Apply and interpret limits of accuracy - Higher tier
When A measurement rounded to some degree of accuracy. are combined, careful consideration of the possibilities is needed.
Example
A piece of A4 paper is 21 cm by 29.7 cm, both measurements correct to one decimal place. What are the lower and upper bounds of its area, in square centimetres, correct to one decimal place?
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
\(21.05~\text{cm} \times 29.75~\text{cm} = 626.2375~\text{cm}^2\)
\(626.2375~\text{cm}^2 \approx 626.2~\text{cm}^2 \:\text{(to one decimal place)}\)
Minimum area
\(20.95~\text{cm} \times 29.65~\text{cm} = 621.1675~\text{cm}^2\)
\(621.1675~\text{cm}^2 \approx 621.2~\text{cm}^2 \:\text{(to one decimal place)}\)
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 one decimal place.
C = 200 cm to one 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 \(A \times C\): \(33.5~\text{cm} \times 150~\text{cm} = 5,025~\text{cm}^2\)
- The upper bound of \(C \div B\): \(250~\text{cm} \div 11.15~\text{cm} = 22.42~\text{(two decimal places)}\)