Upper and lower bounds – Higher tier
All measurements are approximate. Everything we measure comes with a small margin of error, meaning the measured value is very close to – but not completely identical to – the real value. This could be down to human error or a slight sizing error on a measuring instrument. For example, if you measured your height to be 156 cm, your actual height may be 156.2 cm or 155.9 cm.
When a number is rounded, there is a group of measurements that the original number may be between called the limits of accuracy.
Consider a weight of 70 kilograms (kg) which has been measured to the nearest 10. Think about what could be the smallest number which would still round up to 70 kg. In this case it would be 65 kg, which is the lower bound.
Now think of the largest number which would still round down to 70 kg. It is \(74. \dot{9}\) kg. As the 9 is recurring, it is always possible to put an extra 9 at the end of the decimal number, giving a higher number that still rounds to 70 kg. Due to how close this final figure is to 75 kg, we use 75 kg as the upper bound.
It is important to note that whilst the upper bound is 75 kg, 75 kg itself does not round to 70 kg. This can be shown as an inequality \(65~\text{kg} \leq \text{weight} \textless 75~\text{kg}\).
Examples
Work out the upper bound and lower bound for the following measurements.
32 cm, measured to the nearest cm.
The degree of accuracy is to the nearest 1 cm.
\(1~\text{cm} \div 2 = 0.5~\text{cm}\)
Upper bound = \(32 + 0.5 = 32.5~\text{cm}\)
Lower bound = \(32 - 0.5 = 31.5~\text{cm}\)
140 cm, measured to the nearest 10 cm.
The degree of accuracy is to the nearest 10 cm.
\(10~\text{cm} \div 2 = 5~\text{cm}\)
Upper bound = \(140 + 5 = 145~\text{cm}\)
Lower bound = \(140 - 5 = 135~\text{cm}\)
8.4 cm, measured to the nearest 0.1 cm.
The degree of accuracy is to the nearest 0.1 cm.
\(0.1~\text{cm} \div 2 = 0.05~\text{cm}\)
Upper bound = \(8.4 + 0.05 = 8.45~\text{cm}\)
Lower bound = \(8.4 - 0.05 = 8.35~\text{cm}\)
Question
What is the upper bound and lower bound of 62 kg, measured to the nearest kg?
The degree of accuracy is to the nearest 1 kg.
\(1~\text{kg} \div 2 = 0.5~\text{kg}\)
Upper bound = \(62 + 0.5 = 62.5~\text{kg}\)
Lower bound = \(62 - 0.5 = 61.5~\text{kg}\)
Question
What is the upper bound and lower bound of 390 grams, measured to the nearest 10 grams?
The degree of accuracy is to the nearest 10 g.
\(10~\text{g} \div 2 = 5~\text{g}\)
Upper bound = \(390 + 5 = 395~\text{g}\)
Lower bound = \(390 - 5 = 385~\text{g}\)
Question
What is the upper bound and lower bound of 15.89 seconds (s), measured to the nearest 0.01 s?
The degree of accuracy is to the nearest 0.01 s.
\(0.01~\text{s} \div 2 = 0.005~\text{s}\)
Upper bound = \(15.89 + 0.005 = 15.895~\text{s}\)
Lower bound = \(15.89 - 0.005 = 15.885~\text{s}\)