Rounding
🎯 In this topic you will
- Round numbers to a given number of significant figures
🧠 Key Words
- round
- degree of accuracy
- decimal places (d.p.)
- significant figures (s.f.)
Show Definitions
- round: To adjust a number to a nearby value with fewer digits, making it simpler to work with.
- degree of accuracy: A measure of how close an approximation is to the true value, often linked to rounding.
- decimal places (d.p.): The number of digits shown after the decimal point in a number.
- significant figures (s.f.): The digits in a number that show its precision, starting from the first non-zero digit.
When you are asked to round a number, you will be told how accurate your answer should be. This is called the degree of accuracy.
For any degree of accuracy, the method is always the same.
Look at the digit in the position of the required degree of accuracy. What you do to this digit depends on the value of the digit to the right of it.
- If the value of the digit to its right is $5$ or more, increase the original digit by $1$.
- If the value of the digit to the right is less than $5$, leave the original digit as it is.
❓ EXERCISES
1. Round each of these numbers to two decimal places (2 d.p.). The first one has been done for you.
🔎 Reasoning Tip
Decimal precision: In question 1, make sure all your answers have exactly two digits after the decimal point.
a. $5.673 = 5.67$ (to 2 d.p.)
b. $8.421 = \square$
c. $39.555 = \square$
d. $0.487 = \square$
e. $138.2229 = \square$
f. $0.06901 = \square$
👀 Show answer
b. $8.42$ c. $39.56$ d. $0.49$ e. $138.22$ f. $0.07$
2. Arun and Sofia round the number $34.8972$ to two decimal places.
Arun: “I think the answer is $34.9$.”
Sofia: “I think the answer is $34.90$.”
a. Who is correct? Explain how they got this answer.
b. Explain the mistake that the other learner has made.
👀 Show answer
a. Sofia is correct — when rounding to two decimal places, we keep exactly two digits after the decimal, so $34.8972 \to 34.90$.
b. Arun only rounded to one decimal place ($34.9$) instead of two decimal places.
🧠 Think like a Mathematician
Scenario: A pedometer measures the distance that you walk. Liam’s pedometer shows 9.55 km. This distance is given to two decimal places.
Question: What distances might Liam have actually walked?
Discussion task: Consider how rounding to two decimal places affects the possible range of actual distances. Discuss in pairs or groups.
Follow-up prompts:
👀 show answer
- The reading 9.55 km to 2 decimal places means the actual distance could be any value from:
- Lower bound:$9.545$ km (rounded up to 9.55 km)
- Upper bound:$9.555$ km (just below this would still round down to 9.55 km)
- Possible range:$9.545 \leq \text{actual distance} < 9.555$ km
- This method of determining bounds works for any measurement given to a set number of decimal places — you take ± half of the smallest unit shown.
❓ EXERCISES
4. Round each of these numbers to three decimal places (3 d.p.).
a. $12.8943 = \square$
b. $127.99652 = \square$
c. $0.20053 = \square$
d. $9.349612 = \square$
👀 Show answer
a. $12.894$ b. $127.997$ c. $0.201$ d. $9.350$
5. Fina explains her method to round $17.825684$ to four decimal places (4 d.p.).
|
First, I draw a line after the digit in the fourth decimal place. |
$17.8256$|$84$ |
|
Then I circle the digit in the fifth decimal place. |
$17.8256$$8$$4$ |
|
The digit I have circled is $8$, so I must increase the digit before the line by $1$, so the $6$ becomes a $7$. |
$17.8257$($4\ \text{d.p.}$) |
a. Do you like this method that Fina uses?
b. What are the advantages and disadvantages of this method?
c. Can you think of a better/easier method to use? If you can, then write down an explanation of your method.
d. Explain how you would use Fina’s method to round another number to six decimal places.
👀 Show answer
a. Opinion-based; many may like it because it is visual and step-by-step.
b. Advantages: clear, systematic, reduces mistakes. Disadvantages: slower, requires extra writing.
c. Alternative: look directly at the digit after the desired decimal place and decide whether to round up or leave as is; faster and requires fewer steps.
d. Draw a line after the sixth decimal digit, circle the seventh, and apply the same rounding rule: if the circled digit is $\ge 5$, add $1$ to the sixth decimal place; if not, keep it the same.
6. Choose the correct answer, A, B or C, for each of the following. Round each number to four decimal places (4 d.p.).
a. $5.662198$ — A $5.6621$, B $5.6622$, C $5.6623$
b. $197.020549$ — A $197.0206$, B $197.0215$, C $197.0205$
c. $0.0089732$ — A $0.0090$, B $0.009$, C $0.0089$
👀 Show answer
a. A ($5.6621$) b. A ($197.0206$) c. A ($0.0090$)
7. Round each of these numbers to the given degree of accuracy.
a. $126.99231$ (4 d.p.) = $\square$
b. $0.7785$ (1 d.p.) = $\square$
c. $782.02972$ (3 d.p.) = $\square$
d. $3.141592654$ (7 d.p.) = $\square$
e. $3.9975$ (2 d.p.) = $\square$
f. $99.9961$ (1 d.p.) = $\square$
👀 Show answer
a. $126.9923$ b. $0.8$ c. $782.030$ d. $3.1415927$ e. $4.00$ f. $100.0$
8. Match the original number on the left to the rounded number in the middle, to the degree of accuracy on the right. The first one has been done for you.
Original numbers:
- A: $32.7819045$
- B: $32.8729045$
- C: $32.7189045$
- D: $32.7891045$
- E: $32.8792045$
- F: $32.8179045$
Rounded numbers:
- a: $32.873$
- b: $32.789105$
- c: $32.7819$
- d: $32.81790$
- e: $32.7$
- f: $32.88$
Degree of accuracy:
- i: 1 d.p.
- ii: 2 d.p.
- iii: 3 d.p.
- iv: 4 d.p.
- v: 5 d.p.
- vi: 6 d.p.
Zara makes the following statement.
The method I am going to use is to start with an original number and round it to $1\ \text{d.p.}$. If my answer is in the rounded number list, I will join it with a line. If not, I’ll round it to $2\ \text{d.p.}$ and see if this number is in the list. If not, I’ll continue rounding to more decimal places until I find the answer.
a. Use Zara’s method to answer the question.
b. Critique Zara’s method by explaining the advantages and disadvantages.
c. Can you improve her method or suggest a better method?
👀 Show answer
Matching:
A → c → iv (4 d.p.)
B → a → iii (3 d.p.)
C → e → i (1 d.p.)
D → b → vi (6 d.p.)
E → f → ii (2 d.p.)
F → d → v (5 d.p.)
b. Advantages: simple, step-by-step, minimal confusion. Disadvantages: potentially slow if many decimal places need to be checked; not efficient for large datasets.
c. Improved method: determine the degree of accuracy required first, then round directly to that degree without multiple steps. This reduces time and eliminates repeated rounding.
9. Work out the answers to these questions on a calculator. Round each of your answers to the given degree of accuracy.
a. $9 \div 7$ (2 d.p.)
b. $4 + \dfrac{8}{15}$ (4 d.p.)
c. $6 - \sqrt{22}$ (3 d.p.)
👀 Show answer
a. $1.29$
b. $4.5333$
c. $1.317$
10. Sofia, Marcus and Arun go out for lunch. The total bill is \$46.48. They decide to share the bill equally between the three of them. They use a calculator to work out how much each of them should pay.
Sofia: "I think we should round the answer on the calculator to the nearest whole number."
Marcus: "I think we should round the answer on the calculator to one decimal place."
Arun: "I think we should round the answer on the calculator to two decimal places."
a. Work out how much Sofia, Marcus and Arun think they should each pay.
b. Who do you think has made the best rounding decision? Explain your answer.
c. Can you think of a better way to round the answer to help decide how much they should each pay?
👀 Show answer
Total per person (exact) = $\dfrac{46.48}{3} \approx 15.493\overline{3}$.
a.
Sofia: \$15 (nearest whole number)
Marcus: \$15.5 (1 d.p.)
Arun: \$15.49 (2 d.p.)
b. Arun’s decision is the best, as it gives the most accurate and fair split, especially when dealing with money, where cents are relevant.
c. The best way is to use two decimal places (nearest cent), since currency is measured to two decimal places. This ensures the total amount paid matches the bill exactly when combined.
🍬 Learning Bridge
Now that you can round to a set number of decimal places and judge a sensible degree of accuracy, you’re ready to round by significant figures. The same “look one digit to the right” rule still applies, but the rounding starts at the first non‑zero digit so it works consistently for both very large and very small numbers (and links neatly to standard form). This sets you up to choose the most appropriate accuracy in real problems.
You already know how to round decimal numbers to a given number of decimal places (d.p.).
You also need to know how to round numbers to a given number of significant figures (s.f.).
The first significant figure in a number is the first non-zero digit in the number.
For example:
- In the number $450$, $4$ is the first significant figure, $5$ is the second significant figure and $0$ is the third significant figure.
- In the number $0.00806$, $8$ is the first significant figure, $0$ is the second significant figure and $6$ is the third significant figure.
To round a number to a given number of significant figures, follow these steps:
- Look at the digit in the position of the degree of accuracy. The ‘degree of accuracy’ is the number of significant figures you are working to. So, if you have been asked to round to $3$ significant figures, look at the third significant figure in the number.
- If the number to the right of this digit is $5$ or more, increase the digit by $1$. If the number is less than $5$, leave the digit as it is.
❓ EXERCISES
11. Round each of these numbers to one significant figure (1 s.f.). Choose the correct answer: A, B or C.
a. $352$ — A: $4$, B: $40$, C: $400$
b. $7.291$ — A: $7$, B: $7.3$, C: $7.29$
c. $11540$ — A: $12000$, B: $10000$, C: $11000$
d. $0.0087$ — A: $9$, B: $0.09$, C: $0.009$
👀 Show answer
a. C ($400$) b. A ($7$) c. A ($12000$) d. C ($0.009$)
12. Round each of these numbers to two significant figures (2 s.f.). All the answers are in this list: $0.0024$, $0.24$, $2.4$, $24$, $240$, $2400$.
a. $243$
b. $0.235$
c. $24.15$
d. $0.0023801$
e. $2396$
f. $2.3699$
👀 Show answer
a. $240$ b. $0.24$ c. $24$ d. $0.0024$ e. $2400$ f. $2.4$
🧠 Think like a Mathematician
Scenario: Harry has been asked to round $45150$ and $0.03284$ to two significant figures. He wrote:
Question: Round these numbers to 2 s.f.
a) 45150 → Harry’s answer: 45
b) 0.03284 → Harry’s answer: 0.03
Both answers are wrong.
- a) Explain the mistakes he made. Write the correct answers.
- b) What must you remember to do when rounding a large number to a given number of significant figures?
- c) What must you remember to do when rounding a small number to a given number of significant figures?
Follow-up prompts:
👀 show answer
- a: - For $45150$, the first two significant figures are 4 and 5. The correct rounded value is $45000$ (or $4.5 \times 10^4$ in standard form). Harry incorrectly wrote 45, which is 1000 times smaller. - For $0.03284$, the first two significant figures are 3 and 2. The correct rounded value is $0.032$ (or $3.2 \times 10^{-2}$ in standard form). Harry’s answer 0.03 only has one significant figure.
- b: Keep the correct place value. After choosing the correct significant figures, replace other digits with zeros (for large numbers) to keep the scale the same.
- c: For small numbers, count significant figures starting from the first non-zero digit, and keep zeros that are needed to show the correct value’s scale.
❓ EXERCISES
14. Round each number to the stated number of significant figures (s.f.).
a. $135$ (1 s.f.)
b. $45678$ (2 s.f.)
c. $18.654$ (3 s.f.)
d. $0.0931$ (1 s.f.)
e. $0.7872$ (2 s.f.)
f. $1.40948$ (4 s.f.)
g. $985$ (1 s.f.)
h. $0.697$ (2 s.f.)
i. $8.595$ (3 s.f.)
👀 Show answer
a. $100$ b. $46000$ c. $18.7$ d. $0.09$ e. $0.79$ f. $1.409$
g. $1000$ h. $0.70$ i. $8.60$
15. Which answer is correct: A, B, C or D?
a. $2569$ rounded to 1 s.f. — A: $2$, B: $3$, C: $2000$, D: $3000$
b. $47.6821$ rounded to 3 s.f. — A: $47.6$, B: $47.682$, C: $47.7$, D: $48.0$
c. $0.0882$ rounded to 2 s.f. — A: $0.08$, B: $0.088$, C: $0.09$, D: $0.1$
d. $3.08962$ rounded to 4 s.f. — A: $3.089$, B: $3.0896$, C: $3.09$, D: $3.090$
e. $19.963$ rounded to 3 s.f. — A: $2$, B: $20$, C: $20.0$, D: $19.96$
👀 Show answer
a. C b. A c. B d. B e. B
16. Round the number $209.095046$ to the stated number of significant figures (s.f.).
a. 1 s.f.
b. 2 s.f.
c. 3 s.f.
d. 4 s.f.
e. 5 s.f.
f. 6 s.f.
👀 Show answer
a. $200$ b. $210$ c. $209$ d. $209.1$ e. $209.10$ f. $209.095$
17.
a. Use a calculator to work out the answer to $26^{2} + \sqrt{58}$. Write all the numbers on your calculator display.
b. Round your answer to part a to the stated number of significant figures (s.f.).
i $1$ s.f. ii $2$ s.f. iii $3$ s.f. iv $4$ s.f. v $5$ s.f. vi $6$ s.f.
👀 Show answer
a. $26^{2} + \sqrt{58} \approx 683.6157731059$ (calculator display may vary slightly).
b. i $700$ ii $680$ iii $684$ iv $683.6$ v $683.62$ vi $683.616$
18. At a football match there were $63\,475$ Barcelona supporters and $32\,486$ Arsenal supporters.
How many supporters were there altogether?
Give your answer correct to two significant figures.
👀 Show answer
Total $= 63\,475 + 32\,486 = 95\,961$.
To $2$ s.f.: $\; 95\,961 \approx 96\,000$.
19. Ahmad has a bag of peanuts that weighs $150\,\text{g}$. There are $335$ peanuts in the bag.
Work out the average (mean) mass of one peanut. Give your answer correct to one significant figure.
👀 Show answer
Mean mass $= \dfrac{150}{335}\,\text{g} \approx 0.4478\,\text{g} \approx 0.4\,\text{g}$ (to $1$ s.f.).
20. The speed of light is approximately $670\,616\,629$ miles per hour. Use the formula
Work out the speed of light in metres per second. Give your answer correct to three significant figures.
👀 Show answer
$\dfrac{670\,616\,629}{2.25}\ \text{m/s} \approx 298\,051\,835\ \text{m/s} \approx 2.98\times 10^{8}\ \text{m/s}$ (to $3$ s.f.).
21. Zara and Sofia are looking at this question:
Work out the area of a rectangle with length $9.6 \ \text{m}$ and width $0.87 \ \text{m}$. Give your answer to an appropriate degree of accuracy.
🔎 Reasoning Tip
Degree of accuracy: Choose a sensible degree of accuracy for the context. Think about how precise your answer needs to be.
Zara’s comment: “$\text{area} = 9.6 \times 0.87 = 8.352 \ \text{m}^2$. I think we should give $8.352 \ \text{m}^2$ as our answer.”
Sofia’s comment: “$9.6$ and $0.87$ are both written to $2$ s.f., so I think we should round our answer to $2$ s.f. I think our answer should be $8.4 \ \text{m}^2$.”
👀 Show answer
Exact calculation: $9.6 \times 0.87 = 8.352 \ \text{m}^2$.
Since both measurements are given to $2$ significant figures, the result should be rounded to $2$ s.f.
Answer: $8.4 \ \text{m}^2$ (2 s.f.)
Sofia’s reasoning is correct for significant figure rounding.
22. A rugby club sells, on average, $12{,}600$ tickets to a match each week. The average cost of a ticket is $\$26.80$.
How much money does the club get from ticket sales, on average, each week?
Round your answer to an appropriate degree of accuracy.
👀 Show answer
Total revenue = $12{,}600 \times 26.80 = 337{,}680$.
To 3 significant figures: $\$338{,}000$.
23. This formula is often used in science: $F = m a$
Work out the value of $a$ when $F = 32$ and $m = 15$.
Round your answer to an appropriate degree of accuracy.
🔎 Reasoning Tip
Formula rearrangement: Change the subject of the formula first before substituting values.
👀 Show answer
$a = \dfrac{F}{m} = \dfrac{32}{15} \approx 2.133\ldots$
To 3 significant figures: $2.13$.
24. This is part of Jake’s homework. He works out an estimate by rounding each number to one significant figure.
🔎 Reasoning Tip
Approximation symbol: The symbol \( \approx \) means “is approximately equal to.”
Example:
Estimate $\dfrac{0.238 \times 576}{39.76}$
$0.238 \approx 0.2,\ \ 576 \approx 600,\ \ 39.76 \approx 40$
$0.2 \times 600 = 120,\quad 120 \div 40 = 3$
Estimate $= 3$
Accurate calculation: $0.238 \times 576 = 137.088$, then $137.088 \div 39.76 = 3.45$ (3 s.f.)
The estimate is close to the accurate value, so the accurate answer is probably correct.
Follow these steps for each calculation below:
- Use Jake’s method to work out an estimate of the answer.
- Use a calculator to work out the accurate answer. Give this answer correct to three significant figures.
- Compare your estimate with the accurate answer. Decide if your accurate answer is correct.
a. $\dfrac{0.3941 \times 196}{4.796}$
b. $\dfrac{4732 + 9176}{19.5166}$
c. $\dfrac{2.764 \times 84.695}{9.687 - 4.19}$
d. $\dfrac{58432 \times 0.08}{0.2 \times 348}$
👀 Show answer
a. Estimate: $\frac{0.4 \times 200}{5} = 16$
Accurate: $\frac{0.3941 \times 196}{4.796} \approx 16.1$
b. Estimate: $\frac{5000 + 9000}{20} = 700$
Accurate: $\frac{4732 + 9176}{19.5166} \approx 713$
c. Estimate: $\frac{3 \times 85}{10 - 4} = \frac{255}{6} \approx 42.5$
Accurate: $\frac{2.764 \times 84.695}{9.687 - 4.19} \approx 42.3$
d. Estimate: $\frac{60000 \times 0.08}{0.2 \times 350} = \frac{4800}{70} \approx 68.6$
Accurate: $\frac{58432 \times 0.08}{0.2 \times 348} \approx 67.2$
⚠️ Be careful!
When rounding to significant figures, never just chop off the extra digits — you must check the next digit to decide whether to round up. For example, rounding $0.07856$ to 2 s.f. gives $0.078$, not $0.07$.