Chapter 3.3

Chapter 3

Scientific Measurement

Chapter Three: Table of Contents

3.1 Using and Expressing Measurements

3.2 Units of Measurement

3.3 Solving Conversion Problems

Section 3.3

Solving Conversion Problems

What happens when a measurement is multiplied by a conversion factor?

When a measurement is multiplied by a conversion factor, the numerical value is generally changed, but the actual size of the quantity measured remains the same.

A conversion factor is a ratio of equivalent measurements. 

The two parts of a conversion factor, the numerator and the denominator, are equal.

Dimensional analysis is a way to analyze and solve problems using the units, or dimensions, of the measurements.

Let's practice!

Example

How many seconds are in a workday that lasts exactly eight hours?

1 Analyze: List the knowns and the unknown

To convert time in hours to time in seconds, you'll need two conversion factors.

1. You must convert hours to minutes:

h → min

2. You must convert minutes to seconds:

min → s

3. Identify the proper conversion factors based on the relationships:

1 h = 60 min and 1 min = 60 s

KNOWNS

time worked = 8 h

1 hour = 60 min

1 minute = 60 s

UNKNOWN

seconds worked = ? s

2 Calculate Solve for the unknown

The first conversion factor is based on 1 h = 60 min. The unit hours must be in the denominator so that the known unit will cancel: 

 $\frac{60\ \min}{1\ hr}$  

Make sure that the units cancel and that the numerator and denominator of each conversion factor are equal to each other.

2 Calculate cont'd.

The second conversion factor is based on 1 min = 60 s. The unit minutes must be in the denominator so that the desired units (seconds) will be in your answer.

 $\frac{60\ s}{1\ \min}$  

Multiply the time worked by the conversion factors

 $8\ hr\ \times\frac{60\ \min}{1\ hr}\ \times\frac{60\ s}{1\ \min}\ =\ 28,000\ s\ =\ 2.8800\times10^4s$  

See how the units cancel. Are you left with the unit that you wanted?

3 Evaluate Does the result make sense? 

The answer has the desired unit (s).

Since the second is a small unit of time, you should expect a large number of seconds in 8 hours.

The answer is exact since the given measurement and each of the conversion factors is exact.

Multiple Choice

How many seconds are in an exactly 40-hour work week?

• $1.44000\times10^5s$

• 144000 s

• $1.44000\times10^5$

• 144000

Multiple Choice

What is the mass, in grams, of a sample of cough syrup that has a volume of 50.0 cm³? The density of cough syrup is 0.950 g/cm³.

• 47.5 g

• 47.5 cm³

• 47.5 g/cm³

Multiple Choice

The diameter of Earth is 1.3 × 10⁴ km. What is the diameter expressed in decimeters?

• 1.3 dm

• 1.3 Dm

• $1.3\times10^8$ dm

• $1.3\times10^8$ Dm

Multiple Choice

Light travels at a speed of 3.00 × 10¹⁰ cm/s. What is the speed of light in kilometers/hour?

• $1.08\times10^9$ km/h

• $2.13\times10^7$ km/h

• $1.08\times10^9\$ km

• $2.13\times10^7$ h