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 easily taken the wrong way round
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
Exam tip
Write each measurement's bounds down first
Questions on the bounds of a calculation were asked in 10 of the 21 papers, and every one of those schemes gave a mark for a correct bound of a measurement seen in the working, such as 7.5 for 8 cm to the nearest centimetre. Write each bound down before you combine them: if you then pair them the wrong way or slip in the arithmetic, that mark is still yours.
Worked example
Lower bound of a density
A metal block has a mass of 90 g, correct to the nearest 5 g, and a volume of 12 cm³, correct to the nearest cm³. Calculate the lower bound of its density. [density = mass ÷ volume]
Solution:
- Half of 5 g is 2.5 g, so the mass lies in 87.5 ⩽ m < 92.5
- Half of 1 cm³ is 0.5 cm³, so the volume lies in 11.5 ⩽ V < 12.5
- Density is a division, so the smallest density takes the smallest mass over the largest volume
- Lower bound = 87.5 ÷ 12.5 = 7 g/cm³
- Using both lower bounds instead gives 87.5 ÷ 11.5 = 7.6…, which is bigger than 7 and so cannot be the lower bound