A1.13 Report calculations with suitable precision and significant figures

Ontario Grade 12 Chemistry

Scientific Investigation Skills and Career Exploration

SCH4U study topic: A1.13

A balance may display a mass as 2.46 g, while another balance may display 2.463 g. The extra digit communicates a finer measurement, not necessarily a more accurate one. In chemistry, calculated answers depend on the measurements used to obtain them. Reporting too many digits can suggest that a result is more precise than the measurements support. This lesson focuses on how to choose and report suitable precision in calculations.

What you will learn

Explain the difference between a measured value’s precision and the precision of a calculated result.
Identify significant figures in measured quantities.
Apply significant-figure rules for multiplication, division, addition, and subtraction.
Report calculated answers with appropriate units and precision.

1. From measurements to reported answers

A measurement is a value obtained using an instrument or a defined measurement process. A measured value includes a number and a unit, such as 12.4 mL. Its digits communicate the precision of the measurement. Precision describes how finely a value is reported; it does not by itself prove that the value is close to the true value.

A calculated result is built from measured values. For example, a chemist might use a measured solution volume and concentration to calculate an amount of substance. The calculator may display many digits, but those digits do not all represent supported information. The reported result should reflect the precision of the inputs.

A significant figure is a digit that communicates the precision of a measured value. Non-zero digits are significant. Zeros between non-zero digits are significant. Zeros at the beginning of a decimal number only locate the decimal point, so they are not significant. For example, 0.00420 has three significant figures: 4, 2, and the final 0.

A final zero after a decimal point can be significant because it communicates reported precision. In 6.30, the final zero indicates that the value was reported to the hundredths place. Scientific notation can make the intended number of significant figures clear:

6.30 = 6.30 \times 10^0

has three significant figures.

Measured digits carry information about precision.
Leading zeros are not significant; zeros between non-zero digits and indicated final decimal zeros are significant.
A calculator display does not determine how many digits should be reported.

2. Rules for calculations

For multiplication and division, the result should have the same number of significant figures as the input with the fewest significant figures. This rule applies because a result cannot reliably carry more precision than its least precise measured input.

For addition and subtraction, use decimal places instead of counting significant figures. The result should be rounded to the same decimal place as the input with the fewest decimal places. For example, adding a value reported to the tenths place to one reported to the hundredths place gives a result reported to the tenths place.

Rounding means reducing the number of reported digits. Keep the last digit when the next digit is less than 5. Increase it by 1 when the next digit is 5 or greater. Round only after completing the calculation when possible. Rounding intermediate values can change the final result.

Exact numbers do not limit significant figures. An exact number comes from counting or a defined relationship, rather than an instrument measurement. For example, the conversion

1\ \mathrm{L}=1000\ \mathrm{mL}

is exact. It does not reduce the precision of a calculation. Measured quantities, by contrast, do limit the reported precision.

Units are part of a chemistry result. Carry them through the calculation, and include the appropriate unit with the final value. A number without its unit may not say what quantity was calculated.

\text{reported precision: multiplication/division} \rightarrow \text{fewest significant figures};\quad \text{addition/subtraction} \rightarrow \text{fewest decimal places}

Multiplication and division: match the fewest significant figures.
Addition and subtraction: match the fewest decimal places.
Keep extra digits during intermediate steps and round the final result.
Exact values do not limit significant figures; measured values do.

3. Worked chemistry calculation

Suppose a student calculates the amount of dissolved substance in a measured sample of solution. The concentration is

0.125\ \mathrm{mol/L}

, and the sample volume is

25.0\ \mathrm{mL}

. The concentration has three significant figures, and the measured volume has three significant figures. The conversion from millilitres to litres is exact.

Use amount = concentration multiplied by volume. Convert the volume to litres so the units cancel correctly. The result of multiplication must be reported to three significant figures because both measured inputs have three significant figures.

The unrounded calculator result is

0.003125\ \mathrm{mol}

. The fourth significant digit is 5, so the third significant digit rounds up. Report the result as

0.00313\ \mathrm{mol}

. The leading zeros locate the decimal point and do not count as significant figures.

Convert the volume before multiplying so that units are consistent.
The exact unit conversion does not affect the significant-figure limit.
The final amount is reported to three significant figures.

4. A reliable reporting routine

Before calculating, record the measured values with their units and identify how many significant figures or decimal places each value has. Choose the rule based on the operation. If a calculation contains more than one operation, follow the mathematical order of operations and keep unrounded digits until the end.

After calculating, check whether the units make sense and apply the appropriate rounding rule. For multiplication or division, count significant figures in the measured inputs. For addition or subtraction, compare the decimal places. Then write the rounded answer with its unit.

A common mistake is to copy every digit from a calculator. Those extra digits are not new measurements. Another mistake is to use the multiplication rule for addition. A third is to remove a zero that communicates precision. For example, changing 2.50 g to 2.5 g changes the reported precision, even though the numerical amount is the same.

Suitable precision is not the same as always using a fixed number of digits. The correct report depends on the measurements and the calculation. A clear final answer gives the quantity, a defensible number of digits, and a unit.

Identify the measured inputs and the operation before rounding.
Use units to check the meaning of the result.
Report only digits justified by the measurement precision.

Worked example

Amount of substance in a solution sample

A solution has a concentration of

0.125\ \mathrm{mol/L}

. A sample volume is measured as

25.0\ \mathrm{mL}

. Calculate the amount of dissolved substance in the sample and report it with suitable precision.

Choose the relationship
Amount of substance equals concentration multiplied by volume. Convert the measured volume to litres because the concentration is given per litre.
$n=cV$
Convert the volume
Use the exact millilitre-to-litre conversion. The conversion factor does not limit the significant figures.
$25.0\ \mathrm{mL}\times\frac{1\ \mathrm{L}}{1000\ \mathrm{mL}}=0.0250\ \mathrm{L}$
Calculate and round
Multiply the concentration by the volume. Both measured inputs have three significant figures, so report three significant figures. Round the unrounded result at the end.
$0.125\ \mathrm{mol/L}\times0.0250\ \mathrm{L}=0.003125\ \mathrm{mol}\approx0.00313\ \mathrm{mol}$

Answer:

3.13\times10^{-3}\ \mathrm{mol}

Check: The litre units cancel, leaving moles. The reported answer has three significant figures, matching the measured inputs.

Common mistakes and how to avoid them

Reporting every digit shown on a calculator.

Correction: Use the calculation rule to limit the final digits to the precision supported by the measured inputs.

Applying the significant-figure counting rule to addition and subtraction.

Correction: For addition and subtraction, round to the least precise decimal place.

Treating leading zeros as significant figures.

Correction: Leading zeros only locate the decimal point. Count the non-zero digits and any relevant zeros between them or at the end of a decimal value.

Dropping a final decimal zero without considering what it communicates.

Correction: Keep a final zero when it indicates the precision of the reported measurement or result.

Lesson summary

Significant figures communicate the precision of measured values.
For multiplication and division, the result uses the fewest significant figures among measured inputs.
For addition and subtraction, the result uses the fewest decimal places among inputs.
Keep extra digits during intermediate calculations, then round the final answer.
Include units and report no more precision than the measurements support.

Check your understanding

Question 1

How many significant figures are in

0.006080

?

2
3
4
6

Show answer and explanation

4

The leading zeros do not count. The digits 6, 0, 8, and the final decimal zero are significant, giving four significant figures.

Question 2

A multiplication uses measured values with four and two significant figures. How many significant figures should the final result have?

Two
Three
Four
The number of digits on the calculator

Show answer and explanation

Two

For multiplication, report the result with the same number of significant figures as the measured input with the fewest. That is two.

Question 3

Calculate

12.11+0.3

with suitable precision.

$12.41$
$12.4$
$12.410$
$12$

Show answer and explanation

12.4

The least precise input is reported to the tenths place. The sum is 12.41 before rounding, so the suitable result is 12.4.

Key terms

Measurement: A value obtained using an instrument or a defined measurement process.
Precision: How finely a value is reported, as shown by its digits.
Significant figure: A digit that communicates the precision of a measured value.
Exact number: A value from counting or a defined relationship that does not limit significant figures.
Rounding: Reducing reported digits according to the value of the next digit.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Chemistry (SCH4U), expectation A1.13. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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A1.13 · Report calculations with suitable precision and significant figures

Ontario Grade 12 Chemistry

SCH4U study topic: A1.13

What you will learn

1. From measurements to reported answers

2. Rules for calculations

3. Worked chemistry calculation

4. A reliable reporting routine

Amount of substance in a solution sample

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SCH4U

About this lesson and its review