When a measurement has been rounded, its true value lies within half a unit either side. The lower bound is the smallest value that rounds to it and the upper bound is the largest. In calculations, choose the bounds that give the biggest or smallest result: adding and multiplying use like bounds, while subtracting and dividing use opposite bounds.
Bounds in calculations
- Bounds: given value ± half the rounding unit
- Maximum of a + b or a × b: upper × upper
- Minimum of a + b or a × b: lower × lower
- Maximum of a − b: upper a − lower b
- Maximum of a ÷ b: upper a ÷ lower b; minimum of a ÷ b: lower a ÷ upper b
Finding upper and lower bounds
Find the size of the rounding unit, then go half a unit either side. 15 m to the nearest metre has a unit of 1 m, so the bounds are 14.5 m and 15.5 m. A mass of 6.4 kg to 1 decimal place has a unit of 0.1 kg, so the bounds are 6.35 kg and 6.45 kg.
Bounds for sums and products
For addition and multiplication, the result is biggest when both inputs are biggest, and smallest when both are smallest. So the upper bound of an area is the upper bound of the length times the upper bound of the width.
Bounds for differences and quotients
Subtraction and division need opposite bounds. To make a − b as big as possible, use the biggest a and the smallest b. To make a ÷ b as big as possible, divide the biggest a by the smallest b. Mixing these up is the most common way to lose marks.
Speed, density and other formulae
Formulae like speed = distance ÷ time are divisions, so the upper bound of the speed uses the upper bound of the distance and the lower bound of the time. Write down every bound you use before calculating.
Giving a suitable answer
Some questions ask for a value to an appropriate degree of accuracy. Work out both the upper and lower bound of the result; the answer is the value that both round to, at the most accurate level they agree.
Worked examples
Example 1. A length is 15 m, to the nearest metre. Write down its lower and upper bounds.
- The unit is 1 m, so go 0.5 m either side.
Answer: Lower bound 14.5 m, upper bound 15.5 m.
Example 2. A rectangle is 12 cm by 7 cm, each to the nearest cm. Work out the upper bound of its area.
- Upper bounds: 12.5 cm and 7.5 cm.
- 12.5 × 7.5.
Answer: 93.75 cm².
Example 3. For the same rectangle, work out the lower bound of its area.
- Lower bounds: 11.5 cm and 6.5 cm.
- 11.5 × 6.5.
Answer: 74.75 cm².
Example 4. Two masses are 50 kg and 32 kg, each to the nearest kg. Work out the maximum possible difference between them.
- Maximum difference: upper of the larger minus lower of the smaller.
- 50.5 − 31.5.
Answer: 19 kg.
Example 5. A runner covers 200 m, to the nearest 10 m, in 25 s, to the nearest second. Work out the upper bound of the average speed, to 2 decimal places.
- Distance upper bound: 205 m. Time lower bound: 24.5 s.
- 205 ÷ 24.5 = 8.367…
Answer: 8.37 m/s.
Common mistakes
- Using a whole unit instead of half. 15 m to the nearest metre has bounds 14.5 m and 15.5 m.
- Using upper bounds for every calculation. Subtraction and division need one upper and one lower bound.
- Getting the rounding unit wrong. To the nearest 10 m, the bounds are 5 m either side.
- Dividing upper by upper for a maximum. Maximum of a ÷ b is upper a ÷ lower b.
- Not writing the bounds down. List each bound before calculating, so the method is clear.
How upper and lower bounds are examined
Bounds calculations are a Higher tier topic on Edexcel GCSE (1MA1) and International GCSE (4MA1) calculator papers, often worth 3 or 4 marks. Questions ask you to:
- write down the upper and lower bounds of a measurement
- find the upper or lower bound of an area, perimeter or total
- find bounds for a speed, density or other quotient
- decide on a suitable degree of accuracy for an answer
The error intervals and bounds worksheet has full answers, and the GCSE number revision engine creates more questions.
Bounds are written as error intervals, and they are used with speed, distance and time.
Frequently asked questions
What is the upper bound of a rounded value?
The given value plus half of the rounding unit.
How do you find the maximum of a ÷ b?
Divide the upper bound of a by the lower bound of b.
Why do subtraction and division use opposite bounds?
Because making b bigger makes a − b and a ÷ b smaller.