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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
Check

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

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.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

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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.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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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

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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