# Upper and Lower Bounds

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

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This is level 4: discrete and continuous quantities rounded to a number of significant figures. You can earn a trophy if you get at least 7 questions correct.

 1. The number of matches in a box, M, is given as 50. This number has been rounded to one significant figure. What is the least and most matches that could be in the box? ≤ M ≤ 2. The number of trees in a forest, F, is given as 70. This number has been rounded to one significant figure. What is the least and most trees that could be in the forest? ≤ F ≤ 3. The number of points (integers) scored in a game, P, is given as 700. This number has been rounded to one significant figure. What is the lowest and highest number of points that could have been scored? ≤ P ≤ 4. A train's speed (v kph) is rounded to two significant figures. If the speed is given as 140 kph, what is the slowest and fastest it could have actually been travelling? ≤ v < 5. Another train's speed (w kph) is rounded to three significant figures. If the speed is also given as 140 kph, what is the slowest and fastest it could have actually been travelling? ≤ w < 6. The length of a curtain, L, is measured to be 1.22m to three significant figures. What is the shortest and longest the curtain could be? (give your answers in metres) ≤ L < 7. The length of a train, T, is listed as 64.0m to three significant figures. What, in metres, are the upper and lower bounds of the train's length? ≤ T < 8. The probability, p, of a certain event occuring has been calculated to be 0.0987 to three significant figures. What are the limits of accuracy? ≤ p < 9. The time taken (t seconds), for a particular insect to react to a sound was measured to be 0.800 seconds to three significant figures. What are the limits of accuracy? ≤ t < 10. A number, N, is 7.00 to three significant figures. What are the limits of accuracy? ≤ N <
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This is Upper and Lower Bounds level 4. You can also try:
Level 1 Level 2 Level 3 Level 5 Level 6

## Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

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## Description of Levels

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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.

Exam Style questions are in the style of GCSE or IB/A-level exam paper questions and worked solutions are available for Transum subscribers.

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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.

## Help Video

Some of the questions in level 4 are about descrete quantities such as the number of matches in a matchbox. In this case it is not possible to have a fraction of a match, the number of matches must be an integer.

Example: The number of matches in a box, $$n$$, is given as 80 to the nearest 10. The error bounds would be written as:

$$75 \le n \le 84$$

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