Learning Objectives
- Round numbers to a stated number of decimal places or significant figures.
- Use estimation to check calculations and obtain approximate answers.
- Choose a sensible degree of accuracy in context.
- Understand the difference between rounding and truncation.
Key Terms
- Decimal place
- A place-value position after the decimal point.
- Significant figure
- A digit that contributes to the precision of a number, beginning with the first non-zero digit.
- Estimate
- An approximate value obtained using convenient rounded numbers.
- Degree of accuracy
- The precision to which a value is stated.
- Truncation
- Cutting off digits without rounding.
Rounding To Decimal Places
To round to a given number of decimal places, identify the required digit and inspect the next digit. If the next digit is 5 or more, increase the required digit by 1; otherwise leave it unchanged. Remove all following digits.
Worked Example: Decimal Places
Question: Write 18.74693 correct to 3 decimal places.
- The third decimal digit is 6.
- The next digit is 9, so increase 6 to 7.
Answer: 18.747
Rounding To Significant Figures
Significant figures begin at the first non-zero digit. Zeros between non-zero digits are significant. Leading zeros are not significant, while trailing zeros after a decimal point may be significant because they show precision.
Worked Example: Significant Figures
Question: Write 0.0040967 correct to 3 significant figures.
- The first significant digit is 4.
- The first three significant digits are 4, 0 and 9.
- The next digit is 6, so 9 rounds up, causing a carry.
Answer: 0.00410
The final zero in 0.00410 is important because it shows that the value has been rounded to 3 significant figures.
Rounding Whole Numbers
When rounding a whole number to tens, hundreds or thousands, locate the place value and inspect the digit immediately to its right. Replace following digits by zeros.
Worked Example: Whole Number Rounding
Question: Write 5764 correct to the nearest thousand.
- The thousands digit is 5.
- The hundreds digit is 7, so round upward.
Answer: 6000.
Estimation
Estimation uses convenient rounded values, often 1 significant figure, to obtain a quick approximate result. It is especially useful for checking calculator answers and deciding whether the decimal point is sensible.
Worked Example: Estimate A Calculation
Question: Estimate \frac{9.79\times0.765}{41.3} by rounding each number to 1 significant figure.
- 9.79\approx10, 0.765\approx0.8 and 41.3\approx40.
- Calculate \frac{10\times0.8}{40}=\frac{8}{40}=0.2.
Answer: Approximately 0.2.
Choosing A Reasonable Accuracy
The context determines how an answer should be rounded. A number of people must be a whole number. Money is commonly given to two decimal places. A measured length may be given to a precision consistent with the original data. Avoid claiming greater accuracy than the information supports.
| Context | Typical final form |
|---|---|
| Number of buses required | Round up to a whole number because a fraction of a bus is impossible. |
| Money | Usually 2 decimal places. |
| Population estimate | Often nearest whole number, thousand or million. |
| Angle measured with a protractor | Often nearest degree. |
| Calculator answer with no stated accuracy | Usually 3 significant figures, unless an exact answer is appropriate. |
Rounding Versus Truncation
Truncation simply removes digits. For example, 8.769 truncated to 2 decimal places is 8.76, while rounded to 2 decimal places it is 8.77. Questions may test the difference indirectly through calculator displays or bounds.
8th Edition Chapter Map
- Rounding to place value, decimal places and significant figures
- Limits of accuracy and bounds
- Approximation errors in context
- Estimation and reasonableness
Upper And Lower Bounds
A rounded value represents an interval rather than one exact value. If a length is 8.4 cm correct to the nearest 0.1 cm, the actual length L satisfies 8.35\le L<8.45. The lower bound is included; the upper bound is excluded because 8.45 would round to 8.5.
Worked Example: Bounds Of A Calculation
A rectangle has length 12.6 cm and width 4.8 cm, each correct to the nearest 0.1 cm. Find bounds for its area.
- 12.55\le L<12.65 and 4.75\le W<4.85.
- All quantities are positive, so minimum area uses both lower bounds: 12.55\times4.75=59.6125.
- Maximum area uses both upper bounds: 12.65\times4.85=61.3525.
- 59.6125\le A<61.3525 cm^2.
Bounds In Division
For positive quantities, the smallest possible quotient comes from the smallest numerator and largest denominator. The largest possible quotient comes from the largest numerator and smallest denominator. This is different from multiplication, so write the intended extreme values before calculating.
Worked Example: Speed Bounds
A distance is 240 km correct to the nearest 10 km and a time is 3.2 h correct to the nearest 0.1 h. Then 235\le d<245 and 3.15\le t<3.25. The lower speed bound is 235\div3.25\approx72.3 km/h, and the upper speed bound is 245\div3.15\approx77.8 km/h.
Approximation Error
The absolute error is |\text{approximate value}-\text{actual value}|. The percentage error compares this with the actual value. When the exact actual value is unknown but a rounding interval is known, the maximum absolute rounding error is half of the rounding unit.
Estimation Strategies
- Round values to one significant figure for a quick order-of-magnitude check.
- Use compatible numbers that divide easily.
- Estimate before using a calculator, not after, so the estimate can detect keying errors.
- Do not over-round intermediate results in a multi-stage calculation.
Worked Example: Estimating A Complex Expression
Estimate \frac{19.7\times0.493}{3.96}. Using one significant figure gives \frac{20\times0.5}{4}=2.5. A calculator answer should therefore be near 2.5.
Extended Practice
A. A mass is 3.70 kg correct to the nearest 0.01 kg. State its error interval.
3.695\le m<3.705 kg.
B. x=16.2 and y=5.4, each correct to the nearest 0.1. Find the upper bound of x/y.
x_U=16.25, y_L=5.35, so upper bound =16.25/5.35\approx3.037.
C. Round 0.0067849 to three significant figures.
0.00678. The first significant digit is 6.
Examination Guidance
- Write the required zeros when they communicate significant figures, e.g. 2.50 has 3 significant figures.
- Do not round intermediate values unless instructed; keep full calculator precision and round the final answer.
- When estimating, show the rounded numbers before calculating.
- Use the approximation symbol \approx rather than = when values have been rounded.
Common Mistakes
- Counting zeros before the first non-zero digit as significant.
- Rounding 0.0040967 to 0.0041 and then claiming it is 3 significant figures; the written zero is needed: 0.00410.
- Using 1 decimal place when the question asks for 1 significant figure.
- Rounding every line of a multi-step calculation and accumulating avoidable error.
Knowledge Check
1. Round 62.9478 to 2 decimal places.
Answer
62.95.
2. Round 708450 to 3 significant figures.
Answer
708000.
3. Round 0.00097384 to 2 significant figures.
Answer
0.00097.
4. Estimate \frac{19.6\times4.08}{0.198} using 1 significant figure values.
Answer
\frac{20\times4}{0.2}=400.
5. State a sensible final answer if a calculation gives 7.12 minibuses.
Answer
8 minibuses, because the number must be a whole number and all passengers need transport.
Mixed Review With Full Solutions
1. A distance is 7.3 km correct to the nearest 0.1 km. State the lower and upper bounds in metres.
7.25\le d<7.35 km, so 7250\le d<7350 m.
2. The sides of a rectangle are 8.2 cm and 5.6 cm, each to the nearest 0.1 cm. Find the lower bound for its perimeter.
Lower side bounds are 8.15 and 5.55. Lower perimeter =2(8.15+5.55)=27.4 cm.
3. Estimate \frac{597\times0.0487}{19.8} using one significant figure.
\frac{600\times0.05}{20}=1.5.
4. A calculator gives 0.004986273. Give the answer to 2 significant figures and 4 decimal places.
To 2 significant figures: 0.0050. To 4 decimal places: 0.0050. The same written value has different accuracy descriptions.
