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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?
Fact | 1 foot is equal to 12 inches. | 1 hour is equal to 60 minutes. |
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Ratio | 12 in.1 ft | 60 min1 h |
Write the given ratio in the indicated form.
Write a ratio using the fact that 1 inch on the map represents 0.075 miles in real distance.
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1a=a
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.
The 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.
Customary Units | ||
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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 |
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.
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Cross out common units
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Calculate quotient
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 |
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 |
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 |
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.
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Cross out common units
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1a=a
Multiply
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.
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 | ||||
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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 |
Remember, 1 kilogram is about 2.2 pounds and 1 inch is 2.54 centimeters.
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.
a⋅cb=ca⋅b
Cross out common units
Cancel out common units
a⋅1=a
ca⋅b=ca⋅b
Calculate quotient
Round to 2 decimal place(s)
a⋅cb=ca⋅b
Cross out common units
Cancel out common units
1a=a
Multiply
a⋅cb=ca⋅b
Cross out common units
Cancel out common units
1a=a
Multiply
a⋅cb=ca⋅b
Cross out common units
Cancel out common units
a⋅1=a
ca⋅b=ca⋅b
Calculate quotient