GCSE MATHS • HIGHER TIER

Bounds Made Simple

Understand the smallest and largest possible values when numbers and measurements have been rounded.

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Bounds in Four Moves

1

Identify

Identify the rounding level or interval.

2

Halve

Find half of the rounding interval.

3

Calculate

Subtract for the lower bound and add for the upper bound.

4

Write

Express the bounds correctly using inequalities.

What You’ll Learn

Understand Bounds

Know why rounded values represent a range of possible exact values.

Find Lower and Upper Bounds

Calculate the smallest and largest possible values.

Use Inequality Notation

Write bounds such as 45 ≤ x < 55.

Apply Bounds to Measures

Use bounds with decimals, perimeter and area.

Understand Accuracy

Connect bounds with rounding and estimation.

Key Rules

Lower Bound

The smallest possible value.

Upper Bound

The largest possible limit.

Use Half the Interval

Add and subtract half of the rounding interval.

Correct Inequality Signs

Include the lower bound using ≤, but exclude the upper bound using <.

Worked Proof: Bounds

Step-by-Step Method

  • 1. Identify the rounding interval.
  • 2. Find half of the rounding interval.
  • 3. Subtract half the interval to find the lower bound.
  • 4. Add half the interval to find the upper bound.
  • 5. Write the answer using lower bound ≤ x < upper bound.

More Worked Examples

1

A length is 30 cm correct to the nearest centimetre .

Lower Bound = 29.5 cm

Upper Bound = 30.5 cm
29.5 ≤ x < 30.5

2

A mass is 8 kg correct to the nearest kilogram .

Lower Bound = 7.5 kg

Upper Bound = 8.5 kg
7.5 ≤ x < 8.5

3

A value is 2.4 correct to 1 decimal place.

Lower Bound = 2.35

Upper Bound = 2.45
2.35 ≤ x < 2.45

4

A rectangle measures 5 cm by 8 cm correct to the nearest cm.

Minimum Area:

4.5 × 7.5 = 33.75 cm²
Maximum Area :
5.5 × 8.5 = 46.75 cm²
  • Common GCSE Mistakes
  • Rounded value: Do not use the rounded value itself as a bound.
  • Interval: Remember to use half of the rounding interval.
  • Upper bound: Do not include the upper bound.
  • Direction: Do not mix up lower and upper bounds.
  • Exam Tips
  • Identify exactly what the value was rounded to first.
  • Find half of that rounding interval before calculating the bounds.
  • Subtract half the interval for the lower bound.
  • Add half the interval for the upper bound.
  • Use ≤ for the lower bound and < for the upper bound.
  • For perimeter and area, choose the appropriate bounds for maximum or minimum values.

Skills Used in Bounds

Rounding

Estimation

Inequalities

Lower Bounds

Upper Bounds

Decimals

Perimeter

Area

Accuracy and Measures

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Printable Worksheet

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Exam-Style Questions

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Watch the Bounds Tutorial

Step-by-step walkthroughs of Bounds with clear methods and exam tips.

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Frequently Asked Questions

Bounds give the smallest and largest possible values when an exact value has been rounded.

The lower bound is the smallest possible original value.

The upper bound is the largest possible limit, but that upper endpoint is not included.

The upper-bound value would round to the next rounded value rather than the stated one.

Identify the rounding interval, find half of it, then subtract and add that amount.

Yes. Bounds can be used to calculate maximum and minimum possible perimeters and areas.