Worked example · Easy
Estimate from a known height
A signpost appears about three times as tall as a 1.7 m adult. Estimate the signpost height.
\[3\times1.7=5.1\text{ m}\]
A sensible estimate is about \(\boxed{5\text{ m}}\).
Chapter 9
Estimation
Round accurately, estimate intelligently and choose a sensible level of precision. The aim is not just to get an approximate answer, but to know whether that answer is reasonable.
By the end of this chapter, you should be able to
- make sensible estimates of lengths using familiar benchmarks;
- round values to a stated number of decimal places;
- round values to a stated number of significant figures;
- estimate calculations by replacing values with convenient approximations;
- judge whether an estimate is likely to be above or below the exact value;
- choose convenient approximations for division and square roots;
- state answers to a sensible degree of accuracy for the context.
1
Subtopic 1
Estimating lengths
An estimate should be realistic rather than exact. Compare an unknown length with a familiar length, then scale up or down.
About 1 madult waist height
About 1.7 madult height
About 30 cmlength of a school ruler
About 2 mheight of a typical door
Units matter
A motorway length should normally be in kilometres, a room height in metres, and the thickness of a book in millimetres or centimetres. An unreasonable unit can make a sensible number meaningless.
Worked example · Medium
Choose sensible units
Estimate the length of a classroom.
A classroom is several adult heights long, so metres are sensible. A reasonable estimate might be \(\boxed{8\text{ m}}\), not 8 cm or 8 km.
Additional exam-focused worked example
Worked example · Exam-style
A bus is roughly 12 adult strides long. Each stride is about 0.75 m. Estimate the bus length.
Step-by-step solution
\[12\times0.75=9\text{ m}\] So a sensible estimate is about \(\boxed{9\text{ m}}\).
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
Estimate the height of a classroom door in metres. Explain your benchmark.
Show Answer
Steps: A typical door is slightly taller than an adult (about 1.7 m). A sensible estimate is \(\boxed{2\text{ m}}\).
2Easy practice
Which is the most sensible estimate for a pencil length: 18 mm, 18 cm or 18 m?
Show Answer
Steps: A pencil is shorter than a 30 cm ruler but much longer than 18 mm. Choose \(\boxed{18\text{ cm}}\).
3Medium practice
A tree in a picture is roughly four times the height of a 1.7 m person. Estimate its height.
Show Answer
Steps: Multiply the known height by 4: \[4\times1.7=6.8\text{ m}\] Give a rough answer: \(\boxed{7\text{ m}}\).
4Medium practice
A truck appears 2.5 times the length of a 4 m car. Estimate the truck length.
Show Answer
Steps: \[4\times2.5=\boxed{10\text{ m}}\] The answer is an estimate, so do not imply centimetre accuracy.
5Exam-style practice
A corridor is approximately 13 strides long. A stride is about 0.8 m. Estimate the corridor length to a sensible degree of accuracy.
Show Answer
Steps: \[13\times0.8=10.4\text{ m}\] Round a rough estimate appropriately: \(\boxed{10\text{ m}}\).
2
Subtopic 2
Rounding to a given number of decimal places
Decimal places are counted from the decimal point. To round, locate the final required decimal place and inspect the next digit.
Rounding rule
- Find the digit in the required decimal place.
- Look at the next digit.
- If it is 0–4, keep the required digit unchanged.
- If it is 5–9, increase the required digit by 1.
- Remove all digits after it.
Worked example · Easy
Round to 2 decimal places
Round \(7.4836\) to 2 d.p.
The hundredths digit is 8. The next digit is 3, so 8 stays unchanged.
\[7.4836\approx\boxed{7.48}\text{ (2 d.p.)}\]Worked example · Medium
Rounding causes a carry
Round \(12.996\) to 2 d.p.
The third decimal digit is 6, so round the second decimal digit up.
\[12.996\to12.100\text{? No.}\]Rounding 9 hundredths up causes a carry:
\[\boxed{13.00}\text{ (2 d.p.)}\]Common mistake
For 2 d.p., the answer must show two decimal places. For example, write \(13.00\), not just 13, when the stated accuracy matters.
Additional exam-focused worked example
Worked example · Exam-style
A measurement is 4.9996 cm. Round it to 3 decimal places, showing any carrying.
Step-by-step solution
The fourth decimal digit is 6, so increase the third decimal place. The carry passes through 999: \[4.9996\approx\boxed{5.000\text{ cm}}\quad(3\text{ d.p.})\]
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
Round 6.734 to 2 decimal places.
Show Answer
Steps: The third decimal digit is 4, so retain the hundredths digit: \(\boxed{6.73}\).
2Easy practice
Round 0.0987 to 3 decimal places.
Show Answer
Steps: The fourth digit is 7. Round the third decimal digit 8 up to 9: \(\boxed{0.099}\).
3Medium practice
Round 27.995 to 2 decimal places.
Show Answer
Steps: The third decimal digit is 5: a carry passes through both 9s. \(\boxed{28.00}\) (2 d.p.).
4Medium practice
Calculate 4.6 × 3.47 and give the result to 2 decimal places.
Show Answer
Steps: \[4.6\times3.47=15.962\] The third decimal digit is 2, so \(\boxed{15.96}\).
5Exam-style practice
A square has side length 5.64 cm. Work out its area and round to 1 decimal place.
Show Answer
Steps: \[A=5.64^2=31.8096\text{ cm}^2\] The second decimal digit is 0, so \(\boxed{31.8\text{ cm}^2}\) (1 d.p.).
3
Subtopic 3
Rounding to a given number of significant figures
Significant figures are counted from the first non-zero digit. Zeros before that first non-zero digit are placeholders and are not significant.
| Number | First 3 significant digits | 3 s.f. |
|---|---|---|
| 52 684 | 5, 2, 6 | 52 700 |
| 0.007284 | 7, 2, 8 | 0.00728 |
| 17 082 | 1, 7, 0 | 17 100 |
Decimal places vs significant figures
Decimal places start counting immediately after the decimal point. Significant figures start at the first non-zero digit, wherever it occurs.
Worked example · Easy
Large number to 2 s.f.
Round \(63\,482\) to 2 significant figures.
The first two significant digits are 6 and 3. The next digit is 4.
\[63\,482\approx\boxed{63\,000}\text{ (2 s.f.)}\]Worked example · Medium
Small decimal to 3 s.f.
Round \(0.0047862\) to 3 significant figures.
Ignore the leading zeros. The first three significant digits are 4, 7, 8. The next digit is 6, so 8 rounds up.
\[0.0047862\approx\boxed{0.00479}\text{ (3 s.f.)}\]Worked example · Exam-style
Identify an interior zero
Round \(20.049\) to 4 significant figures.
The significant digits are 2, 0, 0, 4. The next digit is 9, so 4 rounds up.
\[20.049\approx\boxed{20.05}\text{ (4 s.f.)}\]Additional exam-focused worked example
Worked example · Exam-style
Round 0.00099946 to 3 significant figures.
Step-by-step solution
The significant digits begin at the first 9. The fourth significant digit is 4, so keep the first three 9s: \[0.00099946\approx\boxed{0.000999}\quad(3\text{ s.f.})\]
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
Round 83 624 to 2 significant figures.
Show Answer
Steps: Keep 8 and 3; the next digit is 6. \(\boxed{84\,000}\).
2Easy practice
Round 0.006482 to 2 significant figures.
Show Answer
Steps: Ignore initial zeros. Keep 6 and 4; next digit 8 rounds up: \(\boxed{0.0065}\).
3Medium practice
Round 7.996 to 3 significant figures.
Show Answer
Steps: Keep 7,9,9 and look at 6. Carry through the nines: \(\boxed{8.00}\), with the zeros showing 3 s.f.
4Medium practice
Round 51 059 to 3 significant figures.
Show Answer
Steps: Keep 5,1,0; next digit 5 rounds 0 up: \(\boxed{51\,100}\).
5Exam-style practice
The mass is 0.040958 kg. Give it to 3 significant figures and explain which zeros count.
Show Answer
Steps: The leading zeros are not significant; digits 4,0,9 are. Next digit 5 rounds 9 up: \(\boxed{0.0410\text{ kg}}\) (3 s.f.; the final zero is significant).
4
Subtopic 4
Estimating answers to problems
A reliable first method is to round each value to about 1 significant figure, then perform the easier calculation. Your final estimate should normally be given to only 1 or 2 significant figures.
Estimation workflow
- Round each number to a convenient value, usually 1 s.f.
- Write the approximate calculation using \(\approx\).
- Calculate mentally or with simple arithmetic.
- Ask whether the answer has a sensible size and unit.
Worked example · Easy
Estimate a product
Estimate \(38.7\times6.2\).
\[38.7\approx40,\qquad6.2\approx6\]\[38.7\times6.2\approx40\times6=\boxed{240}\]
Worked example · Medium
Estimate a real cost
A club buys 47 shirts at $18.90 each. Estimate the total cost.
\[47\approx50,\qquad18.90\approx20\]\[50\times20=\boxed{\$1000}\]
Worked example · Exam-style
Is the estimate above or below?
Estimate \(4.8\times3.7\) using 1 s.f. State whether your estimate is greater or less than the exact answer.
\[4.8\approx5,\qquad3.7\approx4\]\[5\times4=20\]
Both values were rounded up, so the estimate is greater than the exact product.
Worked example · Exam-style
Estimate a quotient and judge direction
Estimate \(78.6\div24.3\) using convenient values. Is the estimate likely to be above or below the exact value?
\[78.6\approx80,\qquad24.3\approx20\]\[80\div20=\boxed4\]
The numerator was rounded up and the denominator down, so the estimate is likely to be above the exact quotient.
Estimating divisions and square roots with convenient numbers
Rounding everything mechanically to 1 significant figure is not always best. Choose nearby numbers that make the calculation easy.
For division
Choose numbers with useful common factors. For example, \(83.7\div2.87\) is easier as \(90\div3\) or \(81\div3\).
For square roots
Round to a nearby perfect square such as 25, 36, 49, 64, 81 or 100.
Worked example · Medium
Convenient division
Estimate \(294\div49.2\).
Choose nearby values that divide exactly:
\[294\approx300,\qquad49.2\approx50\]\[300\div50=\boxed6\]Worked example · Medium
Estimate a square root
Estimate \(\sqrt{83}\).
The nearest convenient perfect square is 81.
\[\sqrt{83}\approx\sqrt{81}=\boxed9\]Worked example · Exam-style
Mixed estimation with a root
Estimate \(0.62\sqrt{47.8}\).
Use convenient values:
\[0.62\approx0.6,\qquad47.8\approx49\]\[0.6\sqrt{49}=0.6\times7=\boxed{4.2}\]Additional exam-focused worked example
Worked example · Exam-style
Estimate \(\dfrac{29.6\times4.91}{0.49}\) using easy numbers.
Step-by-step solution
Choose \(30\), \(5\) and \(0.5\): \[\frac{29.6\times4.91}{0.49}\approx\frac{30\times5}{0.5}=\boxed{300}\]
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
Estimate 49.2 × 2.9 by rounding to 1 significant figure.
Show Answer
Steps: \[49.2\approx50,\quad2.9\approx3\] \(50\times3=\boxed{150}\).
2Easy practice
Estimate 318 ÷ 7.8 using convenient numbers.
Show Answer
Steps: \[318\approx320,\quad7.8\approx8\] \(320\div8=\boxed{40}\).
3Medium practice
Estimate the area of a circle with radius 5.1 m by using \(\pi\approx3\).
Show Answer
Steps: \[A=\pi r^2\approx3\times5^2=\boxed{75\text{ m}^2}\]
4Medium practice
Estimate \(0.79\sqrt{50}\) using a nearby square.
Show Answer
Steps: Use \(0.79\approx0.8\) and \(50\approx49\): \(0.8\sqrt{49}=0.8\times7=\boxed{5.6}\).
5Exam-style practice
Estimate \(198\div9.7\). Is using \(200\div10\) guaranteed to give an overestimate?
Show Answer
Steps: \[200\div10=\boxed{20}\] No. Both the numerator and denominator increased, so their combined effect must be checked. The exact value is about \(20.41\), making 20 an underestimate.
5
Subtopic 5
Working to a sensible degree of accuracy
An answer should suit both the context and the accuracy of the data used. Giving many extra digits does not make an answer more trustworthy.
General rule
Do not normally give a calculated answer to a greater degree of accuracy than the measurements or data used in the calculation.
Worked example · Easy
Area from measured lengths
A tabletop measures 1.82 m by 0.74 m, both to 2 decimal places. A calculator gives an area of 1.3468 m². Give a sensible answer.
The measurements were given to 2 d.p., so \(1.3468\text{ m}^2\) is unnecessarily precise.
\[\boxed{1.35\text{ m}^2}\]Worked example · Exam-style
Context controls the answer
879 kg is the total mass of 13 people. Calculate a sensible average mass.
\[879\div13=67.61538\ldots\]
Individual body mass would not sensibly be reported to many decimal places here. A reasonable answer is \(\boxed{68\text{ kg}}\).
Think before rounding
Keep full calculator accuracy during intermediate steps. Round the final answer unless the question specifically tells you to estimate first.
Additional exam-focused worked example
Worked example · Exam-style
A measured length of 6.2 m is divided into 7 equal pieces. State a sensible quoted length for one piece.
Step-by-step solution
\[6.2\div7=0.885714\ldots\text{ m}\] Since the length was given to 1 decimal place, quoting \(\boxed{0.89\text{ m}}\) is more sensible than a long decimal; the chosen accuracy should suit the purpose.
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
A measured mass is 4.6 kg. Is reporting it as 4.600000 kg justified?
Show Answer
Steps: No. The given value is only to 1 d.p.; adding unsupported zeros suggests extra precision.
2Easy practice
A calculated length is 7.836 m. Give it to 1 decimal place.
Show Answer
Steps: The second decimal digit is 3, so \(\boxed{7.8\text{ m}}\).
3Medium practice
A room measures 4.8 m by 3.1 m. Calculate its area to a sensible 1 d.p.
Show Answer
Steps: \[4.8\times3.1=14.88\text{ m}^2\] To 1 decimal place, \(\boxed{14.9\text{ m}^2}\).
4Medium practice
A journey covers 178 km in 2.3 h. Calculate its speed to the nearest whole km/h.
Show Answer
Steps: \[178\div2.3=77.3913\ldots\] The required rounded speed is \(\boxed{77\text{ km/h}}\).
5Exam-style practice
A shop needs 28 m of ribbon. Rolls each contain 6 m. How many whole rolls are needed, and why is ordinary rounding unsuitable?
Show Answer
Steps: \[28\div6=4.666\ldots\] You cannot buy part of a roll and 4 rolls are insufficient. Round UP: \(\boxed{5\text{ rolls}}\).
Chapter 9 practice
The early questions build rounding confidence. The later questions require you to choose sensible approximations and justify the direction or accuracy of an estimate.
A. Build confidence
1Decimal places[4 marks]
Round a) \(8.374\) to 1 d.p. and b) \(0.04678\) to 3 d.p.
Show Answer
a) The next digit after 3 is 7, so \(8.374\to\boxed{8.4}\).
b) The first three decimal digits are 046; the next digit is 7, so \(0.04678\to\boxed{0.047}\).
2Significant figures[4 marks]
Round a) \(58\,746\) to 2 s.f. and b) \(0.003864\) to 2 s.f.
Show Answer
a) First two significant digits 5, 8; next digit 7 → \(\boxed{59\,000}\).
b) First two significant digits 3, 8; next digit 6 → \(\boxed{0.0039}\).
3Decimal places or s.f.?[4 marks]
Write \(0.006748\) to a) 2 d.p. and b) 2 s.f.
Show Answer
a) To 2 d.p., inspect the third decimal digit 6: \(\boxed{0.01}\).
b) Significant figures start at 6: 6, 7; next digit 4 → \(\boxed{0.0067}\).
4Sensible unit[2 marks]
Which is the most sensible estimate for the height of a classroom door: 2 mm, 2 cm, 2 m or 2 km?
Show Answer
A door is a little taller than an adult, so \(\boxed{2\text{ m}}\) is sensible.
B. Estimate calculations
5Product estimate[3 marks]
Estimate \(62.4\times19.7\) by rounding each number to 1 s.f.
Show Answer
\[62.4\approx60,\qquad19.7\approx20\]\[60\times20=\boxed{1200}\]
6Area estimate[4 marks]
A rectangle is 7.8 cm by 4.2 cm. Estimate its area.
Show Answer
\[7.8\approx8,\qquad4.2\approx4\]\[8\times4=\boxed{32\text{ cm}^2}\]
7Cost estimate[4 marks]
Estimate the cost of 31 cinema tickets at $12.70 each.
Show Answer
\[31\approx30,\qquad12.70\approx10\]\[30\times10=\boxed{\$300}\]
Other sensible nearby-number estimates may also be acceptable if the working is clear.
8Convenient division[4 marks]
Estimate \(356\div58.4\) using convenient numbers.
Show Answer
Use \(360\div60\), which is easy and close to the original values.
\[360\div60=\boxed6\]9Square-root estimate[3 marks]
Estimate \(\sqrt{67}\).
Show Answer
64 is a nearby perfect square.
\[\sqrt{67}\approx\sqrt{64}=\boxed8\]C. Explain the estimate
10Above or below?[4 marks]
Estimate \(6.8\times4.6\) as \(7\times5\). Is the estimate greater or less than the exact product? Explain.
Show Answer
\[7\times5=35\]
Both original values were rounded up, so the estimate is \(\boxed{\text{greater than the exact product}}\).
11Division direction[5 marks]
Estimate \(191\div12.4\) using \(200\div10\). Is the estimate likely to be above or below the exact answer?
Show Answer
\[200\div10=20\]
The numerator has increased and the denominator has decreased, both of which make the quotient larger. So the estimate is \(\boxed{\text{above the exact value}}\).
12Newspaper rounding[4 marks]
A stadium attendance is 16 240 adults, 10 780 children and 8 190 seniors. A report says “about 35 000 people attended.” Explain why this is reasonable.
Show Answer
Round to convenient thousands:
\[16\,240\approx16\,000,\quad10\,780\approx11\,000,\quad8\,190\approx8\,000\]\[16\,000+11\,000+8\,000=\boxed{35\,000}\]D. Exam-style challenge
13Volume estimate[5 marks]
A cuboid measures 3.82 cm by 5.13 cm by 1.94 cm. Estimate its volume.
Show Answer
\[3.82\approx4,\quad5.13\approx5,\quad1.94\approx2\]\[4\times5\times2=\boxed{40\text{ cm}^3}\]
14Mixed root estimate[5 marks]
Estimate \(1.18\sqrt{63.5}\).
Show Answer
Choose convenient values:
\[1.18\approx1.2,\qquad63.5\approx64\]\[1.2\sqrt{64}=1.2\times8=\boxed{9.6}\]15Reasonable accuracy[4 marks]
A rectangle is measured as 4.7 m by 2.3 m, each to 1 decimal place. A calculator gives 10.81 m². State a sensible area and explain.
Show Answer
The measurements are only to 1 d.p., so giving many extra digits would suggest false precision. A sensible result is \(\boxed{10.8\text{ m}^2}\).
16Mean and accuracy[5 marks]
The masses of 9 parcels total 137 kg, measured to the nearest kilogram. Find the mean mass and give a sensible final answer.
Show Answer
\[137\div9=15.222\ldots\]
Since the total was measured to the nearest kilogram, \(\boxed{15\text{ kg}}\) is a sensible level of accuracy for the mean.
17Two rounding instructions[6 marks]
For \(0.094768\), give a) 3 decimal places and b) 3 significant figures. Explain why the answers differ.
Show Answer
a) Decimal places count from the decimal point: \(0.094768\to\boxed{0.095}\).
b) Significant figures start at the first non-zero digit 9: digits 9,4,7; next digit 6, so \(\boxed{0.0948}\).
They differ because the two counting systems start in different places.
18Estimate and check[7 marks]
A journey of 487 km takes 6.2 hours. a) Estimate the average speed. b) The calculator value is 78.548… km/h. Give a sensible final answer.
Show Answer
a) Use convenient values:
\[487\approx480,\qquad6.2\approx6\]\[480\div6=\boxed{80\text{ km/h}}\]b) A sensible final value is \(\boxed{79\text{ km/h}}\) (or 78.5 km/h if the context specifically calls for 1 d.p.).
Chapter 9 high-grade checklist
Length estimates
I compare with familiar benchmarks and always use sensible units.
Decimal places
I count from the decimal point and inspect the next digit before rounding.
Significant figures
I start at the first non-zero digit and keep placeholder zeros where necessary.
Estimation
I show the rounded values I used and keep the final estimate suitably rough.
Convenient values
For division I choose useful factors; for roots I choose nearby perfect squares.
Sensible accuracy
I do not present a final answer as more precise than the information used to calculate it.