Rounding, Estimation and Bounds

Number

Choosing which bound to use

A calculation using rounded data has bounds of its own, and each one comes from a particular combination of the input bounds
Adding or multiplying: the largest result uses both upper bounds, the smallest uses both lower bounds
Subtracting or dividing: the largest result uses the upper bound of the first value with the lower bound of the second, and the smallest reverses that
Dividing by a smaller number gives a larger answer, which is the step most often taken the wrong way round

Exam tip

The instinct to pair "lower" with "lower" all the way through is what loses this mark: a bigger divisor makes a smaller result. For the smallest possible speed, take the shortest distance over the longest time. This specification asks only for upper and lower bounds — "error interval" appears in none of these 43 papers.

Worked example

Bounds of an area

A rectangle measures 8 cm by 5 cm, each correct to the nearest centimetre. Find the upper and lower bounds of its area.

Solution:

The half-unit for the nearest centimetre is 0.5 cm
The length lies in 7.5 ⩽ l < 8.5 and the width in 4.5 ⩽ w < 5.5
Area is a product, so the upper bound uses both upper bounds: 8.5 × 5.5 = 46.75 cm²
The lower bound uses both lower bounds: 7.5 × 4.5 = 33.75 cm²