If you've ever measured something, you'll know it's almost impossible to be perfectly exact. Did you run 100 metres, or was it 100.4 metres? Is that bag of flour 1.5 kg, or 1.52 kg? In GCSE Maths, we handle this uncertainty using bounds and error intervals. This topic is a favourite of exam boards like AQA, Edexcel, and OCR, and once you understand the core ideas, it's surprisingly straightforward.
What Are Bounds?
When a number has been rounded or truncated, we can state the range of possible values it could have been before that process. The lower bound is the smallest possible value it could have been, and the upper bound is the largest.
Key Definitions
Lower Bound: The smallest value that would round up to the given number.
Upper Bound: The smallest value that would round up to the next number. (We often say "the largest value that would round down," but the first definition is more precise for finding it).
- Error Interval: The range of possible values, written as an inequality (e.g., 15 ≤ x < 20).
A Simple Process for Finding Bounds
Follow this flowchart to determine the bounds for any rounded number.
flowchart TD
A["Start: Identify the rounded value
and degree of accuracy"] --> B["How was the value
produced?"];
B -->|Rounded| C["Add and subtract
HALF the degree of accuracy"]
B -->|Truncated| D["Add the FULL degree of accuracy
to find upper bound"]
C --> E["Is it a discrete
or continuous quantity?"];
D --> E;
E -->|Continuous
(e.g., weight, length)| F["Upper bound is NOT included
Use < or >"]
E -->|Discrete
(e.g., people, coins)| G["Upper bound IS included
Use ≤ or ≥"]
F --> H["State Error Interval:
Lower Bound ≤ x < Upper Bound"]
G --> I["State Error Interval:
Lower Bound ≤ x ≤ Upper Bound"]
Step-by-Step Walkthrough with Examples
Example 1: Basic Rounding
> "The length of a pencil, l, is 12 cm to the nearest centimetre. Write down the error interval for l."
Identify the precision: "To the nearest cm" means the degree of accuracy is 1 cm.
Calculate half of this: 1 cm ÷ 2 = 0.5 cm.
Find the bounds:
Lower Bound: 12 - 0.5 = 11.5 cm
Upper Bound: 12 + 0.5 = 12.5 cm
Determine the inequality: Length is a continuous quantity. Could the pencil be exactly 12.5 cm? If it were, it would round to 13 cm, not 12 cm. So 12.5 is not included.
- State the error interval: 11.5 ≤ l < 12.5
Example 2: Different Degrees of Accuracy
> "The mass of a bag of potatoes, m, is 3.4 kg to the nearest 0.1 kg. Write down the error interval."
Precision: Nearest 0.1 kg.
Half of precision: 0.1 ÷ 2 = 0.05 kg.
Find bounds:
Lower Bound: 3.4 - 0.05 = 3.35 kg
Upper Bound: 3.4 + 0.05 = 3.45 kg
Inequality: Mass is continuous. 3.45 would round to 3.5.
- Answer: 3.35 ≤ m < 3.45
Example 3: Truncation
Truncation means "cutting off" digits without rounding.
> "A number, x, is truncated to 1 decimal place, giving 7.2. Write the error interval."
Identify Process: Truncated to 1 d.p. The smallest number that truncates to 7.2 is 7.2 itself.
Find bounds:
Lower Bound: 7.2
Upper Bound: The largest number that would still truncate to 7.2 is just below 7.3. The degree of accuracy is 0.1, so we add this to the lower bound: 7.2 + 0.1 = 7.3.
Inequality: 7.3 is NOT included, as 7.30 truncates to 7.3.
- Answer: 7.2 ≤ x < 7.3
Applying Bounds in Calculations
A common GCSE question asks you to find the maximum or minimum possible result of a calculation.
Golden Rule:
To find the maximum value of a calculation, use the combination of bounds that gives the largest result.
- To find the minimum value, use the combination that gives the smallest result.
Example: Division
> "A rectangle has an area, A, of 350 cm² to the nearest 10 cm². Its length, l, is 25 cm to the nearest cm. Calculate the upper bound for the width, w, where w = Al."
Find bounds for A: Nearest 10 cm² → Half is 5 cm².
A lower bound: 350 - 5 = 345 cm²
A upper bound: 350 + 5 = 355 cm²
Find bounds for l: Nearest 1 cm → Half is 0.5 cm.
l lower bound: 25 - 0.5 = 24.5 cm
l upper bound: 25 + 0.5 = 25.5 cm
Maximise w = A / l: To make a division as large as possible, maximise the numerator (A) and minimise the denominator (l).
Use A upper bound (355) and l lower bound (24.5).
- Calculate: Upper bound for w = 355 ÷ 24.5 ≈ 14.4898... cm
Common Pitfalls and Exam Tips
< vs ≤: Always ask: "Could the value be exactly the upper bound?" If yes, use ≤. For continuous measures (length, mass, time) from rounding, it's almost always <. For discrete counts, it's ≤.
Truncation vs Rounding: Know the difference! Truncation gives a different lower bound.
Stating the Final Answer: In the exam, always write your error interval in the form: Lower Bound ≤ x < Upper Bound. Use the correct inequality symbols.
- Units: Don't forget to include units (cm, kg, etc.) in your final answer unless it's a pure number.
Practice for Your Revision
Try this typical question:
> "The population of a town, P, is 56,000 to the nearest thousand. The area of the town, A, is 14.5 km² to 1 decimal place. Work out the lower bound for the population density (people per km²)."
(Hint: Density = Population ÷ Area. Find the bounds for P and A first, then combine them to minimise the result).
Answer:
P bounds: 55,500 ≤ P < 56,500
A bounds: 14.45 ≤ A < 14.55
To minimise Density, use smallest P (55,500) and largest A (just under 14.55).
- Lower Bound ≈ 55,50014.55 ≈ 3814.4 people/km².
- Mastering bounds is about careful reading and methodical steps. Use the flowchart, remember the half-the-accuracy rule for rounding, and always think about whether your upper bound is included. Good luck with your revision!