Apply and interpret upper and lower bounds
If a value is rounded, its upper or lower bound as appropriate should be used when finding the largest or smallest possible answers to a calculation.
Examples
A bag of flour has a mass of 1500 g (to the nearest 10 g). Find the upper bound for the mass of 10 bags of flour.
The upper bound of 1500 g rounded to the nearest 10 g is 1505 g. The upper bound for 10 bags of flour is therefore 10 x 1505 g = 15050 g ( = 15.05 kg).
Example
Jack is 1.8 m tall (rounded to the nearest 10 cm). Ella is 1.63 m tall (rounded to the nearest cm). What is the smallest possible difference in their heights?
To find the smallest difference, we need to use the lower bound for Jackâs height and the upper bound for Ellaâs height. The lower bound of 1.8 m (rounded to the nearest 10 cm) is 1.75 m. The upper bound of 1.63 m (rounded to the nearest cm) is 1.635 m.
The smallest possible difference in Jack and Ellaâs height is 1.75 m â 1.635 m = 0.115 m = 11.5 cm.
Example
A piece of A4 paper is 21.1 cm by 29.7 cm measured to the nearest 0.1 cm. What are the lower and upper bounds of its area, in square centimetres, to 1 dp?
This problem requires the biggest and smallest areas possible. What are the upper and lower bounds of the measurements and how should they be combined to achieve the correct outcomes?
In this example, the maximum area is found by multiplying the two upper bounds. The minimum area is found by multiplying the two lower bounds.
Maximum area
\(21.15~\text{cm} \times 29.75~\text{cm} = 629.2125~\text{cm}^2\)
\(629.2125~\text{cm}^2 \approx 629.2~\text{cm}^2 \:\text{(1 dp)}\)
Minimum area
\(21.05~\text{cm} \times 29.65~\text{cm} = 624.1325~\text{cm}^2\)
\(624.1325~\text{cm}^2 \approx 624.1~\text{cm}^2 \:\text{(1 dp)}\)
The following rules help to decide which bounds to use when doing combinations and calculations.
| Operation | Rule |
| Adding | \(\text{Upper bound} + \text{upper bound} = \text{upper bound}\)\(\text{Lower bound} + \text{lower bound} = \text{lower bound}\) |
| Subtracting | \(\text{Upper bound} - \text{lower bound} = \text{upper bound}\)\(\text{Lower bound} - \text{upper bound} = \text{lower bound}\) |
| Multiplying | \(\text{Upper bound} \times \text{upper bound} = \text{upper bound}\)\(\text{Lower bound} \times \text{lower bound} = \text{lower bound}\) |
| Dividing | \(\text{Upper bound} \div \text{lower bound} = \text{upper bound}\)\(\text{Lower bound} \div \text{upper bound} = \text{lower bound}\) |
| Adding | |
|---|---|
| Rule | \(\text{Upper bound} + \text{upper bound} = \text{upper bound}\)\(\text{Lower bound} + \text{lower bound} = \text{lower bound}\) |
| Subtracting | |
|---|---|
| Rule | \(\text{Upper bound} - \text{lower bound} = \text{upper bound}\)\(\text{Lower bound} - \text{upper bound} = \text{lower bound}\) |
| Multiplying | |
|---|---|
| Rule | \(\text{Upper bound} \times \text{upper bound} = \text{upper bound}\)\(\text{Lower bound} \times \text{lower bound} = \text{lower bound}\) |
| Dividing | |
|---|---|
| Rule | \(\text{Upper bound} \div \text{lower bound} = \text{upper bound}\)\(\text{Lower bound} \div \text{upper bound} = \text{lower bound}\) |
Question
A = 34 cm to the nearest cm.
B = 11.2 cm to 1 decimal place.
C = 200 cm to 1 significant figure.
Calculate:
- the upper bound for \(A + B\)
- the lower bound for \(C - B\)
- the lower bound for \(A \times C\)
- the upper bound for \(C \div B\)
Upper bound of A = 34.5 cm
Lower bound of A = 33.5 cm
Upper bound of B = 11.25 cm
Lower bound of B = 11.15 cm
Upper bound of C = 250 cm
Lower bound of C = 150 cm
- The upper bound of \(A + B\): \(34.5~\text{cm} + 11.25~\text{cm} = 45.75~\text{cm}\)
- The lower bound of \(C - B\): \(150~\text{cm} - 11.25~\text{cm} = 138.75~\text{cm}\)
- The lower bound of \(A \times C\): \(33.5~\text{cm} \times 150~\text{cm} = 5025~\text{cm}^2\)
- The upper bound of \(C \div B\): \(250~\text{cm} \div 11.15~\text{cm} = 22.42 \:\text{(2 dp)}\)