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. A number, A, if not a multiple of ten, is rounded down to the previous multiple of ten. The result is 100. What is the smallest the number could have been? What is the largest the number could have been? Type you answers into the boxes on the right to make the inequalities true.	≤ A <
2. A number, B, if not a multiple of fifty, is rounded up to the next multiple of fifty. The result is 250. What are the limits of accuracy?	< B ≤
3. Bluehound buses take an average of 380 minutes to travel from Northpoint to South Pier. This time has been rounded up to a multiple of ten if not already a multiple of ten. What are the error bounds of this time?	< t ≤
4. A number, D, is truncated to become a whole number. The result is 34. What is the smallest the number D could have been? What is the largest it could have been? Type you answers into the boxes on the right to make the inequalities true.	≤ D <
5. A number, E, is truncated to one decimal place. The result is 2.4. What is the smallest the number E could have been? What are the limits of accuracy?	≤ E <
6. You can receive points in a game for the number of minutes, m, you keep the trolls at bay. The time is rounded down to a multiple of five minutes. If a player's time is displayed as 660 what is the shortest and longest the time could have actually been?	≤ m <
7. A number, G, if not already a multiple of five is rounded up to the next multiple of five. The result is 735. What are the limits of accuracy?	< G ≤
8. A number, H, is rounded down to a multiple of twenty. The result is 800. What are the limits of accuracy?	≤ H <
9. A number, I, if not already a multiple of twenty is rounded up to the next multiple of twenty. The result is 960. What are the limits of accuracy?	< I ≤
10. A number, J, is rounded down to a multiple of fifty. The result is 1050. 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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