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

Learn to report calculations with suitable precision and significant figures through clear examples and targeted practice.

Ontario Grade 11 Chemistry

Scientific Investigation Skills and Career Exploration

Ontario Grade 11 Chemistry — A1.13

A digital balance might display a mass as

12.4\ \mathrm{g}

. Writing that same measurement as

12.4000\ \mathrm{g}

would make it look more precise than the balance reading. In chemistry, calculated results often depend on measurements. The digits in a reported result should match the precision supported by those measurements. This lesson reviews how to choose those digits, how to round, and how to keep units in a calculation.

What you will learn

Explain why a calculated result should not suggest more certainty than the measurements support.
Use significant figures to report results from multiplication and division.
Use decimal places to report results from addition and subtraction.
Keep units with measurements and round a final answer appropriately.

1. Measurements and reported precision

A measurement is a value obtained using an instrument or another measuring method. Its recorded digits show how finely the value was measured or recorded. For example, a mass written as

12.4\ \mathrm{g}

has three significant figures. Significant figures are the digits that communicate the meaningful precision of a measured value.

The last recorded digit matters. It marks the smallest place value included in the measurement. A calculator can display many digits after a calculation, but those extra digits do not make the original measurements more precise.

Precision describes how finely a value is recorded. Accuracy describes how close a measurement is to an accepted value. This lesson focuses on precision when reporting calculated results.

Keep each value with its unit. A unit tells what the number measures and helps show whether a final answer is reported in a useful form. For example, dividing a mass in grams by a volume in millilitres gives units of grams per millilitre. Units can be treated like the numbers in the calculation: they are carried through the operations.

Before using significant-figure rules, notice whether the calculation adds or subtracts, or multiplies or divides. The operation determines which rule to use. Do not apply one general rule to every calculation.

Do not add digits to a reported answer just because a calculator displays them.
Significant figures communicate the precision of a measured value.
Keep units attached to values throughout the calculation.

2. Counting significant figures

Use these rules to count significant figures. Every non-zero digit is significant. A zero between non-zero digits is also significant. For example,

4.07

has three significant figures because the zero is between 4 and 7.

Zeros at the beginning of a decimal value are not significant. They locate the decimal point. For example,

0.0062

has two significant figures: 6 and 2. The zeros before the 6 do not show measured precision.

Zeros at the end of a decimal value are significant when they are written. For example,

3.40

has three significant figures. The final zero shows that the value was recorded to the hundredths place. In contrast, a whole number written without a decimal point can leave the significance of ending zeros unclear.

Scientific notation is a way to write a number as a value from 1 up to, but not including, 10 multiplied by a power of 10. The digits in the first value show the significant figures. For example,

5.20 \times 10^3

has three significant figures. This form can make the intended precision of ending zeros clear.

Rounding changes a value to fewer digits. Look at the first digit that will be removed. If it is 5 or greater, increase the last kept digit by one. If it is less than 5, leave the last kept digit unchanged. Remove the unneeded digits, but keep any zero needed to show place value or intended precision.

Non-zero digits and zeros between non-zero digits count.
Leading zeros do not count.
Ending zeros in a decimal value count.
Scientific notation can clarify how many significant figures are intended.

3. Choose the rule that matches the operation

For multiplication and division, count the significant figures in each measured value. Report the result with the same number of significant figures as the measured value with the fewest. For example, if a product uses one value with three significant figures and another with two, report the product with two significant figures.

For addition and subtraction, compare decimal places instead. Report the result to the same decimal place as the value with the fewest decimal places. For example, if one value is recorded to the tenths place and another to the hundredths place, the sum is reported to the tenths place. Counting significant figures would not give the correct rule for these operations.

A decimal place is a digit position after the decimal point. The first position is tenths, the second is hundredths, and the third is thousandths. These place values tell how finely each value was recorded for addition or subtraction.

Some calculations have more than one operation. Follow the order of operations, but keep extra digits in intermediate results rather than rounding them early. At the end, apply the rule for the operation that produces the final result, while respecting the precision of the measured values used. Exact counted quantities, such as a stated number of objects, do not limit significant figures in the same way as measured values.

For a final check, ask two separate questions: which operation rule applies, and what unit belongs in the answer? Addition and subtraction use decimal places. Multiplication and division use significant figures. The precision of the result is therefore not described by one rule for every operation.

Multiplication and division: use the fewest significant figures among the measured values.
Addition and subtraction: use the fewest decimal places among the measured values.
Keep extra digits in intermediate steps and round the final result.
Include the unit in the final answer.

4. Report and check the result

A suitable report includes the calculated value, an appropriate number of digits, and the correct unit. A calculator display is useful for carrying out arithmetic, but it does not decide how many digits belong in the final report.

In the worked example, two measured masses are added and then the sum is divided by a measured volume. The addition rule identifies the place to which the sum is supported. The unrounded sum is used in the division so early rounding does not change the calculation. The final division determines the significant-figure rule for the reported answer.

When checking a multi-step calculation, also check that the units make sense. In the example, grams divided by millilitres gives grams per millilitre. The unit is part of the result, not an optional label.

Identify the operation that determines the final result.
Do not round intermediate values prematurely.
Check both the unit and the number of reported digits.

Worked example

Combining measured masses before dividing

A sample has a measured mass of

12.4\ \mathrm{g}

. A second portion has a measured mass of

3.26\ \mathrm{g}

. The combined mass is divided by a measured volume of

5.0\ \mathrm{mL}

. Calculate and report the mass per volume with suitable precision.

Add the measured masses
The masses are added, so use the addition rule. The first mass is recorded to the tenths place, while the second is recorded to the hundredths place. The sum is supported to the tenths place. Keep the unrounded sum for the next step.
$12.4\ \mathrm{g} + 3.26\ \mathrm{g} = 15.66\ \mathrm{g}$
Divide by the measured volume
Use the unrounded sum in the division. The original mass measurements each have three significant figures, while the volume has two. The division rule means the final reported answer must have two significant figures.
$15.66\ \mathrm{g} \div 5.0\ \mathrm{mL} = 3.132\ \mathrm{g/mL}$
Round and report
Keep two significant figures. The first two digits are 3 and 1. The next digit is 3, which is less than 5, so the 1 stays unchanged. Report the result in grams per millilitre.
$3.132\ \mathrm{g/mL} \approx 3.1\ \mathrm{g/mL}$

Answer: The mass per volume is

3.1\ \mathrm{g/mL}

Check: The result has two significant figures, matching the measured volume with the fewest significant figures. Its unit is mass divided by volume.

Common mistakes and how to avoid them

Reporting every digit shown by a calculator.

Correction: Use the calculator to find the value, then round the reported result according to the measurements and operation.

Using the fewest significant figures for addition.

Correction: For addition and subtraction, match the fewest decimal places. Use the significant-figure rule for multiplication and division.

Rounding a partial result before finishing a multi-step calculation.

Correction: Keep extra digits in intermediate work. Round the final answer to suitable precision.

Leaving off units in the final answer.

Correction: Carry units through the calculation and include the resulting unit in the report.

Lesson summary

Significant figures show the meaningful precision of a measured value.
For multiplication and division, report the answer with the fewest significant figures among the measured values.
For addition and subtraction, report the answer to the fewest decimal places among the measured values.
Keep extra digits during intermediate calculations, round the final result, and include its unit.

Check your understanding

Question 1

How many significant figures are in

0.0405

Two
Three
Four
Five

Show answer and explanation

Three

The leading zeros do not count. The zero between 4 and 5 does count, so the significant digits are 4, 0, and 5.

Question 2

A measured value is multiplied by another measured value. One has three significant figures and the other has two. How many significant figures should the reported product have?

One
Two
Three
Five

Show answer and explanation

Two

For multiplication, use the fewest significant figures in the measured values. The answer has two significant figures.

Question 3

A sum is calculated from values recorded to the tenths and hundredths places. To which decimal place should the sum be reported?

Ones place
Tenths place
Hundredths place
Thousandths place

Show answer and explanation

Tenths place

For addition, use the least precise decimal place. The tenths place is less precise than the hundredths place.

Key terms

Measurement: A value obtained using an instrument or another measuring method.
Precision: How finely a value is recorded.
Significant figures: Digits that communicate the meaningful precision of a measured value.
Decimal place: A digit's position after the decimal point, such as tenths or hundredths.
Scientific notation: A way to write a number as a value from 1 up to, but not including, 10 multiplied by a power of 10.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Chemistry (SCH3U), expectation A1.13. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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