, 2.9→3, 10.3→10. (500×3)/10 = 1500/10 = 150.

> **考试提示:** Estimation questions are usually worth 2 marks: 1 mark for rounding all values to 1 sf correctly, 1 mark for the final approximate result.

*计算器:* allowed

## Identifying Upper and Lower Bounds

Any value given to a specific degree of accuracy has a range of possible original values. The minimum possible value is the lower bound (LB), the maximum possible is the upper bound (UB). For a value rounded to the nearest unit u, bounds are value ± u/2.

**Bounds Interval** — The range of values a rounded measurement can take, written as LB ≤ x < UB. Note the upper bound is not included, as values equal to the UB would round up.

*例:* A length given as 6 cm to the nearest cm has bounds 5.5 cm ≤ x < 6.5 cm.

**例题:** Find the upper and lower bounds of a mass given as 2.4 kg, rounded to 2 significant figures.

1. First identify the precision of the rounding: 2 sf for 2.4 kg means the value is rounded to the nearest 0.1 kg, so u = 0.1 kg.
2. $$LB = 2.4 - 0.05 = 2.35 kg$$
3. $$UB = 2.4 + 0.05 = 2.45 kg$$
4. Final bounds: 2.35 kg ≤ m < 2.45 kg

> **warning**
>
> For significant figure bounds, always work out the size of the smallest unit of precision first, e.g. 3 sf for 1200 is nearest 100, so u=100, bounds 1150 ≤ x < 1250.

> **考试提示:** Always state the precision unit u first when calculating bounds, as this will help you avoid mistakes with significant figure bound questions.

*计算器:* allowed

## Higher Tier: Solving Multi-Step Bound Problems

For Higher tier questions, you will combine bounds for calculations. To find the maximum result of a product, multiply upper bounds of both values. For minimum product, multiply lower bounds. For quotients, maximum result = UB of numerator / LB of denominator, minimum result = LB of numerator / UB of denominator.

**例题:** A rectangle has length 6.2 cm and width 4.8 cm, both rounded to 2 significant figures. Calculate the maximum possible area of the rectangle, giving your answer to 3 significant figures.

1. First find bounds for length and width: both rounded to 2 sf, so precision 0.1 cm.
2. Length bounds: 6.15 cm ≤ l < 6.25 cm, UB of l = 6.25 cm
3. Width bounds: 4.75 cm ≤ w < 4.85 cm, UB of w = 4.85 cm
4. $$Maximum area = UB(l) × UB(w) = 6.25 × 4.85 = 30.3125 cm²$$
5. Round final answer to 3 sf: 30.3 cm²

> **考试提示:** Always state the bounds you are using first in your working, as examiners award marks for correct bound identification even if your final calculation is wrong.

*计算器:* allowed

## 常见错误

- **错误做法:** Rounding intermediate steps of a calculation before getting the final answer
  - 原因: This introduces rounding errors that can make your final answer inaccurate, leading to lost marks
  - 正确做法: Keep full precision for all intermediate working, only round the final answer to the requested degree of accuracy.
- **错误做法:** Including the upper bound in the possible value range (e.g. writing 5.5 ≤ x ≤ 6.5 for a value rounded to nearest 1 cm)
  - 原因: A value of exactly 6.5 would round up to 7 cm, so it is not part of the range for values rounded to 6 cm
  - 正确做法: Always use strict inequality for the upper bound: 5.5 ≤ x < 6.5
- **错误做法:** Using upper bounds for both numerator and denominator when calculating the minimum value of a quotient
  - 原因: Dividing by a larger number gives a smaller result, so you need the upper bound of the denominator to get the minimum quotient
  - 正确做法: Minimum quotient = LB(numerator) / UB(denominator); Maximum quotient = UB(numerator) / LB(denominator)
- **错误做法:** Counting leading zeros as significant figures, e.g. saying 0.0023 has 4 significant figures
  - 原因: Leading zeros only show the place value of the number, they do not contribute to its precision
  - 正确做法: Start counting significant figures from the first non-zero digit from the left: 0.0023 has 2 significant figures.
- **错误做法:** Calculating maximum area as UB(length) × LB(width) for a rectangle
  - 原因: Area is a product, so to maximise the result you need to multiply the largest possible values of both sides
  - 正确做法: Maximum product = UB of both factors; Minimum product = LB of both factors.

## 速查表

| Concept | Rule | Example |
| --- | --- | --- |
| Rounding to nearest u | Value < u/2 above midpoint: round down; ≥ u/2: round up | Round 346 to nearest 10: 350 |
| 1 sf Estimation | Round all values to 1 sf first, then calculate | (39×0.21) → (40×0.2) = 8 |
| Bounds of rounded value | LB = value - u/2, UB = value + u/2, u = precision unit | 2.7 cm (1 dp): 2.65 ≤ x < 2.75 |
| Max product | Multiply UB of all factors | Max area = UB(length) × UB(width) |
| Min quotient | Divide LB of numerator by UB of denominator | Min speed = LB(distance) / UB(time) |

## 下一步

Now that you have mastered degree of accuracy, you can apply these skills to a wide range of Edexcel IGCSE Math A topics, including measurement problems, geometry calculations, and statistical data analysis. Rounding and bound reasoning are frequently tested across both Foundation and Higher papers, so practice these skills regularly to avoid easy marks lost to small errors. For Higher tier students, you will use bound reasoning in more complex topics such as trigonometry and kinematics problems where measurement precision is important.

- [Standard Form (Numbers and the Number System)](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s1-standard-form/)

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