Upper and lower bounds

If a number or measurement has been rounded, it can be important to consider what possible values the exact value could have been. For example, if a kitchen unit is 600 mm wide to the nearest 10 mm, it could actually be any width from 595 mm up to 605 mm – so it might not fit into a gap that is exactly 600 mm wide. To describe all the possible values that a rounded number could be, we use lower and upper bounds.

Lower and Upper Bounds

The lower bound is the smallest value that would round up to the estimated value.

The upper bound is the smallest value that would round up to the next estimated value.

For example, a mass of 70 kg, rounded to the nearest 10 kg, has a lower bound of 65 kg, because 65 kg is the smallest mass that rounds to 70 kg. The upper bound is 75 kg, because 75 kg is the smallest mass that would round up to 80kg.

This can be shown as an error interval (using inequality symbols): 65~\text{kg} \leq \text{weight} \textless 75~\text{kg}.

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A quick way to calculate upper and lower bands is to halve the degree of accuracy specified, then add this to the rounded value for the upper bound and subtract it from the rounded value for the lower bound.

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 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 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 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 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}