Rounding, Estimation & Bounds Flashcards

All 13 cards in this deck

What are the steps to round a number to a given number of significant figures?

e.g. write $7357$ correct to 3 significant figures

1. Count significant figures from the first non-zero digit: $7, 3, 5$
2. Look at the next digit ($7$): 5 or more, so round up
3. Fill with zeros to keep the place value: $7360$

Which digit is the first significant figure of $0.00408$?

$4$ — the first non-zero digit.

What are the steps to work out an estimate for a calculation?

e.g. estimate $513 \times 0.81$

1. Round each number to 1 significant figure: $500$ and $0.8$
2. Do the easier calculation: $500 \times 0.8$
3. State the estimate: $\approx 400$

How do you estimate a square root that is not exact?

e.g. estimate $\sqrt{80}$

Round to the nearest square number: $\sqrt{81}=9$, so $\sqrt{80}\approx 9$.

True or false? When estimating a division, rounding the number you are dividing by down makes the estimate smaller.

False. Dividing by a smaller number gives a larger answer, so it makes the estimate bigger (an over-estimate).

How do you write the error interval for a value that has been rounded?

e.g. a length is $90$ m correct to the nearest metre

Go half the rounding unit either side: $89.5 \text{ m} \le \text{length} < 90.5 \text{ m}$.

How do you write the error interval for a value that has been truncated?

e.g. $N$ truncated to 1 digit gives $7$

From the truncated value up to the next value: $7 \le N < 8$.

To find the lower bound of $a-b$, which bounds of $a$ and $b$ do you use?

$\text{LB}(a)-\text{UB}(b)$ — smallest $a$ take away largest $b$.

Which bounds do you use to test whether an average speed could have been greater than a stated value, when the distance and time are both rounded?

The upper bound of the speed: $\text{UB(distance)} \div \text{LB(time)}$, then compare it with the stated value.

What are the steps to give a calculated value to a suitable degree of accuracy by considering bounds?

e.g. $d=\dfrac{c^3}{8}$ where $c=10.9$ correct to 3 s.f.

1. Use $\text{LB}(c)=10.85$ and $\text{UB}(c)=10.95$ in the formula: $159.66\ldots$ and $164.11\ldots$
2. Round both bounds until they agree: both give $160$
3. State it with the reason: $d=160$ since both bounds round to $160$

True or false? If a distance is measured to the nearest 5 km instead of the nearest km, the maximum possible average speed for a fixed time interval increases.

True. The bounds are wider ($\pm 2.5$ km instead of $\pm 0.5$ km), so the upper bound of the distance is larger.

In an estimate, both numbers in a multiplication are rounded up to 1 significant figure. Is the estimate an over-estimate or an under-estimate?

e.g. $513 \times 0.81 \approx 600 \times 0.9$

An over-estimate.
Both factors are larger than the true values, so the product is larger.

True or false? When estimating a division, rounding the number being divided up and rounding the number you are dividing by down gives an over-estimate.

True.
A larger number divided by a smaller number gives a bigger answer.