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.
Show less Show more expand_more| 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.
1in. = 0.075 mi ⇓ 0.075 mi/1in.
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 need a conversion factor convert the given quantity from ounces to pounds. Let's recall how pounds and ounces are related.
1pound is16ounces. or 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 the given quantity in ounces to pounds by multiplying 84 ounces by 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 6 pounds allowed by the competition.
5.25 lb < 6 lb 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?
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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.
1meter is100centimeters. or 1m = 100 cm In this case, the target unit is centimeters, so this will be in the numerator of the conversion factor. The denominator will be its equivalent length in meters so that the meters are canceled out when we multiply by 0.35 meters. Conversion Factor Meters → Centimeters [0.8em] 100 cm/1 m Now convert the given quantity from meters to centimeters by multiplying it by the 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.
35 cm < 40 cm 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 the relationship between these units. 1kilogram is about2.2pounds. or 1kg ≈ 2.2lb 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. Conversion Factor Pounds → Kilograms [0.8em] 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. Conversion Factor Kilograms → Pounds [0.8em] 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. 1inch is2.54centimeters. or 1in. = 2.54cm
Finding B
Car 1 is 15 inches long. The conversion factor of 2.54cm1in. will convert this length to centimeters. Conversion Factor Inches → Centimeters [0.8em] 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 the length of Car 2 to inches by multiplying its length by 1 in.2.54cm. Conversion Factor Centimeters → Inches [0.8em] 1 in./2.54cm Multiply 35 cm 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. A = & 2.4 B = & 38.1 C = & 5.5 D = & 13.8
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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.
r = d/t 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. Alternatively, 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, use the fact that 1 inch is 2.54 centimeters. 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. Car A will likely win this race!
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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 1.1
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Let's take a look at the given statement. 64 fl oz = c The units in the statement are the units used by the customary system. The symbol c stands for cups and fl oz stands for fluid ounces. We know that 1 cup is equal to 8 fluid ounces. 1c = 8fl oz Converting between cups and fluid ounces involves using a conversion factor. We will use the conversion factor 1c8 fl oz. Multiplying with this factor will eliminate the unit that appears in the given quantity. 1c/8fl oz Let's multiply 64 fl oz and the conversion factor.
We need to write 8 into the blank space. 64 fl oz = 8 c
We want to convert from tons (T) to pounds (lb).
7/20 T = lb
Remember that 1 ton is equal to 2000 pounds.
1T = 2000lb
We will use the conversion factor 2000lb1 T to convert between tons and pounds. Multiplying the given measure by this factor will result in pounds.
Therefore, the given measure is equal to 700 pounds. 7/20 T = 700 lb
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Hint & Answer
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Solution
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Let's take a look at the given statement. 3.4 L = cL The letter L stands for liters. The liter is the base unit of capacity measurement in the metric system. We also know that 1 liter is equal to 100 centiliters. 1L = 100cL We want to convert from liters to centiliters. The appropriate conversion factor in this case will be 100cL1 L. Multiplying by this factor will leave us the unit we need. 3.4 L * 100cL/1 L Let's perform the multiplication.
We can fill in the blank with 100. 3.4 L = 340 cL
In this case, we are asked to convert from grams (g) to kilograms (kg).
10 000 g = kg
We will use the fact that 1 kilogram is equal to 1000 grams.
1kg = 1000g
We will use the conversion factor 1kg1000 g. This factor will allow us to cancel out the common unit when we multiply the given weight by it.
10 000 g * 1kg/1000 g
Let's perform the multiplication.
This means that 10 000 grams is equivalent to 10 kilograms. 10 000g = 10 kg
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Hint & Answer
S
Solution
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We want to convert between miles (mi) and kilometers (km). 22 mi ≈ km We will use a conversion factor to do so. Recall that 1 mile is about 1.61 kilometers. The appropriate conversion factor in the given case is 1.61km1 mi. 1mi ≈ 1.61km ⇓ 1.61km/1 mi Multiplying 22 miles by this conversion factor will convert it to kilometers.
We can fill in the blank with 35.42. 22 mi ≈ 35.42 km
We are asked to convert from kilograms (kg) to ounces (oz).
35 kg ≈ oz
We will use the fact that 1 kilogram is about 35.27 ounces. The conversion factor 35.27oz1 kg will allow us to cancel out the common unit.
1 kg ≈ 35.27 oz
⇓
35.27oz/1 kg
Let's multiply 35 kg by the conversion factor.
This means that 35 kilograms are about 1234.45 ounces. 35kg ≈ 1234.45 oz
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Hint & Answer
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Solution
{"type":"choice","form":{"alts":[">","<","="],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":2}
{"type":"choice","form":{"alts":[">","<","="],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
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We want to compare the given values by writing a sign <, >, or = between them. First, take a look at the units in the given statement. 3 12 yd 10 12 ft Both yards (yd) and feet (ft) are customary units of length. We need to convert one of them to the other unit to compare these values. Recall how yards and feet are related. 1 yd = 3 ft We will write a conversion factor using this equation. Let's convert 3 12 yards to feet. Since we want to convert yards to feet, our ratio will have yards in the denominator. We need to write 3 feet in the numerator and 1 yard in the denominator. Conversion Factor 3ft/1yd Let's multiply 3 12 yards and the conversion factor. We will start with writing the mixed number as improper fraction. Then, we will divide out the common units. This will give us the desired unit, feet.
We can rewrite the fraction as a mixed number.
We obtained the value on the other side of the box. Therefore, 3 12 yards is equal to 10 12 feet. Knowing that we can fill in the blank. 3 12 yd = 10 12 ft
In this case, we need to compare two quantities measured in different systems. The gallon (gal) is a customary unit, and the liter (L) is a metric unit. Both are used to measure capacity or volume.
26 gal 100 L
We need to convert one of them to the other unit to compare these values. Recall the relationships between these units.
1 gal ≈ 3.79 L
Let's convert 26 gal to liters. We need to write an appropriate conversion factor to do so. Since we want to convert gallons to liters, we write 3.79L in the numerator and 1 gal in the denominator.
3.79L/1gal
Let's multiply 26 gal and the conversion factor.
This is less than 100 L. We can fill in the square with <. 26 gal < 100 L ⇕ 98.54 L < 100 L
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Hint & Answer
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":"yd\/min","answer":{"text":["720"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":"L\/min","answer":{"text":["5.7"]}}
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We want to convert 12 yards per second to yards per minute. Let's first write 12 yards per second as a fraction. 12 yards per second ⇒ 12 yd/1 sec We will use a conversion factor between seconds and minutes. We know that 1 minute is 60 seconds. Our conversion factor will have minutes in the denominator because we want to convert per second to per minute. Conversion Factor 60sec/1min We can now multiply 12 yards per second by the conversion factor.
We have found that 12 yards per second is 720 yards per minute.
Let's start by writing 342 liters per hour as a fraction.
342 liters per hour ⇒ 342 L/1 h
We want to convert liters per hour to liters per minute. This requires the use of a conversion factor between minutes and hours. Let's recall that 1 hour is 60 minutes. The conversion factor will have minutes in the denominator so that minutes can cancel out.
Conversion Factor
1h/60min
Now let's multiply 342 liters per hour by this conversion factor.
Therefore, 342 liters per hour is 5.7 liters per minute.
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Convert Measurement Units
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