# 3.17: Uncertainty in Multiplication and Division

### Introduction

Calculators do just what you ask of them, no more and no less. However, they sometimes can get a little out of hand. If I multiply 2.48 times 6.3, I get an answer of 15.687, a value that ignores the number of significant figures in either number. Division with a calculator is even worse. When I divide 12.2 by 1.7, the answer I obtain is 7.176470588. Neither piece of data is accurate to nine decimal places, but the calculator doesn't know that. The human being operating the instrument has to make the decision about how many places to report.

### Uncertainty in Multiplication and Division

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

$D = \frac{m}{V} = \frac{37.46 \: \text{g}}{12.7 \: \text{cm}^3} = 2.949606299 \: \text{g/cm}^3$

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 least number of significant figures. Applying this rule results in a density of $$2.95 \: \text{g/cm}^3$$, with three significant figures - the same as the volume measurement.

Example 3.17.1

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

A. $$0.048 \: \text{m} \times 32.97 \: \text{m}$$

B. $$14,570 \: \text{kg} \div 5.81 \: \text{L}$$

Solution:

Step 1: Plan the problem.

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 multiplying or dividing, the units are also multiplied or divided.

Step 2: Calculate

A. $$0.048 \: \text{m} \times 32.97 \: \text{m} = 1.6 \: \text{m}^2$$ Round to two significant figures because 0.048 has two.

B. $$14,570 \: \text{kg} \div 5.81 \: \text{L} = 2510 \: \text{kg/L}$$ Round to three significant figures because 5.81 has three.

### Summary

• For multiplication and division problems, the answer should be recorded to the same number of significant figures as the measurement with the least number of significant figures.

### Contributors

• CK-12 Foundation by Sharon Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and Jean Dupon.