7. Convert Measurement Units
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Chapter 3
7.
Convert Measurement Units
Understanding measurement conversion is crucial in various fields, from engineering to daily life. The lesson emphasizes the importance of being proficient in converting units, especially within the metric system. It also delves into the concept of unit ratios, illustrating their practical applications like determining speed or converting dimensions of objects. This knowledge serves as a comprehensive guide for those involved in activities that require a nuanced understanding of unit conversions, making it easier to meet specific criteria or standards.
| Student Learning Objectives: |
|---|
|
| | 14 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Convert Measurement Units
Slide of 14
Measuring is the process of using numbers and units to describe things like length, weight, or space. We use a variety of units to measure things. For example, distance can be measured in kilometers or miles. This lesson explains how to convert from one unit to another.
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Challenge
Different Measures: Comparing Average Speeds
Ignacio and Emily are competing in an international remote control car race.
Ignacio car traveled 0.4 miles in 2 hours. Emily's car traveled at an average speed of 0.4 kilometers per hour. Whose car drove at a higher average speed?
{"type":"choice","form":{"alts":["Ignacio","Emily"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
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Discussion
Unit Ratio
A unit ratio is a ratio with a denominator of 1 unit.
When writing unit ratios using different units of measure, the denominator of the unit ratio is still 1 unit. Note that different ratios can be written using the same relationships.
| Fact | 1 foot is equal to 12 inches. | 1 hour is equal to 60 minutes. |
|---|---|---|
| Ratio | 1ft/12in. | 1h/60min |
The denominators of these ratios are different than 1 unit. Dividing both the numerator and denominator of each ratio by the number in the denominator will give the appropriate unit ratio.
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Pop Quiz
Practice Rewriting Ratios
Write the given ratio in the indicated form. Do not round any decimal numbers.
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Example
Converting the Map's Distance to the Actual Distance
Ignacio found a map of his school's campus.
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Hint
Write a ratio using the fact that 1 inch on the map represents 0.075 miles in real distance.
Solution
We want to find how many miles 1.4 inches on the map represents in reality. We can multiply 1.4 inches by a factor that converts inches to miles on the map. Consider the given information on the map again.
ccc
On the Map & & In Actual Distance 1in. & : & 0.075 mi
Let's write a ratio using this information. Since the given distance 1.4 is in inches and the goal is to convert it to miles, the denominator of the factor will also be in inches. This will cancel out the inch units. The numerator will contain the target unit miles.
0.075 mi/1in.
Let's multiply 1.4 inches by this ratio to get the distance in miles.
1.4 in. * 0.075 mi/1in.
MoveLeftFacToNum
a*b/c= a* b/c
1.4 in. * 0.075 mi/1in.
Cross out common units
1.4 in.* 0.075 mi/1 in.
Cancel out common units
1.4 * 0.075 mi/1
Multiply
Multiply
0.105 mi/1
DivByOne
a/1=a
0.105 mi
This means that 1.4 inches on the map represents 0.105 miles in real life.
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Discussion
Conversion Factor
A conversion factor is a fraction where the numerator and denominator represent the same quantity with different units. Example Conversion Factor [0.5em] 60 minutes/1 hour Recall that 1 hour and 60 minutes represent the same quantity. Multiplying a quantity by a conversion factor changes the quantity to an equivalent quantity in different units.
| Given Quantity | Conversion | Result |
|---|---|---|
| 2 hours | 2 hours * 60 minutes/1 hour | 120 minutes |
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 target units.
Why
The Reason That the Quantities Are EquivalentThe numerator and denominator of the conversion factor represent the same quantity. This means that their quotient equals 1. Then, by the Identity Property of Multiplication, the amount of the given quantity does not change when multiplied by the conversion factor.
Given Amount * 1 = Given Amount
When converting from one unit to another, the target unit needs to be in the numerator of the conversion factor and the given unit needs to be in the denominator. Then when the quantity is multiplied by the conversion factor, the given unit will cancel out and the target unit will remain.
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Discussion
Customary System
The customary system is the measurement system most 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. Examples of Units in the Customary System inch, ounce, and quart The table shows the relationship between units of each measure type in the customary system.
| Customary Units | ||
|---|---|---|
| Type | Unit | Equivalent Unit |
| Length | 1 foot (ft) | 12 inches (in.) |
| 1 yard (yd) | 3ft | |
| 1 mile (mi) | 5280ft | |
| Weight | 1 pound (lb) | 16 ounces (oz) |
| 1 ton (T) | 2000 lb | |
| Volume | 1 cup (c) | 8 fluid ounces (fl oz) |
| 1 pint (pt) | 2c | |
| 1 quart (qt) | 2pt | |
| 1 gallon (gal) | 4qt | |
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Example
Determining the Weight of the Robotic Car
Remote control cars in a robotics competition must weigh less than 6 pounds. Tearrik's car currently weighs 84 ounces.
a
Find the weight of the car in pounds.
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b
Does the weight of this robotic car meet the requirement?
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Hint
a
Write a conversion factor using the fact that 1 pound is 16 ounces.
b
Is the answer from Part A less than 6 pounds?
Solution
a
We will convert the given quantity from ounces to pounds using a conversion factor. To find it, let's first recall that 1 pound is 16 ounces.
1lb = 16oz Because our target unit is pounds, 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, ounces, so we can cancel them out. Conversion Factor Ounces → Pounds [0.8em] 1 lb/16 oz Now let's convert 84 ounces to pounds using our conversion factor.
84 oz * 1lb/16 oz
MoveLeftFacToNum
a*b/c= a* b/c
84 oz * 1lb/16 oz
Cross out common units
84 oz * 1lb/16 oz
Cancel out common units
84 * 1 lb/16
MultByOne
a * 1=a
84 lb/16
MovePartNumRight
a* b/c=a/c* b
84/16 lb
CalcQuot
Calculate quotient
5.25 lb
Therefore, the robotic car weighs 5.25 pounds.
b
We found that the car weighs to 5.25 pounds, which is less than the maximum of 6 pounds allowed by the competition. This means that Terrik's car meets the weight requirement.
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Discussion
Metric System
The metric system is a measurement system commonly in most countries around the world. The base units in the metric system are meters for length, liters for capacity, and kilograms for weight. Base Units in Metric System meter, liter, and kilogram In the metric system, units are related by powers of 10. 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 |
|---|---|
| 1000 millimeters (mm) | 1 meter (m) |
| 100 centimeters (cm) | 1m |
| 10 decimeters (dm) | 1m |
| 1 dekameter (dam) | 10m |
| 1 hectometer (hm) | 100m |
| 1 kilometer (km) | 1000m |
Metric Units of Capacity
For measuring capacity, the metric system uses the liter as the base unit.
| Unit | Equivalent Unit |
|---|---|
| 1000 milliliters (mL) | 1 liter (L) |
| 100 centiliters (cL) | 1L |
| 10 deciliters (dL) | 1L |
| 1 dekaliter (daL) | 10L |
| 1 hectoliter (hL) | 100L |
| 1 kiloliter (kL) | 1000L |
Metric Units of Weight
In the metric system, kilogram, gram, and milligram are some commonly used units for measuring weight.
| Unit | Equivalent Unit |
|---|---|
| 1000 milligrams (mg) | 1 gram (g) |
| 100 centigrams (cg) | 1g |
| 10 decigrams (dg) | 1g |
| 1 dekagram (dag) | 10g |
| 1 hectogram (hg) | 100g |
| 1 kilogram (kg) | 1000g |
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Example
The Length of Emily's Car
Remote control cars in a robotics competition must be no longer than 40 centimeters.
a
Emily's car is 0.35 meters long. Write this length in centimeters.
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b
Does this car meet the length criteria?
{"type":"choice","form":{"alts":["Yes","No"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
Hint
a
Write a conversion factor using the fact that 1 meter is 100 centimeters.
b
Is the answer from Part A longer than 40 centimeters?
Solution
a
We need to compare target length in centimeters to a given length in in meters. We can use the fact that 1 meter is 100 centimeters to convert from meters to centimeters.
1m = 100 cm In this case, the target unit is centimeters, so we will put 100cm in the numerator of the conversion factor. The denominator will be its equivalent length in meters. 100cm/1m Now let's convert the given quantity from meters to centimeters by multiplying it by the above conversion factor.
0.35 m * 100cm/1 m
MoveLeftFacToNum
a*b/c= a* b/c
0.35 m * 100cm/1 m
Cross out common units
0.35 m * 100cm/1 m
Cancel out common units
0.35 * 100 cm/1
DivByOne
a/1=a
0.35 * 100 cm
Multiply
Multiply
35 cm
The length of the car is 35 centimeters.
b
Emily's car is 35 centimeters long. This is less than 40 centimeters, so her car meets the length criteria of the competition.
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Discussion
Converting Measures Between Systems
Units can be converted between the customary system and the metric system using conversion factors. Use the applet below to explore several common conversion factors.
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Example
Unit Conversion Practice
Fill in the table to convert between units.
| Applications | ||||
|---|---|---|---|---|
| Car | Weight | Length | ||
| Car 1 | 5.3 pounds | A kilograms | 15 inches | B centimeters |
| Car 2 | C pounds | 2.5 kilograms | D inches | 35 centimeters |
If necessary, round answers to the nearest tenth.
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Hint
Remember, 1 kilogram is about 2.2 pounds and 1 inch is 2.54 centimeters.
Solution
The measurements given for Car 1 are in customary units, but they are in metric units for Car 2. We want to convert between them. Let's start with the weights.
Finding Missing Weights
The weights are in kilograms and pounds. Let's recall that 1 kilogram is about 2.2 pounds. We can create a conversion factor using this information.
Finding A
Car 1 weighs 5.25 pounds, so our conversion factor should have a numerator in kilograms and a denominator in pounds. 1kg/2.2lb Now let's multiply the weight of Car 1 and the conversion factor.
5.3 lb * 1kg/2.2 lb
MoveLeftFacToNum
a*b/c= a* b/c
5.3 lb * 1kg/2.2 lb
Cross out common units
5.3 lb * 1kg/2.2 lb
Cancel out common units
5.3 * 1 kg/2.2
MultByOne
a * 1=a
5.3 kg/2.2
MovePartNumRight
a* b/c=a/c* b
5.3/2.2 kg
CalcQuot
Calculate quotient
2.409090 ... kg
RoundDec
Round to 1 decimal place(s)
≈ 2.4
Hyperion has a weight of about 2.4 kilograms. In other words, the value of A is 2.4.
Finding C
To find the weight of Car 2 in pounds, we will use the multiplicative inverse of our previous conversion factor. This will put the pounds unit in the numerator and give our conversion in pounds. 2.2lb/1kg Let's use this factor to find the equivalent weight in pounds.
2.5 kg * 2.2lb/1 kg
MoveLeftFacToNum
a*b/c= a* b/c
2.5 kg * 2.2lb/1 kg
Cross out common units
2.5 kg * 2.2lb/1 kg
Cancel out common units
2.5 * 2.2 lb/1
DivByOne
a/1=a
2.5 * 2.2 lb
Multiply
Multiply
5.5 lb
Car 2 weighs 5.5 pounds. The value of C is 5.5.
Finding Missing Lengths
This part requires converting between inches and centimeters. Let's remember that 1 inch is 2.54 centimeters.
Finding B
Car 1 is 15 inches long. To convert this length to centimeters, we can use the following conversion factor. 2.54cm/1in. Multiply the given length by this conversion factor to get the equivalent length in centimeters.
15 in. * 2.54cm/1in.
MoveLeftFacToNum
a*b/c= a* b/c
15 in. * 2.54cm/1 in.
Cross out common units
15 in. * 2.54cm/1 in.
Cancel out common units
15 * 2.54 cm/1
DivByOne
a/1=a
15 * 2.54 cm
Multiply
Multiply
38.1 cm
Car 1 is 38.1 centimeters long. The value of B is 38.1.
Finding D
Let's convert Car 2's length, 35 cm, to inches by multiplying it by 1 in. 2.54cm.
35 cm * 1 in./2.54cm
MoveLeftFacToNum
a*b/c= a* b/c
35 cm * 1in./2.54 cm
Cross out common units
35 cm * 1in./2.54 cm
Cancel out common units
35 * 1 in./2.54
MultByOne
a * 1=a
35 in./2.54
MovePartNumRight
a* b/c=a/c* b
35/2.54 in.
CalcQuot
Calculate quotient
13.779527 ... in.
RoundDec
Round to 1 decimal place(s)
≈ 13.8 in.
Car 2 is about 13.8 inches long, which means that D is 13.8. Let's summarize our findings.
| Applications | ||||
|---|---|---|---|---|
| Car | Weight | Length | ||
| Car 1 | 5.3 pounds | 2.4 kilograms | 15 inches | 38.1 centimeters |
| Car 2 | 5.5 pounds | 2.5 kilograms | 13.8 inches | 35 centimeters |
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Example
Speeds of Robotic Cars
Two remote control cars race around a 80-foot track.
a
Car A completed one lap in 5 minutes. Find the speed of the car in inches per second.
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b
Car B completed 3 laps in 16 minutes. Find the speed of the vehicle in centimeters per second.
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c
Which car will likely win?
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Hint
a
The speed of an object is the distance traveled divided by the time elapsed. Remember that 1 foot is 12 inches.
b
Keep in mind that 1 foot is 30.48 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 80 feet and Car A completes a lap in 5 minutes, its speed can be written as follows.
r = 80 ft/5min Notice that this unit is feet per minute. To convert this to inches per second, we need to combine two conversion factors.
| Equivalent Quantities | Conversion Factor |
|---|---|
| 1 ft = 12 in. | 12 in./1ft |
| 1 min = 60 sec | 1 min/60sec |
Let's multiply the speed by the conversion factors.
80ft/5 min * 12 in./1ft * 1 min/60sec
MultFrac
Multiply fractions
80ft * 12 in. * 1 min/5 min * 1ft * 60sec
Cross out common units
80 ft * 12 in. * 1 min/5 min * 1 ft * 60sec
Cancel out common units
80 * 12 in. * 1/5 * 1 * 60 sec
MultByOne
a * 1=a
80 * 12 in./5 * 60 sec
Multiply
Multiply
960 in./300 sec
CalcQuot
Calculate quotient
3.2 in./sec
The speed of the car is 3.2 inches per second.
b
Car B completed 3 laps in 16 minutes. Since a lap is 80 feet, 3 laps is equal to 240 feet. Let's use this information to find the speed of the car in feet per minute.
r = 240 ft/16min To convert it to centimeters per second, we need two conversion factors.
| Equivalent Quantities | Conversion Factor |
|---|---|
| 1 ft = 30.48 cm | 30.48 cm/1ft |
| 1 min = 60 sec | 1 min/60sec |
Let's multiply the speed by the conversion factors.
240ft/16 min * 30.48 cm/1ft * 1 min/60sec
MultFrac
Multiply fractions
240ft * 30.48 cm * 1 min/16 min * 1ft * 60sec
Cross out common units
240 ft * 30.48 cm * 1 min/16 min * 1 ft * 60sec
Cancel out common units
240 * 30.48 cm * 1/16 * 1 * 60 sec
MultByOne
a * 1=a
240 * 30.48 cm/16 * 60 sec
Multiply
Multiply
7315.2 cm/960 sec
CalcQuot
Calculate quotient
7.62 cm/sec
Car B travels at a speed of 7.62 centimeters per second.
c
The setup gives us enough information to compare the speeds of the cars. Let's make a table showing each car's speed.
| Car A | Car B | |
|---|---|---|
| Speed = Distance/Time | 80 ft/5min | 240 ft/16min |
| Simplify | 16ft/min | 15ft/min |
As we can see, Car A travels at 16 feet in a minute, while Car B travels at 15 feet per minute. Therefore, Car A is faster and will likely win this race!
Alternative Solution
Alternative Solution of Part CWe can solve Part C in a different way. We can use the answers we found in Part A and Part B to compare the cars' speeds, but we will need to convert between inches and centimeters.
| Car A | Car B | |
|---|---|---|
| Speed | 3.2in./sec | 7.62cm/sec |
To compare these quantities, let's use the fact that 1 inch is 2.54 centimeters. We will multiply the speed of Car A by 2.54 cm1 in. to convert it to centimeters per second. 3.2in./1 sec * 2.54 cm/1 in. = 8.128cm/1 sec Car A travels at 8.128 centimeters per second. That is greater than Car B's speed.
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Closure
Different Measures: Comparing Average Speeds
Earlier we were given the distance traveled by two different remote control cars.
| Travels | |
|---|---|
| Ignacio's Car | 0.4 miles in 2 hours |
| Emily's Car | 0.4 kilometers per hour |
Since speed is distance traveled divided by time, the speed of Ignacio's car is 0.4 miles divided by 2 hours. Ignacio's Car [0.7em] 0.4 miles/2 hours = 0.2miles/1 hour Ignacio's car travels at 0.2 miles per hour. Emily's car's speed is given in kilometers per hour, however. To compare two speeds with different units, one of them must be rewritten in terms of the other. Let's use the conversion factor between kilometers and miles. 1mile is about1.6kilometers. ⇓ Conversion Factor 1 mi/1.6 km Use this factor to convert kilometers per hour to miles per hour.
0.4 km/1 h * 1mi/1.6 km
▼
Simplify
MultFrac
Multiply fractions
0.4 km * 1mi/1 h* 1.6 km
Cross out common units
0.4 km * 1mi/1 h* 1.6 km
Cancel out common units
0.4 * 1mi/1 h* 1.6
MultByOne
a * 1=a
0.4 mi/1.6 h
ReduceFrac
a/b=.a /1.6./.b /1.6.
0.25 mi/1 h
RoundDec
Round to 1 decimal place(s)
0.25 mi/1 h
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Convert Measurement Units
Exercise 3.1
A painter needs 1 liter of paint to cover an area of 90 square feet. How many gallons of paint does the painter need to cover a room whose walls have an area of 100 square meters?
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We know that the painter uses 1 liter of paint to cover an area of 90 square feet. Let's express it as a rate. 90 ft^2/1L We want to find the number of gallons of paint it will take to cover a room whose walls have an area of 100 square meters. First, we will use 90 ft^21L to find out how many square feet the painter can cover with 1 gallon. To do so, remember the conversion factor between gallons and liter. Conversion Factor 3.79 L/1 gal Let's multiply the ratio by the conversion factor.
This ratio means that one gallon of paint covers about 341.1 square feet. We will multiply it by another conversion factor to convert square feet to square meters. We know that 1 square meter is about 10.76 square feet. Then, the conversion factor we need is the following. Conversion Factor 1 m^2/10.76 ft^2 Let's multiply 341.1ft^21 gal by this conversion factor.
This means that 1 gallon of paint covers about 31.7 square meters. Now, we will construct a ratio table for this relationship.
| Gallons | 1 | ||
|---|---|---|---|
| Square Meters | 31.7 |
We can divide each quantity by 31.7 to obtain a unit rate. This will helps us to calculate the gallons per 100 square meters.
| Gallons | 1 | 0.031 | |
|---|---|---|---|
| Square Meters | 31.7 | 1 |
Finally, we can multiply each quantity in the second row by 100.
| Gallons | 1 | 0.031 | 3.1 |
|---|---|---|---|
| Square Meters | 31.7 | 1 | 100 |
We got that 3.1 gallons of paint are needed to cover 100 square meters. Since the room has 4 walls, we need to multiply the 3.1 by 4 to obtain the total amount of paint. 4(3.1)≈ 12.4 Therefore, we need about 12.4 gallons of paint to cover a room whose walls have an area of 100 square meters.
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Convert Measurement Units
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