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Upper and Lower Bounds

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

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This is level 6: upper and lower bounds of algebraic expressions. You can earn a trophy if you get at least 9 questions correct.

For this exercise you will need the following information:

$$v=401 \text{ to the nearest integer}\\ w=424 \text{ to three significant figures}\\ x=400 \text{ to the nearest hundred}\\ y=482.5 \text{ to one decimal place}\\ z=465.28 \text{ to two decimal places}$$

Give answers correct to the number of significant figures indicated.

1. Find the upper bound of \(w + v\). (3sf) Correct Wrong
2. Find the lower bound of \(y - w\). (4sf) Correct Wrong
3. Find the upper bound of \(y - z\). (5sf) Correct Wrong
4. Find the lower bound of \(w - (x - y)\). (5sf) Correct Wrong
5. Find the upper bound of \(\dfrac{x}{y}\). (4sf) Correct Wrong
6. Find the lower bound of \(\dfrac{v}{y-z}\). (5sf) Correct Wrong
7. Find the upper bound of \(\dfrac{2x-5}{500-y}\). (5sf) Correct Wrong
8. Find the lower bound of \(\dfrac{w^2}{xy}\). (5sf) Correct Wrong
9. Find the upper bound of \(5-\dfrac{2w-1}{z^2}\). (5sf) Correct Wrong
10. Find the lower bound of \(\dfrac{y}{x} - \dfrac{x}{y}\). (5sf) Correct Wrong
11. Find the upper bound of \( \sqrt{x^2-(v+w)}\). (6sf) Correct Wrong
12. Find the lower bound of \( \sqrt{ \dfrac{v^2}{y-w} \div \dfrac{5z-2x}{w}} \). (6sf) Correct Wrong

This is Upper and Lower Bounds level 6. You can also try:
Level 1 Level 2 Level 3 Level 4 Level 5


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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.

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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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.

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