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.
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| | 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 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.
Challenge
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 0.4 miles in 2 hours. Emily's robotic car, Photon, traveled at an average speed of 0.4 kilometers per hour. Which robotic car drove at a higher average speed?
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Discussion
Unit Ratio
A unit ratio is a ratio with a denominator of 1 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 1 unit.
The ratio 1 ft12 in., for example, can be reduced by changing the numerator to 1 foot. The ratio then becomes 1 ft1 ft. The units are omitted once they are the same. As a result, the ratio is reduced to 1. Also, note that different ratios can be written using the same relationships.
The denominators of these ratios are different than 1 unit. Divide both the numerator and denominator of each ratio by the number in the denominator. Then, the denominators will be equal to 1 unit.
These resulting unit ratios mean that 1 inch is approximately 0.083 feet and 1 minute is approximately 0.017 hours. Unit ratios are useful when converting measurement units.
| Fact | 1 foot is equal to 12 inches. | 1 hour is equal to 60 minutes. |
|---|---|---|
| Ratio | 12 in.1 ft | 60 min1 h |
Pop Quiz
Practice Rewriting Ratios
Write the given ratio in the indicated form.
Example
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 1 inch on the map corresponds to 0.075 miles of actual distance at the school. How many miles does 1.4 inches on the map represent in reality? Round the answer to three decimal places if necessary.
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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
The goal is to find how many miles 1.4 inches on the map represents in reality. This requires multiplying 1.4 of an inch by a factor that converts inches to miles. Consider the given information on the map.
This means that 1.4 inches on the map represent 0.105 miles in real life.
On the Map1 in.=In Actual Distance0.075 mi
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 should also be in inches. Doing this ensures that the inches will cancel out. The numerator should contain the needed unit, which is miles.
1 in.=0.075 mi⇓1 in.0.075 mi
Now multiply 1.4 inches by this ratio to get the distance in miles.
1.4 in.×1 in.0.075 mi
MoveLeftFacToNum
a⋅cb=ca⋅b
1 in.1.4 in.×0.075 mi
Cross out common units
1 in.1.4 in.×0.075 mi
Cancel out common units
11.4×0.075 mi
Multiply
Multiply
10.105 mi
DivByOne
1a=a
0.105 mi
Discussion
Conversion Factor
A conversion factor is a fraction where the numerator and denominator represent the same quantity with different units.
Example Conversion Factor1 hour60 minutes
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. Examine how to convert 2 hours to minutes using the above conversion factor. | Given Quantity | Conversion | Result |
|---|---|---|
| 2 hours | 2 hours⋅1 hour60 minutes | 120 minutes |
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 60 minutes1 hour. If the task was to convert 120 minutes to hours, 120 minutes would be multiplied by this conversion factor.
| Given Quantity | Conversion | Result |
|---|---|---|
| 120 hours | 120 minutes⋅60 minutes1 hour | 2 hours |
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 |
|---|---|---|
| 3 pounds | 3 pounds⋅1 pound16 ounces | 48 ounces |
| 160 ounces | 160 ounces⋅16 ounces1 pound | 10 pounds |
| 1 mile | 1 mile⋅1 mile1760 yards | 1760 yards |
Some common conversions involve distance, mass, area, volume, time, and temperature.
Why
The Reason That the Quantities Are EquivalentThe numerator and denominator of the conversion factor represent the same quantity. That means 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.
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.
Discussion
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.
Converting measures requires using the appropriate conversion factors.
Examples of Units in the Customary Systeminch, 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) | 3 feet | |
| 1 mile (mi) | 5280 feet | |
| Weight | 1 pound (lb) | 16 ounces (oz) |
| 1 ton (T) | 2000 pounds | |
| Volume | 1 cup (c) | 8 fluid ounces (fl oz) |
| 1 pint (pt) | 2 cups | |
| 1 quart (qt) | 2 pints | |
| 1 gallon (gal) | 4 quarts | |
Example
Determining the Weight of the Robotic Car
The robotics competition that each team will join requires that the robotic cars weigh less than 6 pounds. The robotic car designed by Ignacio's team, Hyperion, currently weighs 84 ounces.
a Find the weight of the robotic 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 Compare the answer from Part A with 6 pounds.
Solution
a A conversion factor is needed to convert the given quantity from ounces to pounds. Recall how pounds and ounces are related.
1 pound is 16 ounces. or 1 lb=16 oz
Because the required 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, allowing them to cancel out.
Conversion FactorOunces→Pounds16 oz1 lb
Now convert the given quantity in ounces to pounds by multiplying 84 ounces by the conversion factor.
84 oz⋅16 oz1 lb
MoveLeftFacToNum
a⋅cb=ca⋅b
16 oz84 oz⋅1 lb
Cross out common units
16 oz84 oz⋅1 lb
Cancel out common units
1684⋅1 lb
MultByOne
a⋅1=a
1684 lb
MovePartNumRight
ca⋅b=ca⋅b
1684 lb
CalcQuot
Calculate quotient
5.25 lb
b In Part A, the weight of the robotic car was found. It is equal to 5.25 pounds, which is less than 6 pounds.
5.25 lb<6 lb
This means that Ignacio's team is eligible to participate in the event.
Discussion
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.
Note that each relationship in the tables can be written as a ratio. These ratios can be considered as conversion factors.
Base Units in Metric Systemmeter, liter,and gram
In the metric system, multiples of units follow a decimal pattern. That is, they are powers of 10. 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 |
|---|---|
| 1000 millimeters (mm) | 1 meter (m) |
| 100 centimeters (cm) | 1 meter |
| 10 decimeters (dm) | 1 meter |
| 1 dekameter (dam) | 10 meters |
| 1 hectometer (hm) | 100 meters |
| 1 kilometer (km) | 1000 meters |
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) | 1 liter |
| 10 deciliters (dL) | 1 liter |
| 1 dekaliter (daL) | 10 liters |
| 1 hectoliter (hL) | 100 liters |
| 1 kiloliter (kL) | 1000 liters |
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) | 1 gram |
| 10 decigrams (dg) | 1 gram |
| 1 dekagram (dag) | 10 grams |
| 1 hectogram (hg) | 100 grams |
| 1 kilogram (kg) | 1000 grams |
Example
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 40 centimeters.
a Photon, Emily's team's robotic 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 Compare the answer from Part A with 40 centimeters.
Solution
a The given length is in meters. To convert from meters to centimeters, use the fact that 1 meter is 100 centimeters.
1 meter is 100 centimeters. or 1 m=100 cm
In this case, the required unit is centimeters, so it will be in the numerator of the conversion factor. The denominator will be its equivalent in meters so that meters are canceled out when multiplied by 0.35 meters.
Conversion FactorMeters→Centimeters1 m100 cm
Now convert the given quantity from meters to centimeters by multiplying it by the conversion factor.
0.35 m⋅1 m100 cm
MoveLeftFacToNum
a⋅cb=ca⋅b
1 m0.35 m⋅100 cm
Cross out common units
1 m0.35 m⋅100 cm
Cancel out common units
10.35⋅100 cm
DivByOne
1a=a
0.35⋅100 cm
Multiply
Multiply
35 cm
b The car has a length of 35 centimeters. This is less than 40 centimeters.
35 cm<40 cm
It follows that Photon meets this criteria.
Discussion
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.
Example
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 | 5.25 pounds | A kilograms | 15 inches | B centimeters |
| Photon | C pounds | 2.5 kilograms | D inches | 35 centimeters |
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Hint
Remember, 1 kilogram is about 2.2 pounds and 1 inch is 2.54 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.1 kilogram is about 2.2 pounds. or 1 kg≈2.2 lb
The required conversion factor will be obtained using this information. Finding A
The weight of Hyperion is 5.25 pounds. In this case, the conversion factor should have a numerator in kilograms and a denominator in pounds.Conversion FactorPounds→Kilograms2.2 lb1 kg
After determining the factor, multiply the weight of Hyperion and the conversion factor.
5.25 lb⋅2.2 lb1 kg
MoveLeftFacToNum
a⋅cb=ca⋅b
2.2 lb5.25 lb⋅1 kg
Cross out common units
2.2 lb5.25 lb⋅1 kg
Cancel out common units
2.25.25⋅1 kg
MultByOne
a⋅1=a
2.25.25 kg
MovePartNumRight
ca⋅b=ca⋅b
2.25.25 kg
CalcQuot
Calculate quotient
2.386363… kg
RoundDec
Round to 2 decimal place(s)
≈2.39
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.Conversion FactorKilograms→Pounds1 kg2.2 lb
Use this factor to find the equivalent weight in pounds.
2.5 kg⋅1 kg2.2 lb
MoveLeftFacToNum
a⋅cb=ca⋅b
1 kg2.5 kg⋅2.2 lb
Cross out common units
1 kg2.5 kg⋅2.2 lb
Cancel out common units
12.5⋅2.2 lb
DivByOne
1a=a
2.5⋅2.2 lb
Multiply
Multiply
5.5 lb
Finding Missing Equivalents of Lengths
This part requires converting between inches and centimeters. At this point, it is useful to remember that 1 inch is 2.54 centimeters.1 inch is 2.54 centimeters. or 1 in.=2.54 cm
Finding B
The length of Hyperion is 15 inches. The conversion factor of 1 in.2.54 cm will convert this length to centimeters.Conversion FactorInches→Centimeters1 in.2.54 cm
Multiply the length of Hyperion by this conversion factor to get the equivalent length in centimeters.
15 in.⋅1 in.2.54 cm
MoveLeftFacToNum
a⋅cb=ca⋅b
1 in.15 in.⋅2.54 cm
Cross out common units
1 in.15 in.⋅2.54 cm
Cancel out common units
115⋅2.54 cm
DivByOne
1a=a
15⋅2.54 cm
Multiply
Multiply
38.1 cm
Finding D
Finally, Photon's length will be converted to inches. This calls for multiplication by 2.54 cm1 in..Conversion FactorCentimeters→Inches2.54 cm1 in.
Multiply 35 cm by 2.54 cm1 in..
35 cm⋅2.54 cm1 in.
MoveLeftFacToNum
a⋅cb=ca⋅b
2.54 cm35 cm⋅1 in.
Cross out common units
2.54 cm35 cm⋅1 in.
Cancel out common units
2.5435⋅1 in.
MultByOne
a⋅1=a
2.5435 in.
MovePartNumRight
ca⋅b=ca⋅b
2.5435 in.
CalcQuot
Calculate quotient
13.779527… in.
RoundDec
Round to 2 decimal place(s)
≈13.78 in.
A=B=C=D= 2.39 38.1 5.5 13.78
Example
Speed of the Robotic Cars on an 80-foot Track
The robotics competition has finally come. Each team will race on an 80-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 5 minutes. Find the speed of the car in inches per second.
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b Photon 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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Solution
a The speed of an object is calculated by dividing the distance traveled by the amount of time spent traveling.
r=td
Since one lap is 80 feet long and Hyperion travels a lap in 5 minutes, its speed can be written as follows.
r=5 min80 ft
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 |
|---|---|
| 1 ft=12 in. | 1 ft12 in. |
| 1 min=60 sec | 60 sec1 min |
5 min80 ft⋅1 ft12 in.⋅60 sec1 min
MultFrac
Multiply fractions
5 min⋅1 ft⋅60 sec80 ft⋅12 in.⋅1 min
Cross out common units
5 min⋅1 ft⋅60 sec80 ft⋅12 in.⋅1 min
Cancel out common units
5⋅1⋅60 sec80⋅12 in.⋅1
MultByOne
a⋅1=a
5⋅60 sec80⋅12 in.
Multiply
Multiply
300 sec960 in.
CalcQuot
Calculate quotient
3.2 sec in.
b It is given that Photon completes 3 laps in 16 minutes. Since a lap is 80 feet, 3 laps is equal to 240 feet. With this information, the speed of the car can be found in feet per minutes.
r=16 min240 ft
To convert it to centimeters per second, two conversion factors are needed. | Equivalent Quantities | Conversion Factor |
|---|---|
| 1 ft=30.48 cm | 1 ft30.48 cm |
| 1 min=60 sec | 60 sec1 min |
16 min240 ft⋅1 ft30.48 cm⋅60 sec1 min
MultFrac
Multiply fractions
16 min⋅1 ft⋅60 sec240 ft⋅30.48 cm⋅1 min
Cross out common units
16 min⋅1 ft⋅60 sec240 ft⋅30.48 cm⋅1 min
Cancel out common units
16⋅1⋅60 sec240⋅30.48 cm⋅1
MultByOne
a⋅1=a
16⋅60 sec240⋅30.48 cm
Multiply
Multiply
960 sec7315.2 cm
CalcQuot
Calculate quotient
7.62 sec cm
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 | |
|---|---|---|
| Speed=TimeDistance | 5 min80 ft | 16 min240 ft |
| Simplify | 16minft | 15minft |
As can be seen, Hyperion can travel 16 feet in a second whereas Photon can travel 15 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 | |
|---|---|---|
| Speed | 3.2secin. | 7.62seccm |
1 sec3.2 in.×1 in.2.54 cm=1 sec8.128 cm
Hyperion traveled at 8.128 centimeters per second. That is greater than Photon's speed. It is likely that Hyperion will win this race!
Closure
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) | 0.4 miles in 2 hours |
| Photon (Emily's) | 0.4 kilometers per hour |
Speed of Hyperion2 hours0.4 miles=1 hour0.2 miles
Hyperion travels at 0.2 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.
1 mile is about 1.6 kilometers.⇓Conversion Factor1.6 km1 mi
Use this factor to convert kilometers per hour to miles per hour.
1 h0.4 km×1.6 km1 mi
▼
Simplify
MultFrac
Multiply fractions
1 h×1.6 km0.4 km×1 mi
Cross out common units
1 h×1.6 km0.4 km×1 mi
Cancel out common units
1 h×1.60.4×1 mi
MultByOne
a⋅1=a
1.6 h0.4 mi
ReduceFrac
ba=b/1.6a/1.6
1 h0.25 mi
RoundDec
Round to 1 decimal place(s)
1 h0.25 mi
Photon0.25 mph>Hyperion0.2 mph
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!
An electric car factory plans to make its new models 39 inches wide.
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We want to determine how wide the new models are in feet. To do so, we first need to recall the measures of length in the customary system, and the relationships between them.
We can see the relationship between feet and inches in the first row. We will now write a conversion factor for these units. Since we want to convert inches into feet, our ratio will have inches in the denominator. We need to write 1 foot in the numerator and 12 inches in the denominator. Conversion Factor 1ft/12in. To find the width of the electric vehicle in feet, we need to multiply the given width, 39 inches, by the conversion factor. We will start with writing the product. Then, we will cancel out the common units. This will give us the desired unit, feet.
New model cars will be 3.25 feet wide.
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The speed of light is about 300000 kilometers per second. The Sun is about 142 million miles away from Mars. How many minutes does it take for sunlight to reach Mars? Round the answer to the nearest integer.
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Consider that the speed of light is about 300 000 kilometers per second. We can write it as a fraction. 300 000kilometers per second ⇓ 300 000 km/1 sec We know that the Sun is about 142 million miles away from Mars. We want to determine the amount of time it takes for sunlight to reach Mars. To do so, we will start by converting the Sun's distance from miles to kilometers using the fact that 1 mile is about 1.61 kilometers. Conversion Factor 1.61 km/1 mi Let's multiply this factor by 142 million miles to convert from miles to kilometers.
This means that the Sun is about 228 620 000 kilometers from Mars. Now that we have the distance in kilometers, we can divide it by 300 000 kilometers per second to obtain the number of seconds it will take for sunlight to reach Mars.
It takes about 762 seconds for sunlight to reach Mars. Recall that 1 minute is equal to 60 seconds. Then, we can use this information as a conversion factor. 1min/60sec Now, let's multiply this conversion factor by 762 seconds.
Therefore, it takes about 13 minutes for sunlight to reach Mars.
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Magdalena is doubling a recipe for a homemade cleaning solution. The new recipe calls for 12 cups of vinegar. Magdalena converts it to pints in the following way.
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It is a given that Magdalena tried to convert 12 cups to pints. 12 c= pt Magdalena multiplies 12 cups by the conversion factor 2 cups1 pint, and she gets 24 pints.
It turns out that she has made some errors. Let's describe those errors.
- Magdalena incorrectly divided out the common units.
- She incorrectly moved a factor in the denominator next to the fraction 241.
- She used the incorrect conversion factor.
The correct conversion factor, in this case, is the reciprocal of 2 cups1 pint, which is 1 pint2 cups. Conversion Factor 1 pt2 c We can conclude that statements I and III are correct. We can perform the conversion correctly by using the factor we wrote.
We showed that 12 cups is 6 pints.
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At a restaurant, a chef uses 14 pounds of butter every day. How many grams of butter does the chef use every day?
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We know a chef uses 14 pounds of butter every day. We want to find out how many grams of butter that is. We are given two conversion factors.
| Conversion Factors | |
|---|---|
| From Pounds to Ounces | 16ounces/1pound |
| From Ounces to Grams | 28.35grams/1ounce |
Using the given conversion factors, we can write the conversion factor we need. When we multiply the conversion factors, the ounces measurement will cancel each other out.
We can use this factor to convert from pounds to grams. Let's do it!
We found that 14 pounds is 6350.4 grams. This means that the chef uses 6350.4 grams of butter every day.
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