CK-12-Chemistry Intermediate

3.3. Uncertainty in Measurements http://www.ck12.org

Significant Figures in Multiplication and Division

The density of a certain object is calculated by dividing the mass by the volume. Suppose that a mass of 37.46 g is
divided by a volume of 12.7 cm^3. The result on a calculator would be:

D=

m

V

=

37 .46 g

12 .7 cm^3

= 2 .94960299 g/cm^3

The value of the mass measurement has four significant figures, while the value of the volume measurement has
only three significant figures.For multiplication and division problems, the answer should be rounded to the same
number of significant figures as the measurement with the lowest number of significant figures. Applying this rule
results in a density of 2.95 g/cm^3 , which has three significant figures—the same as the volume measurement.

Note that the rule for multiplication and division problems is different than the rule for addition and subtraction
problems. For multiplication and division, it is the number of significant figures that must be considered. For
addition and subtraction, it is the position of the decimal place that determines the correct rounding. Review the
sample problem below, paying special attention to this distinction.

Sample Problem 3.9: Significant Figures in Calculations

Perform the following calculations, rounding the answers to the appropriate number of significant figures.

1. 0.048 m×32.97 m


2. 21.9 g –19.417 g


3. 14,570 kg÷5.81 L


4. 71.2 cm + 90 cm

Step 1: Plan the problem.

Decide which calculation rule applies. Analyze each of the measured values to determine how many significant
figures should be in the result. Perform the calculation and round appropriately. Apply the correct units to the
answer. When adding or subtracting, the units in each measurement must be identical and then remain the same in
the result. When multiplying or dividing, the units are also multiplied or divided.

Step 2: Calculate.

1. 0.048 m×32.97 m = 1.6 m^2 - Round to two significant figures because 0.048 has two.


2. 21.9 g - 19.417 g = 2.5 g - Answer ends at the tenths place because of 21.9.


3. 14,570 kg÷5.81 L = 2510 kg/L - Round to three significant figures because 5.81 has three.


4. 71.2 cm + 90 cm = 160 cm - Answer ends at the tens place because of 90.

Practice Problems

2. Solve each problem, rounding each answer to the correct number of significant figures.
   (a) 132.3 g÷29.600 mL
   (b) 3.27 g/cm^3 ×0.086 cm^3
   (c) 125 m + 61.3 m + 310 m
   (d) 3.0× 104 L−1244 L


3. A rectangular prism has dimensions of 3.7 cm by 4.81 cm by 1.90 cm. The mass of the prism is 49.72 g.
   Calculate its density. (Hint: Do the entire calculation without rounding until the final answer in order to
   reduce “rounding error.”)