Bounds and Error Intervals

Understanding the possible range of a rounded value

When a value is rounded, the exact number is not known, but it lies within a range. Bounds describe that range. Students usually study this at GCSE Higher and IGCSE level after learning rounding and estimation. Bounds questions often use lengths, masses, times and other measurements.

Key knowledge and rules

A value rounded to the nearest unit has a half-unit interval. For example, 12 cm to the nearest centimetre means 11.5 ≤ length < 12.5. The lower bound is included, but the upper bound is excluded. For one decimal place, use half of 0.1; for two decimal places, use half of 0.01. Review decimals and wider Number lessons

Step-by-step method

Identify the rounding accuracy, calculate half of that unit, subtract it for the lower bound and add it for the upper bound. Write the interval carefully using ≤ on the left and < on the right. In calculations, choose the combination of bounds that produces the requested maximum or minimum.

Worked examples

Example 1: 7.4 kg correct to one decimal place gives 7.35 ≤ m < 7.45. Example 2: 350 correct to the nearest 10 gives 345 ≤ n < 355. Example 3: A rectangle is 8 cm by 5 cm, each correct to the nearest centimetre. Its maximum possible area is just below 8.5 × 5.5 = 46.75 cm². This connects with Geometry Lessons

Common mistakes

Students often include the upper bound, use the wrong half-unit, or choose upper bounds for every value without considering whether a maximum or minimum is required. Writing the interval before calculating reduces errors. Skills from ratio and standard form may also appear in harder questions.

Practice exercises

Exercise 1: 24 cm is correct to the nearest centimetre. Answer: 23.5 ≤ x < 24.5. Exercise 2: 3.6 kg is correct to one decimal place. Answer: 3.55 ≤ m < 3.65. Exercise 3: A square side is 9 cm to the nearest centimetre. Find the greatest possible area. Answer: use the upper bound 9.5, so the area is less than 9.5² = 90.25 cm².

Connections and revision

Bounds link with percentages, fractions and algebra

Benefits of one-to-one maths tutoring

One-to-one lessons with maths tutor Ryan Harvey can target weak rounding knowledge, improve interval notation and build confidence with exam-style calculations.

Contact and lesson enquiry

Students and parents can contact MasterMaths Tutoring for personalised GCSE or IGCSE support.

Students can also review Ratio Lessons to strengthen proportional reasoning used in bounds calculations.