Giving Answers to the Right Accuracy

How rounding to decimal places differs from rounding to significant figures, which of the two to use, and how accurately to give an answer when the question does not say.

Before you start

To round a number to a given number of significant figures (s.f.), there are four steps. (1) Start at the first non-zero digit and count the figures asked for: the last one counted is the last digit kept. (2) Look only at the next digit. (3) If it is 5 or more, round up by adding 1 to the last digit kept; if it is 4 or less, round down by leaving the last digit kept as it is. (4) Before the decimal point, write a zero for each digit dropped; after the decimal point, leave the dropped digits out. So $38\,472 \approx 38\,000$ and $0.04736 \approx 0.047$ , both correct to 2 significant figures. The sign $\approx$ is read "is approximately equal to".

A zero that is one of the significant figures asked for is written, because it shows the accuracy: $0.03996 \approx 0.0400$ (correct to 3 s.f.), not $0.04$ . Zeros in front of the first non-zero digit (leading zeros) are never significant.

When a 9 is rounded up, write 0 and carry 1 to the place on its left. If the carry makes a new first significant figure, count the significant figures again from it: $9962 \approx 10\,000$ (correct to 2 s.f.).

To round a number to a given number of decimal places (d.p.), count that many digits after the decimal point, look only at the next digit, and round up or down in the same way. The digits dropped are left out: $6.283 \approx 6.28$ (correct to 2 d.p.) and $3.46 \approx 3.5$ (correct to 1 d.p.).

Rounding to Decimal Places or to Significant Figures

A question may ask for a number of decimal places or for a number of significant figures. The two are counted from different starting points, so the same number rounded to 2 decimal places and to 2 significant figures usually gives two different answers. Read which one is asked for.

Rule: where the counting starts

Decimal places are counted from the decimal point, to the right.

Significant figures are counted from the first non-zero digit, to the right.

The number $0.040962$ rounded in eight ways

How many	Correct to that many decimal places	Correct to that many significant figures
1	$0.0$	$0.04$
2	$0.04$	$0.041$
3	$0.041$	$0.0410$
4	$0.0410$	$0.04096$

Correct to 1 decimal place the answer is $0.0$ , which says only that the number is small. Correct to 1 significant figure it is $0.04$ , which still gives its size. For a number between 0 and 1, significant figures keep what decimal places can lose.

The answer $0.0410$ is in the table twice: correct to 4 decimal places and correct to 3 significant figures. Both times the last digit kept is the 9 and the next digit is 6, so the 9 is rounded up, and the zero that takes its place must be written.

The number $6048.5962$ rounded in six ways

How many	Correct to that many decimal places	Correct to that many significant figures
1	$6048.6$	$6000$
2	$6048.60$	$6000$
3	$6048.596$	$6050$

For a number greater than 1 it is the other way round. Rounding to decimal places is done after the decimal point, so here the whole part, $6048$ , stays as it is. (Correct to 2 decimal places, the 9 is rounded up and the zero that takes its place is written: $6048.60$ .) Significant figures start at the 6, so the first few of them round the whole part itself.

Correct to 1 and to 2 significant figures the answer is the same, $6000$ : the second significant figure is the zero after the 6, and the next digit, 4, rounds down. Correct to 4 significant figures the number is $6049$ , and correct to 5 it is $6048.6$ , the same as the answer correct to 1 decimal place.

Choosing between decimal places and significant figures

If the question names one of them, use that one.
Decimal places make every answer stop at the same place. They suit answers in one unit that are compared or added: an amount of money in dollars is given to 2 decimal places, which is to the nearest cent.
Significant figures give every answer the same number of figures, whatever its size. They suit very large numbers, very small numbers, and numbers of very different sizes.

Example 1: Which way of rounding suits both numbers?

A table of average masses, worked out on a calculator, gives 24.9471 kg for a sack of rice and 0.001268 kg for a paper clip.
(a) Round each mass to 2 decimal places.
(b) Round each mass to 3 significant figures.
(c) Which way of rounding is better for these two masses?

Solution

(a) $24.9471$ : the second digit after the decimal point is 4 and the next digit is 7, so round up: $24.95$ . $0.001268$ : the first two digits after the decimal point are 0 and 0, and the next digit is 1, so round down: $0.00$ .

(b) $24.9471$ : the significant figures are 2, 4 and 9, and the next digit is 4, so round down: $24.9$ . $0.001268$ : the significant figures are 1, 2 and 6, and the next digit is 8, so round up: the 6 becomes 7. That gives $0.00127$ .

(c) Correct to 2 decimal places, the paper clip has a mass of 0.00 kg: nothing is left of it. Correct to 3 significant figures, each mass keeps its three most important digits.

(a) The masses are 24.95 kg and 0.00 kg (correct to 2 d.p.).
(b) The masses are 24.9 kg and 0.00127 kg (correct to 3 s.f.).
(c) Rounding to significant figures is better here, because the two masses are of very different sizes.

Answers Correct to 3 Significant Figures

Many calculations do not come out exactly: a division may give a decimal that never ends. If the question says how accurate the answer must be, do as it says. If it does not, there is a rule. The syllabus for Singapore's G3 Mathematics examination states it in these words:

Rule: when the question does not say

"Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question."

In other words:

If the question states an accuracy, such as "correct to 2 decimal places" or "to the nearest cent", use it.
If an answer is not exact, give it correct to 3 significant figures, unless it is an angle in degrees.
If an angle in degrees is not exact, give it correct to 1 decimal place.

The rule is only about answers that are not exact. An exact answer is left as it is: it is not rounded, and it needs no extra zeros.

Exact and non-exact answers

An answer is exact if it can be written down completely: a whole number, a decimal that ends, a fraction, or a multiple of $\pi$ when the question asks for one. Here "exact" is about the calculation. The numbers in a question are taken just as they are given, even when they are measurements.

As a decimal, an answer is not exact (non-exact) if its digits go on without end, so that you have to stop writing somewhere: a recurring decimal such as $4.8333\ldots$ , the value of a root that does not come out exactly, such as $\sqrt{7} = 2.645751\ldots$ , or a value worked out on a calculator with $\pi$ . (The dots of a recurring decimal, as in $4.8\dot{3}$ , show the whole of it in short, but its digits still go on without end, and the rule still asks for the fraction or for 3 significant figures.)

Exact or not exact?

The answer you get	Is it exact?	What to write
$84 \div 7 = 12$	exact	12, not $12.0$
$3 \div 4 = 0.75$	exact	$0.75$ , not $0.750$
$1.25 \times 1.25 = 1.5625$	exact	$1.5625$
$\frac{1}{2} + \frac{1}{6} = \frac{2}{3} = 0.6666\ldots$	exact as a fraction, not exact as a decimal	$\frac{2}{3}$ , or as a decimal $0.667$ (correct to 3 s.f.)
$29 \div 6 = 4\frac{5}{6} = 4.8333\ldots$	exact as a fraction, not exact as a decimal	$4\frac{5}{6}$ , or as a decimal $4.83$ (correct to 3 s.f.)
$\sqrt{7} = 2.645751\ldots$	not exact as a decimal	$2.65$ (correct to 3 s.f.)
$\pi \times 9 = 28.274333\ldots$	not exact as a decimal	$28.3$ (correct to 3 s.f.)

An exact answer is not rounded, even if it has more than 3 significant figures: $1.5625$ stays as it is. And it needs no extra zeros: write 12, not $12.0$ , which would look like a rounded answer. An amount of money in dollars and cents is the one exception. It is written with 2 decimal places, so an exact answer of 5.4 dollars is written $5.40. A question about money nearly always states the accuracy, "to the nearest cent". If it does not, give the amount to the nearest cent, not to 3 significant figures.

The same value can be exact in one form and not exact in another. A fraction such as $\frac{2}{3}$ is exact, so it may be left as a fraction. Written as a decimal it is $0.6666\ldots$ , which never ends, so the decimal has to be rounded: $0.667$ (correct to 3 s.f.).

In the same way, $29 \div 6$ has an exact answer, the fraction $4\frac{5}{6}$ , and a decimal answer that has to be rounded, $4.83$ (correct to 3 s.f.). So read what the question asks for. If it asks for a decimal ("Give your answer as a decimal"), give the rounded decimal. If it asks for a fraction or for the exact value, give the fraction. If it asks for neither, either answer is correct. A length, a mass or another measurement is usually given as a decimal.

A question may also ask for an answer "in terms of $\pi$ ". Then $\pi \times 9$ is written as $9\pi$ , which is exact, and it is not rounded.

A root that does not come out exactly has an exact form too: the root itself, $\sqrt{7}$ . Unless a question asks for the exact value, give a root as a decimal, and the decimal has to be rounded: $\sqrt{7} \approx 2.65$ (correct to 3 s.f.).

Example 2: No accuracy is stated

None of these questions says how accurate the answer should be. Give each answer as a decimal.
(a) A ribbon 25 m long is cut into 7 equal pieces. Find the length of each piece.
(b) A round cake is cut from its centre into 7 equal slices. Find the angle at the centre of the cake that 3 of the slices take up together.
(c) A rectangle is 4.5 cm long and 1.2 cm wide. Find its area.

Solution

(a) $25 \div 7 = 3.571428\ldots$ , a decimal that does not end, so give the answer correct to 3 significant figures. They are 3, 5 and 7, and the next digit is 1, so round down. (The exact answer is the fraction $3\frac{4}{7}$ m. It would be a correct answer too if the question did not ask for a decimal.)

(b) The angles at a point add up to $360^\circ$ , so each slice takes up $\frac{1}{7}$ of $360^\circ$ and 3 slices take up $\frac{3}{7} \times 360^\circ = 154.285714\ldots^\circ$ . This is an angle in degrees and its decimal does not end, so give it correct to 1 decimal place. The first decimal place has 2 and the next digit is 8, so round up: $154.3^\circ$ . (Correct to 3 significant figures it would be $154^\circ$ , which is not what the rule asks for an angle.)

(a) Each piece is about 3.57 m long (correct to 3 s.f.).
(b) The 3 slices take up an angle of about $154.3^\circ$ at the centre (correct to 1 d.p.).
(c) The area of the rectangle is 5.4 $\text{cm}^{2}$ .

Example 3: Square roots and circles

None of these questions says how accurate the answer should be. Find each answer.
(a) A square has an area of 30 $\text{cm}^{2}$ . Find the length of its side.
(b) A circle has a radius of 4 cm. Find its circumference, using the $\pi$ key of your calculator.
(c) A square has an area of 6.25 $\text{cm}^{2}$ . Find the length of its side.

Solution

(a) The side of a square is the square root of its area, and $\sqrt{30} = 5.477225\ldots$ , a decimal that does not end, so it is not exact. Give it correct to 3 significant figures. They are 5, 4 and 7, and the next digit is 7, so round up: the 7 becomes 8.

(b) Circumference $= 2 \times \pi \times \text{radius} = 2 \times \pi \times 4 = 25.132741\ldots$ , a decimal that does not end, so it is not exact. The significant figures are 2, 5 and 1, and the next digit is 3, so round down.

(c) $\sqrt{6.25} = 2.5$ , because $2.5 \times 2.5 = 6.25$ . This is exact, so the answer is left as it is. There is no need to write $2.50$ .

(a) The side of the square is about 5.48 cm long (correct to 3 s.f.).
(b) The circumference of the circle is about 25.1 cm (correct to 3 s.f.).
(c) The side of the square is 2.5 cm long.

In a calculation with several steps, keep the calculator's full value in the working and round only the final answer. Rounding too early can make the final answer wrong, which is the subject of the lesson on rounding errors.

Common Mistakes

Mixing up decimal places and significant figures. $0.00462$ is $0.00$ correct to 2 decimal places and $0.0046$ correct to 2 significant figures.
Rounding an exact answer, or adding a zero to it. $4.5 \times 1.2 = 5.4$ exactly, so write $5.4$ . The value of $5.40$ is the same, but the extra zero makes an exact answer look like a rounded one.
Giving 2 significant figures out of habit. When no accuracy is stated, $0.666666\ldots$ is given as $0.667$ (correct to 3 s.f.), not $0.67$ .
Giving an angle in degrees to 3 significant figures. When no accuracy is stated, $154.285714\ldots^\circ$ is given as $154.3^\circ$ (correct to 1 d.p.), not $154^\circ$ .
Rounding when the exact answer is asked for. If a question asks for a fraction, for the exact value, or for an answer in terms of $\pi$ , give $\frac{2}{3}$ or $9\pi$ , not $0.667$ or $28.3$ .

Practice

Work each question on paper before you open its solution.

Question 1

Round each number to 2 decimal places, and then to 2 significant figures.
(a) $3.04961$
(b) $0.00738$
(c) $517.068$

ShowHide solution

(a) 2 decimal places: the second digit after the decimal point is 4 and the next digit is 9, so round up: $3.05$ . 2 significant figures: they are 3 and 0, and the next digit is 4, so round down: $3.0$ . The 0 is written because it is the second significant figure.

(b) 2 decimal places: the first two digits after the decimal point are 0 and 0, and the next digit is 7, so round up: the second 0 becomes 1. That gives $0.01$ . 2 significant figures: they are 7 and 3, and the next digit is 8, so round up: the 3 becomes 4. That gives $0.0074$ .

(c) 2 decimal places: the second digit after the decimal point is 6 and the next digit is 8, so round up: $517.07$ . 2 significant figures: they are 5 and 1, and the next digit is 7, so round up: the 1 becomes 2. The 7 is before the decimal point, so a zero is written for it: 520.

(a) $3.04961 \approx 3.05$ (correct to 2 d.p.) and $3.04961 \approx 3.0$ (correct to 2 s.f.)
(b) $0.00738 \approx 0.01$ (correct to 2 d.p.) and $0.00738 \approx 0.0074$ (correct to 2 s.f.)
(c) $517.068 \approx 517.07$ (correct to 2 d.p.) and $517.068 \approx 520$ (correct to 2 s.f.)

Question 2

None of these questions says how accurate the answer should be. For each one, say whether the answer has to be rounded, and then give the answer as a decimal.
(a) $7.5 \times 0.24$
(b) $46 \div 9$
(c) $\sqrt{12.25}$
(d) $\sqrt{12}$
(e) $135 \div 8$

ShowHide solution

(a) $7.5 \times 0.24 = 1.8$ , which is exact. It is not rounded, and it needs no extra zero.

(b) $46 \div 9 = 5.111111\ldots$ , a decimal that does not end, so it is rounded to 3 significant figures. They are 5, 1 and 1, and the next digit is 1, so round down. (The exact value is the fraction $5\frac{1}{9}$ , but the question asks for a decimal.)

(d) $\sqrt{12} = 3.464101\ldots$ , a root that does not come out exactly, so it is rounded to 3 significant figures. They are 3, 4 and 6, and the next digit is 4, so round down.

(e) $135 \div 8 = 16.875$ , a decimal that ends, so it is exact. It is not rounded, although it has 5 significant figures.

(a) Not rounded: $7.5 \times 0.24 = 1.8$
(b) Rounded: $46 \div 9 \approx 5.11$ (correct to 3 s.f.)
(c) Not rounded: $\sqrt{12.25} = 3.5$
(d) Rounded: $\sqrt{12} \approx 3.46$ (correct to 3 s.f.)
(e) Not rounded: $135 \div 8 = 16.875$

Question 3

A computer drawing of a running track gives the length of its straight part as 84.3927 m and the width of the white line painted along it as 0.05084 m.
(a) Round each length to 2 decimal places.
(b) Round each length to 3 significant figures.
(c) How many significant figures does each answer to (a) have? Which way of rounding gives both lengths the same number of significant figures?

ShowHide solution

(a) $84.3927$ : the second digit after the decimal point is 9 and the next digit is 2, so round down: $84.39$ . $0.05084$ : the second digit after the decimal point is 5 and the next digit is 0, so round down: $0.05$ .

(b) $84.3927$ : the significant figures are 8, 4 and 3, and the next digit is 9, so round up: the 3 becomes 4. That gives $84.4$ . $0.05084$ : the significant figures are 5, 0 and 8, and the next digit is 4, so round down: $0.0508$ .

(c) $84.39$ has 4 significant figures, but $0.05$ has only 1, because the zeros in front are not counted. Correct to 2 decimal places, the long length keeps 4 significant figures and the short one only 1. Correct to 3 significant figures, each length keeps 3 figures.

(a) The lengths are 84.39 m and 0.05 m (correct to 2 d.p.).
(b) The lengths are 84.4 m and 0.0508 m (correct to 3 s.f.).
(c) The answers to (a) have 4 significant figures and 1 significant figure. Rounding to 3 significant figures gives both lengths the same number of significant figures.

Question 4

A rectangular field is 48.6 m long and 27.35 m wide. Find its area, giving your answer correct to
(a) 1 decimal place
(b) 3 significant figures
(c) 2 significant figures

ShowHide solution

Area $= 48.6 \times 27.35 = 1329.21$ . The product is exact, but the question states an accuracy each time, so use it.

(a) The first digit after the decimal point is 2 and the next digit is 1, so round down: $1329.2$ .

(b) The significant figures are 1, 3 and 2, and the next digit is 9, so round up: the 2 becomes 3. The 9 is before the decimal point, so a zero is written for it. The digits after the decimal point are left out. That gives $1330$ .

(c) The significant figures are 1 and 3, and the next digit is 2, so round down. A zero is written for the 2 and for the 9. That gives $1300$ .

(a) The area is 1329.2 $\text{m}^{2}$ (correct to 1 d.p.).
(b) The area is 1330 $\text{m}^{2}$ (correct to 3 s.f.).
(c) The area is 1300 $\text{m}^{2}$ (correct to 2 s.f.).

Question 5

None of these questions says how accurate the answer should be. Give each answer as a decimal.
(a) 5 kg of flour is shared equally among 6 containers. Find the mass of flour in each container.
(b) A lucky-draw wheel is divided into 11 equal parts by lines from its centre. Find the angle at the centre that 4 of the parts take up together.
(c) 8 identical books have a total mass of 3.4 kg. Find the mass of one book.
(d) 7 friends spent $100 altogether on a meal. Find the average amount that each friend spent.

ShowHide solution

(a) $5 \div 6 = 0.833333\ldots$ , a decimal that does not end, so give the answer correct to 3 significant figures. They are 8, 3 and 3, and the next digit is 3, so round down. (The exact answer is the fraction $\frac{5}{6}$ kg. It would be a correct answer too if the question did not ask for a decimal.)

(b) The angles at a point add up to $360^\circ$ , so each part takes up $\frac{1}{11}$ of $360^\circ$ and 4 parts take up $\frac{4}{11} \times 360^\circ = 130.909090\ldots^\circ$ . This is an angle in degrees and its decimal does not end, so give it correct to 1 decimal place. The first decimal place has 9 and the next digit is 0, so round down: $130.9^\circ$ . (Correct to 3 significant figures it would be $131^\circ$ , which is not what the rule asks for an angle.)

(d) $100 \div 7 = 14.285714\ldots$ , a decimal that does not end, so it has to be rounded. The answer is an amount of money, so give it to the nearest cent, which is 2 decimal places. The second digit after the decimal point is 8 and the next digit is 5, so round up: the 8 becomes 9. That gives $14.29. (Correct to 3 significant figures it would be $14.3, which is not how an amount of money is given.)

(a) Each container has about 0.833 kg of flour (correct to 3 s.f.).
(b) The 4 parts take up an angle of about $130.9^\circ$ at the centre (correct to 1 d.p.).
(c) The mass of one book is 0.425 kg.
(d) Each friend spent about $14.29 on average (correct to the nearest cent).

Question 6

A basketball team scored 451 points in its first 7 games and 530 points in its next 8 games. The question does not say how accurate the answers should be.
(a) Find the average number of points in a game for the first 7 games. Give your answer as a decimal.
(b) Find the average number of points in a game for the next 8 games. Give your answer as a decimal.
(c) Which of the two answers had to be rounded, and why?

ShowHide solution

(a) Average $= 451 \div 7 = 64.428571\ldots$ , a decimal that does not end, so give it correct to 3 significant figures. They are 6, 4 and 4, and the next digit is 2, so round down. (The exact average is the fraction $64\frac{3}{7}$ , but the question asks for a decimal.)

(b) Average $= 530 \div 8 = 66.25$ , a decimal that ends. It is exact, so it is left as it is.

(c) Only the answer to (a) had to be rounded, because its decimal does not end. $66.25$ has 4 significant figures, but an exact answer is not cut down to 3. Writing $66.3$ would throw away part of a value that is known completely.

(a) The average is about 64.4 points in a game (correct to 3 s.f.).
(b) The average is 66.25 points in a game.
(c) Only (a) had to be rounded: $451 \div 7$ does not end as a decimal, and $530 \div 8 = 66.25$ is exact.

Question 7

None of these questions says how accurate the answer should be. Find each answer. Use the $\pi$ key of your calculator where $\pi$ is needed.
(a) A square has an area of 50 $\text{cm}^{2}$ . Find the length of its side.
(b) A round pond has a diameter of 12 m. Find its circumference.
(c) A square has an area of 2.25 $\text{m}^{2}$ . Find the length of its side.
(d) Find the exact value of $\frac{1}{3} + \frac{1}{4}$ .

ShowHide solution

(a) The side of a square is the square root of its area, and $\sqrt{50} = 7.071067\ldots$ , a decimal that does not end, so it is not exact. Give it correct to 3 significant figures. They are 7, 0 and 7, and the next digit is 1, so round down.

(b) Circumference $= \pi \times \text{diameter} = \pi \times 12 = 37.699111\ldots$ , a decimal that does not end, so it is not exact. The significant figures are 3, 7 and 6, and the next digit is 9, so round up: the 6 becomes 7.

(c) $\sqrt{2.25} = 1.5$ , because $1.5 \times 1.5 = 2.25$ . This is exact, so the answer is left as it is. There is no need to write $1.50$ .

(d) $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$ . A fraction is exact, so this is the exact value, and it is left as a fraction. (As a decimal it is $0.583333\ldots$ , which does not end. A question that asked for a decimal would be answered with $0.583$ , correct to 3 significant figures.)

(a) The side of the square is about 7.07 cm long (correct to 3 s.f.).
(b) The circumference of the pond is about 37.7 m (correct to 3 s.f.).
(c) The side of the square is 1.5 m long.
(d) $\frac{1}{3} + \frac{1}{4} = \frac{7}{12}$

Question 8

A school has 84 pupils in Secondary 1. Of these, 31 come to school by bus and 21 walk. A pie chart is drawn in which the whole circle stands for all 84 pupils. No accuracy is stated. Give each answer as a whole number or a decimal.
(a) Find the angle at the centre of the part of the pie chart for the pupils who come by bus.
(b) Find the angle at the centre of the part for the pupils who walk.

ShowHide solution

The angles at the centre of a circle add up to $360^\circ$ . A group of pupils takes the same fraction of $360^\circ$ as it is of the 84 pupils.

(a) $\frac{31}{84} \times 360^\circ = 132.857142\ldots^\circ$ . This is an angle in degrees and its decimal does not end, so give it correct to 1 decimal place. The first decimal place has 8 and the next digit is 5, so round up: $132.9^\circ$ . (Correct to 3 significant figures it would be $133^\circ$ , which is not what the rule asks for an angle.)

(b) $\frac{21}{84} \times 360^\circ = \frac{1}{4} \times 360^\circ = 90^\circ$ . This is exact, so it is left as it is. There is no need to write $90.0^\circ$ .

(a) The angle for the pupils who come by bus is about $132.9^\circ$ (correct to 1 d.p.).
(b) The angle for the pupils who walk is $90^\circ$ .

Question 9

A round table top has a radius of 0.45 m. The area of a circle is $\pi \times \text{radius} \times \text{radius}$ .
(a) Find the area of the table top in terms of $\pi$ .
(b) Find the area of the table top, using the $\pi$ key of your calculator. No accuracy is stated.
(c) Which of your two answers is exact?

ShowHide solution

(a) Area $= \pi \times 0.45 \times 0.45 = 0.2025\pi$ , because $0.45 \times 0.45 = 0.2025$ . In terms of $\pi$ the answer is exact, so $0.2025$ is not rounded, although it has 4 significant figures.

(b) $\pi \times 0.45 \times 0.45 = 0.636172\ldots$ , a decimal that does not end, so give it correct to 3 significant figures. The zero in front is not significant. The significant figures are 6, 3 and 6, and the next digit is 1, so round down.

(c) $0.2025\pi$ is exact. $0.636$ is only correct to 3 significant figures, because the calculator value goes on without end.

(a) The area is $0.2025\pi$ $\text{m}^{2}$ .
(b) The area is about 0.636 $\text{m}^{2}$ (correct to 3 s.f.).
(c) The answer in terms of $\pi$ , $0.2025\pi$ $\text{m}^{2}$ , is exact.

Question 10

Each of these pupils answered a question that did not say how accurate the answer should be. Say whether each answer is given correctly. If it is not, say what is wrong and give the correct answer.
(a) For the area of a rectangle 2.5 m long and 1.3 m wide, Ali writes 3.3 $\text{m}^{2}$ .
(b) For $\sqrt{2.56}$ , Hui Min writes $1.60$ .
(c) A rope 20 m long is cut into 3 equal pieces. For the length of each piece as a decimal, Devi writes 6.67 m.
(d) A wheel has 14 spokes, equally spaced. For the angle between one spoke and the next, as a decimal, Farid writes $26^\circ$ .

ShowHide solution

(a) Not correct. $2.5 \times 1.3 = 3.25$ , which is exact. Ali has rounded an exact answer to 2 significant figures. It should be left as it is.

(b) The value is right, but the zero should not be there. $\sqrt{2.56} = 1.6$ exactly, because $1.6 \times 1.6 = 2.56$ , and an exact answer needs no extra zeros. Written as $1.60$ , it looks like an answer that has been rounded to 3 significant figures.

(c) Correct. $20 \div 3 = 6.666666\ldots$ , a decimal that does not end, so it is given correct to 3 significant figures. They are 6, 6 and 6, and the next digit is 6, so round up: the last 6 becomes 7.

(d) Not correct. The angles at a point add up to $360^\circ$ , and $360^\circ \div 14 = 25.714285\ldots^\circ$ . An angle in degrees whose decimal does not end is given correct to 1 decimal place. The first decimal place has 7 and the next digit is 1, so round down: $25.7^\circ$ . Farid has rounded to the nearest degree.

(a) Not correct: the area is 3.25 $\text{m}^{2}$ , which is exact.
(b) The value is right, but the zero should not be there: $\sqrt{2.56} = 1.6$ , which is exact.
(c) Correct: each piece is about 6.67 m long (correct to 3 s.f.).
(d) Not correct: the angle is about $25.7^\circ$ (correct to 1 d.p.).

Question 11

Calculate $3\frac{4}{7} \times \left(-2\frac{1}{15}\right)$ . Give your answer correct to
(a) 4 decimal places
(b) 3 significant figures

ShowHide solution

Write the mixed numbers as improper fractions: $3\frac{4}{7} = \frac{25}{7}$ and $2\frac{1}{15} = \frac{31}{15}$ .

A positive number times a negative number is negative: $\frac{25}{7} \times \left(-\frac{31}{15}\right) = -\frac{775}{105} = -\frac{155}{21}$

$155 \div 21 = 7.380952\ldots$ , so the product is $-7.380952\ldots$ , a negative number. Round its numerical value (the number without its sign) and keep the sign.

(a) The fourth digit after the decimal point is 9 and the next digit is 5, so round up: $7.3809 + 0.0001 = 7.3810$ . The 0 is the fourth decimal place, so it is written.

(b) The significant figures are 7, 3 and 8. The next digit is 0, so round down.

(a) $3\frac{4}{7} \times \left(-2\frac{1}{15}\right) \approx -7.3810$ (correct to 4 d.p.)
(b) $3\frac{4}{7} \times \left(-2\frac{1}{15}\right) \approx -7.38$ (correct to 3 s.f.)

Key Points

Decimal places are counted from the decimal point. Significant figures are counted from the first non-zero digit.
Decimal places make every answer stop at the same place, as with money. Significant figures suit numbers of very different sizes.
If the question states an accuracy, use it. If it does not, give an answer that is not exact correct to 3 significant figures, or, if it is an angle in degrees, correct to 1 decimal place.
An exact answer (a whole number, a decimal that ends, a fraction, or a multiple of $\pi$ when the question asks for one) is left as it is: it is not rounded, and it needs no extra zeros. Write 12, not $12.0$ , and $5.4$ , not $5.40$ : the extra zero makes an exact answer look like a rounded one.
An amount of money in dollars and cents is the one exception. It is written with 2 decimal places ($5.40), and when no accuracy is stated it is given to the nearest cent, not to 3 significant figures.
As decimals, a recurring decimal, a root that does not come out exactly and a calculator value with $\pi$ are not exact, so the decimal is rounded: $\sqrt{7} \approx 2.65$ (correct to 3 s.f.).
A value such as $29 \div 6$ is exact as a fraction, $4\frac{5}{6}$ , and not exact as a decimal, $4.8333\ldots$ If the question asks for a decimal, give $4.83$ (correct to 3 s.f.). If it asks for neither a decimal nor a fraction, either answer is correct.