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Measuring is the process of using numbers to describe the physical properties of an object or space. Various units are used to measure. For example, many countries measure distance in kilometers, whereas others measure in miles. Converting these units to one another clears this difference. This lesson will teach how to convert measurement units.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

## Different Measures: Comparing Average Speeds

Two student-led teams, one from Canada and the other the US, made remote controlled robotic cars. Ignacio, of the US, controls the his team's car — Hyperion. Emily, of Canada, controls her team's car — Photon. They are both participating in an international competition and are now doing a test run at the competition site. Ignacio races the robotic car as fast as he can through classrooms, the mountain, around the lake, and finally finishes at the Theatre. Emily's follows a similar path but spends more time going through the mountains. How cool! Ignacio's robotic car, Hyperion, traveled miles in hours. Emily's robotic car, Photon, traveled at an average speed of kilometers per hour. Which robotic car drove at a higher average speed?

## Unit Ratio

A unit ratio is a ratio with a denominator of unit. Every ratio can be rewritten as a unit ratio. A ratio is a comparison of two quantities with similar units of measure. Consider writing unit ratios using the relationship between some units of measure. Notice that the denominator of each unit ratio is unit. The ratio for example, can be reduced by changing the numerator to foot. The ratio then becomes The units are omitted once they are the same. As a result, the ratio is reduced to Also, note that different ratios can be written using the same relationships.
 Fact Ratio is equal to is equal to
The denominators of these ratios are different than unit. Divide both the numerator and denominator of each ratio by the number in the denominator. Then, the denominators will be equal to unit. These resulting unit ratios mean that inch is approximately feet and minute is approximately hours. Unit ratios are useful when converting measurement units.

## Practice Rewriting Ratios

Write the given ratio in the indicated form. ## Converting the Map's Distance to the Actual Distance

Ignacio relies on a GPS screen to help navigate his robotic car Hyperion through the school's diverse terrain. Notice the sign on the map's bottom right corner. This means that inch on the map corresponds to miles of actual distance at the school. How many miles does inches on the map represent in reality? Round the answer to three decimal places if necessary.

### Hint

Write a ratio using the fact that inch on the map represents miles in real distance.

### Solution

The goal is to find how many miles inches on the map represents in reality. This requires multiplying of an inch by a factor that converts inches to miles. Consider the given information on the map.
Write a ratio using this information. Since the given distance is in inches and the goal is to convert it to miles, the denominator of the factor should also be in inches. Doing this ensures that the inches will cancel out. The numerator should contain the needed unit, which is miles.
Now multiply inches by this ratio to get the distance in miles.

Cross out common units

Cancel out common units

This means that inches on the map represent miles in real life.

## Conversion Factor

A conversion factor is a fraction where the numerator and denominator represent the same quantity with different units.
Recall that hour and minutes represent the same quantity. Multiplying a quantity by a conversion factor changes the quantity to an equivalent quantity in different units. Examine how to convert hours to minutes using the above conversion factor.
Given Quantity Conversion Result

Although the final result is in minutes, both quantities represent the same amount of time. Note that the opposite conversion, from minutes to hours, has a conversion factor of If the task was to convert minutes to hours, minutes would be multiplied by this conversion factor.

Given Quantity Conversion Result

As shown in the examples above, the process of including units of measurement as factors is called dimensional analysis. Dimensional analysis can also be used when deciding which conversion factor will produce the desired units. In the table, some common conversion factors are used to convert the given measures.

Given Quantity Conversion Result

Some common conversions involve distance, mass, area, volume, time, and temperature.

### Why

The Reason That the Quantities Are Equivalent

The numerator and denominator of the conversion factor represent the same quantity. That means their quotient equals Then, by the Identity Property of Multiplication, the amount of the given quantity does not change when multiplied by the conversion factor. When converting from one unit to another, the desired unit needs to be in the numerator of the conversion factor while the given unit needs to be in the denominator. That way when the quantity is multiplied by the conversion factor, the given unit will cancel out and the desired unit will remain. Keep in mind that, despite the given quantity and the new quantity have different values, they represent the same amount.

## Customary System

The customary system is the system of measurement commonly used in the United States. This system of measurement contains units for length, capacity, and weight. For example, the inch is a unit of length, the ounce is a unit of weight, and the quart is a unit of volume.
The table shows the relationship between units of each measure type in the customary system.
Customary Units
Type Unit Equivalent Unit
Length foot (ft) inches (in.)
yard (yd) feet
mile (mi) feet
Weight pound (lb) ounces (oz)
ton (T) pounds
Volume cup (c) fluid ounces (fl oz)
pint (pt) cups
quart (qt) pints
gallon (gal) quarts
Converting measures requires using the appropriate conversion factors.

## Determining the Weight of the Robotic Car

The robotics competition that each team will join requires that the robotic cars weigh less than pounds. The robotic car designed by Ignacio's team, Hyperion, currently weighs ounces.

a Find the weight of the robotic car in pounds.
b Does the weight of this robotic car meet the requirement?

### Hint

a Write a conversion factor using the fact that pound is ounces.
b Compare the answer from Part A with pounds.

### Solution

a A conversion factor is needed to convert the given quantity from ounces to pounds. Recall how pounds and ounces are related.
Because the required unit is it will be written in the numerator of the conversion factor. The unit in the denominator will be the same as the unit of the given amount, allowing them to cancel out.
Now convert the given quantity in ounces to pounds by multiplying ounces by the conversion factor.

Cross out common units

Cancel out common units

Therefore, the robotic car is pounds.
b In Part A, the weight of the robotic car was found. It is equal to pounds, which is less than pounds.
This means that Ignacio's team is eligible to participate in the event.

## Metric System

The metric system is the system of measurement used in almost all countries. The base units in the metric system are meters for length, liters for capacity, and grams for weight.
In the metric system, multiples of units follow a decimal pattern. That is, they are powers of Other metric units are named by adding metric prefixes to the base units.

### Metric Units of Length

The table shows the commonly used metric units of length.

Unit Equivalent Unit
millimeters (mm) meter (m)
centimeters (cm) meter
decimeters (dm) meter
dekameter (dam) meters
hectometer (hm) meters
kilometer (km) meters

### Metric Units of Capacity

For measuring capacity, the metric system uses the liter as the base unit.

Unit Equivalent Unit
milliliters (mL) liter (L)
centiliters (cL) liter
deciliters (dL) liter
dekaliter (daL) liters
hectoliter (hL) liters
kiloliter (kL) liters

### Metric Units of Weight

In the metric system, kilogram, gram, and milligram are some commonly used units for measuring weight.

Unit Equivalent Unit
milligrams (mg) gram (g)
centigrams (cg) gram
decigrams (dg) gram
dekagram (dag) grams
hectogram (hg) grams
kilogram (kg) grams
Note that each relationship in the tables can be written as a ratio. These ratios can be considered as conversion factors.

## The Length of the Robot Car Designed by Emily’s Team

Ignacio's team discovered that their robotic car met the weight criteria. At the same time, Emily's team was putting another criterion to the test. The length criterion requires that the cars are no longer than centimeters.

a Photon, Emily's team's robotic car, is meters long. Write this length in centimeters.
b Does this car meet the length criteria?

### Hint

a Write a conversion factor using the fact that meter is centimeters.
b Compare the answer from Part A with centimeters.

### Solution

a The given length is in meters. To convert from meters to centimeters, use the fact that meter is centimeters.
In this case, the required unit is so it will be in the numerator of the conversion factor. The denominator will be its equivalent in so that meters are canceled out when multiplied by meters.
Now convert the given quantity from meters to centimeters by multiplying it by the conversion factor.

Cross out common units

Cancel out common units

The length of the car is centimeters.
b The car has a length of centimeters. This is less than centimeters.
It follows that Photon meets this criteria.

## Converting Measures Between Systems

Units in the customary system can be converted to units in the metric system and vice versa. This may necessitate recalling a lengthy list of conversion factors. ## Filling Out the Competition Application

Emily and Ignacio are filling out the application form for the robotics competition. They must enter the measurements of their cars in multiple system's units.

Applications
Name of Robotic Car Weight Length
Hyperion pounds kilograms inches centimeters
Photon pounds kilograms inches centimeters
Help them find the equivalent measurements. If necessary, round answers to two decimal places.

### Hint

Remember, kilogram is about pounds and inch is centimeters.

### Solution

Notice that Hyperion's known measurements are in customary units, whereas Photon's known measurements are in metric units. Conversion between the systems is needed. The equivalent weights will be found first followed by the equivalent lengths.

### Finding Missing Equivalents of Weights

The weights are in kilograms and pounds. Recall the relationship between these units.
The required conversion factor will be obtained using this information.

#### Finding A

The weight of Hyperion is pounds. In this case, the conversion factor should have a numerator in kilograms and a denominator in pounds.
After determining the factor, multiply the weight of Hyperion and the conversion factor.

Cross out common units

Cancel out common units

Hyperion has a weight of about kilograms. In other words, the value of A is

#### Finding C

To find the weight of Photon in pounds, the multiplicative inverse of the above conversion factor is needed. This is because the measurement in kilograms is converted to pounds.
Use this factor to find the equivalent weight in pounds.

Cross out common units

Cancel out common units

Photon has a weight of pounds. The value of C is

### Finding Missing Equivalents of Lengths

This part requires converting between inches and centimeters. At this point, it is useful to remember that inch is centimeters.

#### Finding B

The length of Hyperion is inches. The conversion factor of will convert this length to centimeters.
Multiply the length of Hyperion by this conversion factor to get the equivalent length in centimeters.

Cross out common units

Cancel out common units

Hyperion has a length of centimeters. The value of B is

#### Finding D

Finally, Photon's length will be converted to inches. This calls for multiplication by
Multiply by

Cross out common units

Cancel out common units

Photon's length is about inches, which means that D is

## Speed of the Robotic Cars on an 80-foot Track

The robotics competition has finally come. Each team will race on an foot track. Opposing teams are watching live from their computers.

The live camera is not that good. The students watching decide to do some math to get a better idea of who is winning!

a Hyperion completed one lap in minutes. Find the speed of the car in inches per second.
b Photon completed laps in minutes. Find the speed of the vehicle in centimeters per second.
c Which car will likely win?

### Hint

a The speed of an object is the distance traveled divided by the time elapsed. Remember that foot is inches.
b Keep in mind that foot is centimeters.
c Determine how many feet each car travels in one minute.

### Solution

a The speed of an object is calculated by dividing the distance traveled by the amount of time spent traveling.
Since one lap is feet long and Hyperion travels a lap in minutes, its speed can be written as follows.
Notice that its unit is feet per minute. To convert this unit to inches per second, two conversion factors are needed.
Equivalent Quantities Conversion Factor
Multiply the speed by the conversion factors.

Cross out common units

Cancel out common units

The speed of the car is inches per second.
b It is given that Photon completes laps in minutes. Since a lap is feet, laps is equal to feet. With this information, the speed of the car can be found in feet per minutes.
To convert it to centimeters per second, two conversion factors are needed.
Equivalent Quantities Conversion Factor
Multiply the speed by the conversion factors.

Cross out common units

Cancel out common units

Photon travels at a speed of centimeters per second.
c In order to compare the speed of the cars, the information given at the beginning is sufficient. Make a table to show the speed of each car.
Hyperion Photon
Simplify

As can be seen, Hyperion can travel feet in a second whereas Photon can travel feet per second. Therefore, Hyperion is faster. Alternatively, the answers found in Part A and Part B can be used. However, a conversion between inches and centimeters is required here.

Hyperion Photon
To compare these quantities, use the fact that inch is centimeters. Multiply the speed of Hyperion by to convert it to centimeters per second.
Hyperion traveled at centimeters per second. That is greater than Photon's speed. It is likely that Hyperion will win this race!

## Different Measures: Comparing Average Speeds

Now take another look at this lesson's challenge comparing the average speeds of two robotic cars. This problem can be completed with the gained knowledge of converting different measurements. Make a table using the given information.

Given
Hyperion (Ignacio's) miles in hours
Photon (Emily's) kilometers per hour
Since speed is the distance traveled divided by the time elapsed, the speed of Hyperion is miles divided by hours.
Hyperion travels at miles per hour. The speed of Photon is written in kilometers per hour, however. To compare two speeds with different units, either of the speeds can be rewritten in terms of the other's unit. In this case, the conversion factor between kilometers and miles is needed.
Use this factor to convert kilometers per hour to miles per hour.
Simplify

Cross out common units

Cancel out common units

The speed of Photon is miles per hour (mph). This is greater than the speed of Hyperion.
This means Emily's car, Photon, drove at a faster average speed than Ignacio drove Hyperion on test runs. However, the performance of Hyperion in the race was better. All things being said, they must have had a blast driving those robotic cars through the school grounds!
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