GCSE Maths revision · Higher: number and algebra

Upper and lower bounds

Higher only

The idea

A measurement rounded to the nearest unit could be up to half a unit either way. For positive measurements, to make a calculation as big as possible, use the upper bound of what you add or multiply, and the lower bound of what you subtract or divide by.

Write the bounds of each value.
Choose the combination that gives the maximum (or minimum).
Upper bound of $\frac{a}{b}$ is $\frac{a_{\text{upper}}}{b_{\text{lower}}}$ .

Watch out Dividing by the upper bound gives the smaller answer, not the larger.

See it worked

Worked exampleCalculator allowed4 marks

A rectangle has length $12.4$ cm and width $8.7$ cm, each correct to 1 decimal place.

(a) Work out the upper bound for the area of the rectangle.

(b) Work out the lower bound for the perimeter of the rectangle.

(a) Upper bounds: $12.45$ and $8.75$ .
$12.45 \times 8.75 = 108.9375$
(b) Lower bounds: $12.35$ and $8.65$ .
$2(12.35 + 8.65) = 2 \times 21 = 42$

Answer (a) $108.9375\ \text{cm}^2$ ; (b) $42$ cm

Your turn

Question 1Calculator allowed3 marks

A lift has a maximum safe load of 100 kg. Each box has a mass of $2.5$ kg, correct to the nearest $0.1$ kg.

Can the lift definitely carry 40 boxes safely? You must show your working.

Question 2Calculator allowed4 marks

A sample has mass 35.6 g and volume 12.4 cm³, both rounded to the nearest 0.1 of their units. Density = mass $\div$ volume. Calculate the lower bound for the density, giving your answer to 3 decimal places.

Question 3Calculator allowed5 marks

A runner covers a distance of $250$ m, correct to the nearest 10 m, in a time of $32.4$ s, correct to the nearest $0.1$ s.

(a) Work out the lower bound for the runner's average speed. Give your answer to 3 significant figures.

(b) The upper bound for the speed is 7.88 m/s. Give the runner's speed to a suitable degree of accuracy. You must give a reason.

In past papers

40 official questions in 39 papers since 2017. The latest:

All GCSE Maths past papers

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