# 3.3.6: Convert Using Unit Analysis

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Conversion Using Unit Analysis

Munni and Caras were excited about their upcoming whale-watching trip. They did their research and discovered that some whales can weigh up to 150 tons. Munni’s home scale said that he weighed 120. Munni told everyone that he weighed only 30 pounds less than a whale. Was he correct?

In this concept, you will learn to use proportions and unit analysis.

**Using Proportions and Unit Analysis**

A **proportion** is created when two ratios are found to be equivalent or equal.

1/2=3/6 is a proportion.

A **unit rate** is a comparison of two measurements, one of which has a value of 1.

55 miles/1 hour is a unit rate.

A **unit scale** is a ratio that compares the dimensions of an actual object to the dimensions of a scale drawing or model that represents the actual object.

1 inch/11 feet is a unit scale.

A **standard unit** is one that is most commonly used.

Both the customary and metric systems of measurement include weight (or mass), length, and liquid volume standard units.

Some examples are:

Customary standard weight: 16 ounces = 1 pound

Metric standard length: 1 meter = 100 centimeters

Customary standard liquid: 4 quarts = 1 gallon

These are called **conversion factors** and can be defined as the ratio of a measurement in one unit to the equivalent numerical value in another unit.

The ratio for ounces to pounds is 16:1 or 16 ounces/1 pound.

Standard units of measurement and proportions can be used to convert one unit of measurement to another. The process of converting one unit to another is called **unit conversion**. Units can be converted by using proportions, setting ratios equal to one another, or by unit analysis, multiplying times a standard unit of measure.

Here is an example using proportions.

A pitcher holds 4 liters of water. Determine how many milliliters of water the pitcher holds. Use the unit conversion: 1 liter = 1000 milliliters

First, write the unit conversion as a fraction.

1 liter/1000 millimeters

Next, write a ratio that compares the unit as it is given to the unknown converted unit.

Remember that a ratio standing alone can be written with either value in the numerator or denominator, but once it is used in a calculation, it must be formatted accordingly.

Unit to be converted = from 4 liters to x milliliters

This could be written as 4 liters/x millimeters or x millimeters/4 liters.

Then, write a proportion using the correct format. When writing a proportion, the units must be consistent.

1 liter/1000 milliliters=4 liters/x milliliters is correct.

Next, solve by cross multiplication.

x=4000

The answer is x = 4,000 ml.

Another way to solve the same problem is to use unit analysis.

First, write the unit conversion as a fraction.

1 liter/1000 milliliters or 1000 milliliters/1 liter

Next, write a ratio for the unit to be converted.

4 liters/1

When using unit analysis, the correct format will be such that after multiplication, the “from” units will cancel, and the answer will equal the “to” units.

4 liters/1 × 1000milliliters/1 liters

Then, cancel liters, and multiply.

4×1000 ml/1×1=4000 ml

The answer is 4,000 ml.

**Examples**

Example 3.3.6.1

Earlier, you were given a problem about Munni and Caras, who are going on a whale-watching trip.

Munni said a whale that weighed 150 tons was only 30 more than his weight of 120. Munni forgot the whale’s weight was in tons, and his was in pounds. Find the whale’s weight in pounds.

**Solution**

First, write the unit conversion as a fraction.

2000 pounds/1 ton

Next, write a ratio that compares the unit as it is given to the unknown converted unit.

x pounds/150 tons

Then, write a proportion making sure that the units are consistent.

2000 pounds/1 ton=x pounds/150 tons

Next, solve by cross multiplication.

x=300,000 pounds

The answer is the whale weighs 300,000 pounds.

Example 3.3.6.2

How many milliliters are there in 2.5 liters? Write a proportion and solve.

**Solution**

First, write the unit conversion as a fraction.

1 liter/1000 milliliters

Next, write a ratio that compares the unit as it is given to the unknown converted unit.

2.5 liters/x milliliters

Then, write a proportion making sure that the units are consistent.

1 liter/1000 milliliters=2.5 liters/x milliliters

Next, solve by cross multiplication.

X=2500 ml

The answer is x = 2,500 ml

Example 3.3.6.3

How many meters in 11 kilometers? Write a proportion and solve.

**Solution**

First, write the unit conversion as a fraction.

1000 m/km

Next, write a ratio that compares the unit as it is given to the unknown converted unit.

x m/11 km

Then, write a proportion making sure that the units are consistent.

1000 m/km=x m/11 km

Next, solve by cross multiplication.

X=11,000 m

The answer is x = 11,000 m

Example 3.3.6.4

How many inches are there in 18 feet? Use unit analysis.

**Solution**

First, write the unit conversion as a fraction.

12 inches/1 foot

Next, write a ratio for the unit to be converted.

18 feet/1

Then, cancel units, and multiply.

(12 inches/1 foot)×(18 feet/1)=216 inches

The answer is 18 feet = 216 inches.

Example 3.3.6.5

Convert 3 gallons to cups.

**Solution**

First, choose a method.

Since there is more than one standard unit between gallons and cups, setting up one proportion is not possible without intermediate steps. Use unit analysis.

First, write the unit conversions from gallons to cups as fractions.

1 gallon/4quarts 1 quart/2 pints 1 pint/4 cups

Next, write a ratio for the unit to be converted.

3 gallons/1

Then, set up a multiplication problem being sure to cancel unwanted units.

(3 gallons/1)×(4 quarts/1 gallon)×(2 pints/1 quart)×(4 cups/1 pint)

Next, cancel units and multiply.

(3×4×2×4 cups)/(1×1×1×1)=96 cups

The answer is 3 gallons = 96 cups.

### Review

Use unit analysis to solve each problem.

- How many feet in 1 mile?
- How many feet in 18.5 miles?
- How many milliliters in 3.75 liters?
- How many milliliters in 18.25 liters?
- How many pounds in 3 tons?
- How many pounds in 2.5 tons?
- How many pounds in 4.75 tons?
- How many feet in 18 yards?
- How many inches in 4 feet?
- How many inches in 8.75 feet?
- How many milliliters in 29.5 liters?

Solve each problem.

- Fred needs to buy vanilla extract to bake a cake. He could buy a 4-ounce bottle of vanilla extract for $8, or a 6-ounce bottle of vanilla extra for $15. Which bottle is the better buy?
- A rope is 3 yards long. How many inches long is the rope? Use these unit conversions: 1 yard = 3 feet and 1 foot = 12 inches.
- At the farmer's market, Maureen can buy 6 ears of corn for $3. At that price, how much would it cost to buy 9 ears of corn?
- James bought a 128-ounce bottle of apple juice. How many pints of apple juice did James buy? Use these unit conversions: 1 cup = 8 fluid ounces and 1 pint = 2 cups.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.15.

### Additional Resources

PLIX Interactive: **The Price of a Unit**

Video:

Practice: **Convert Using Unit Analysis**