Upper and Lower Bounds

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Description of Levels

Level 1 - Numbers truncated or rounded up or down to a given multiple.

Level 2 - Quantities rounded to the nearest multiple.

Level 3 - Numbers rounded to a number of decimal places.

Level 4 - Discrete and continuous quantities rounded to a number of significant figures.

Level 5 - Mixed calculations involving upper and lower bounds.

Level 6 - Upper and lower bounds of algebraic expressions.

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Extension

Students who are also studying Physics may want to investigate a topic called Propagation of Uncertainties that uses these formulas.

$$ \text{If} \quad y= a \pm b \quad \text{then} \quad \Delta y = \Delta a + \Delta b $$ $$ \text{If} \quad y= \frac{ab}{c} \quad \text{then} \quad \frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c} $$ $$ \text{If} \quad y= a^n \quad \text{then} \quad \frac{\Delta y}{y} = \begin{vmatrix} n \frac{\Delta a}{a} \end{vmatrix} $$

The triangular symbols are the Greek letter delta and represent the errors or, more accurately, uncertainties.

1. The length of a pencil, p millimetres, is rounded to the nearest ten millimetres. The length is given as 180mm. What are the limits of accuracy?	≤ p <
2. A set of kitchen scales displays readings to the nearest ten grams. Find the upper and lower bounds in grams for a weight, w, displayed as 270g. 270g	g ≤ w < g
3. A digital thermometer measures temperatures (T) to the nearest two degrees. Find the range of possible temperatures if the display shows 62^o. 62^o	^o ≤ T < ^o
4. A number, D, is rounded to the nearest ten. The result is 460. What are the limits of accuracy?	≤ D <
5. The weight of a box of bolts (w) is rounded to the nearest 50g. If the weight of the box is given as 2500g What are the limits of accuracy.	g ≤ w < g
6. A number, F, is rounded to the nearest fifty. The result is 3450. What are the limits of accuracy?	≤ F <
7. The length of a fibre-optic cable (L) is rounded to the nearest fifty metres. If the length is given as 4000m find the range of values within which the actual length of the cable could be.	m ≤ L < m
8. A number, H, is rounded to the nearest twenty. The result is 1760. What are the limits of accuracy?	≤ H <
9. A plumbing supplier sells pipes cut to size. The length (L millimetres) of the cut pipe is rounded to the nearest twenty millimetres in order to calculate the cost. If the length of a particular piece of pipe is given as 1840 find the range of values within which the actual length could be.	≤ L <
10. A number, J, is rounded to the nearest one hundred. The result is 10800. What are the limits of accuracy?	≤ J <

Upper and Lower Bounds

Determine the upper and lower bounds when rounding quantities used in calculations.

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