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NumberFoundation and Higher (some Higher only)

Error intervals and bounds GCSE questions with answers

8 error intervals and bounds questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.

An error interval shows the range of values a rounded or truncated number could have come from, written with inequality signs such as 85 ≤ x < 95. On Higher you also find the upper and lower bounds of a calculation, like a speed worked out from rounded measurements.

Quiz yourself: 8 questions

Worked example 1: error intervals

A pencil is x cm long. x = 12.2, correct to 1 decimal place. Write down the error interval for x.

  1. x was rounded to 1 decimal place, so the rounding unit is 0.1. x can be up to half of that either side: 0.1 ÷ 2 = 0.05.
  2. Lower bound: 12.2 − 0.05 = 12.15. Upper bound: 12.2 + 0.05 = 12.25.
  3. 12.15 rounds up to 12.2, so x can equal it: use ≤. 12.25 would round up to 12.3, so x must be less than it: use <.
  4. The error interval is 12.15 ≤ x < 12.25.

Answer 12.15 ≤ x < 12.25

01 / Quiz yourselfTry it, then tap to check

Error intervals and bounds questions to try

Work each one out on paper first. Then open the answer and compare your working with ours, step by step.

01 / Error intervalsBoth tiers

A runner takes x seconds to run 100 m. x = 12.6, correct to 1 decimal place. Write down the error interval for x.

Show the answer and working
  1. x was rounded to 1 decimal place, so the rounding unit is 0.1. x can be up to half of that either side: 0.1 ÷ 2 = 0.05.
  2. Lower bound: 12.6 − 0.05 = 12.55. Upper bound: 12.6 + 0.05 = 12.65.
  3. 12.55 rounds up to 12.6, so x can equal it: use ≤. 12.65 would round up to 12.7, so x must be less than it: use <.
  4. The error interval is 12.55 ≤ x < 12.65.

Answer 12.55 ≤ x < 12.65

02 / Error intervalsBoth tiers

A fish tank holds x litres of water. x = 50, correct to 1 significant figure. Write down the error interval for x.

Show the answer and working
  1. x was rounded to 1 significant figure, so the rounding unit is 10. x can be up to half of that either side: 10 ÷ 2 = 5.
  2. Lower bound: 50 − 5 = 45. Upper bound: 50 + 5 = 55.
  3. 45 rounds up to 50, so x can equal it: use ≤. 55 would round up to 60, so x must be less than it: use <.
  4. The error interval is 45 ≤ x < 55.

Answer 45 ≤ x < 55

03 / Error intervalsBoth tiers

A pencil is x cm long. x = 18.7, correct to 1 decimal place. Write down the error interval for x.

Show the answer and working
  1. x was rounded to 1 decimal place, so the rounding unit is 0.1. x can be up to half of that either side: 0.1 ÷ 2 = 0.05.
  2. Lower bound: 18.7 − 0.05 = 18.65. Upper bound: 18.7 + 0.05 = 18.75.
  3. 18.65 rounds up to 18.7, so x can equal it: use ≤. 18.75 would round up to 18.8, so x must be less than it: use <.
  4. The error interval is 18.65 ≤ x < 18.75.

Answer 18.65 ≤ x < 18.75

04 / Error intervalsBoth tiers

A journey takes x minutes. x = 81, correct to the nearest whole number. Write down the error interval for x.

Show the answer and working
  1. x was rounded to the nearest whole number, so the rounding unit is 1. x can be up to half of that either side: 1 ÷ 2 = 0.5.
  2. Lower bound: 81 − 0.5 = 80.5. Upper bound: 81 + 0.5 = 81.5.
  3. 80.5 rounds up to 81, so x can equal it: use ≤. 81.5 would round up to 82, so x must be less than it: use <.
  4. The error interval is 80.5 ≤ x < 81.5.

Answer 80.5 ≤ x < 81.5

05 / Upper and lower bounds of a calculationHigher only

A car travels 41 km, correct to the nearest kilometre, in 0.8 hours, correct to 1 decimal place. Work out the upper bound for its average speed. Give your answer in km/h to 2 decimal places.

You can use a calculator for this one.

Show the answer and working
  1. Speed = distance ÷ time.
  2. For the biggest speed, divide the biggest distance by the smallest time.
  3. Distance: upper bound 41.5. Time: lower bound 0.75.
  4. 41.5 ÷ 0.75 = 55.33333...
  5. To 2 decimal places: 55.33 km/h.

Answer 55.33 km/h

06 / Upper and lower bounds of a calculationHigher only

A rectangle is 17 cm long and 15 cm wide. Both lengths are correct to the nearest centimetre. Work out the lower bound for the area of the rectangle.

17 cm15 cm
Not drawn accurately

You can use a calculator for this one.

Show the answer and working
  1. Each length could be up to half of 1, which is 0.5, bigger or smaller than it says.
  2. For the lower bound of the area, use the smallest possible length and width: 16.5 cm and 14.5 cm.
  3. Area = 16.5 × 14.5 = 239.25 cm².

Answer 239.25 cm²

07 / Upper and lower bounds of a calculationHigher only

A rectangle is 7.4 cm long and 5.7 cm wide. Both lengths are correct to 1 decimal place. Work out the upper bound for the perimeter of the rectangle.

7.4 cm5.7 cm
Not drawn accurately

You can use a calculator for this one.

Show the answer and working
  1. Each length could be up to half of 0.1, which is 0.05, bigger or smaller than it says.
  2. For the upper bound of the perimeter, use the biggest possible length and width: 7.45 cm and 5.75 cm.
  3. Perimeter = 2 × (7.45 + 5.75) = 26.4 cm.

Answer 26.4 cm

08 / Upper and lower bounds of a calculationHigher only

A runner runs 800 m, correct to the nearest metre, in 119.8 seconds, correct to 1 decimal place. Work out the upper bound for her average speed. Give your answer in m/s to 2 decimal places.

You can use a calculator for this one.

Show the answer and working
  1. Speed = distance ÷ time.
  2. For the biggest speed, divide the biggest distance by the smallest time.
  3. Distance: upper bound 800.5. Time: lower bound 119.75.
  4. 800.5 ÷ 119.75 = 6.68476...
  5. To 2 decimal places: 6.68 m/s.

Answer 6.68 m/s

02 / What's covered2 kinds of question

Error intervals and bounds at GCSE and where marks go

  • Error intervalsBoth tiers
  • Upper and lower bounds of a calculationHigher only

Worked example 2: upper and lower bounds of a calculation

A rectangle is 15.5 cm long and 2.3 cm wide. Both lengths are correct to 1 decimal place. Work out the lower bound for the perimeter of the rectangle.

15.5 cm2.3 cm
Not drawn accurately

You can use a calculator for this one.

  1. Each length could be up to half of 0.1, which is 0.05, bigger or smaller than it says.
  2. For the lower bound of the perimeter, use the smallest possible length and width: 15.45 cm and 2.25 cm.
  3. Perimeter = 2 × (15.45 + 2.25) = 35.4 cm.

Answer 35.4 cm

A common mistake

Using ≤ at both ends. If x = 90 correct to 1 significant figure, the error interval is 85 ≤ x < 95. It must be < 95, because 95 itself would round up to 100.

03 / Practise itiPhone and iPad

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