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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.

Challenge

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?

{"type":"choice","form":{"alts":["Hyperion","Photon"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}

Discussion

A unit ratio is a ratio with a denominator of $1$ unit. Every ratio can be rewritten as a unit ratio.

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.

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 $1ft12in. ,$ for example, can be reduced by changing the numerator to $1$ foot. The ratio then becomes $1ft1ft .$ 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.

Fact | $1$ $foot$ is equal to $12$ $inches.$ | $1$ $hour$ is equal to $60$ $minutes.$ |
---|---|---|

Ratio | $12in.1ft $ | $60min1h $ |

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.

Pop Quiz

Write the given ratio in the indicated form.

Example

Ignacio relies on a GPS screen to help navigate his robotic car Hyperion through the school's diverse terrain. ### Hint

### Solution

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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Write a ratio using the fact that $1$ inch on the map represents $0.075$ miles in real distance.

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 Map1in. = In Actual Distance0.075mi $

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.
$1in.=0.075mi⇓1in.0.075mi $

Now multiply $1.4$ inches by this ratio to get the distance in miles.
$1.4in.×1in.0.075mi $

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$1in.1.4in.×0.075mi $

Cross out common units

$1in.1.4in.×0.075mi $

Cancel out common units

$11.4×0.075mi $

Multiply

Multiply

$10.105mi $

DivByOne

$1a =a$

$0.105mi$

Discussion

A conversion factor is a fraction where the numerator and denominator represent the *same* quantity with different units.

### Why

The Reason That the Quantities Are Equivalent

$Example Conversion Factor1hour60minutes $

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 |
---|---|---|

$2$ $hours$ | $2hours ⋅1hour 60minutes $ | $120$ $minutes$ |

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 $60minutes1hour .$ If the task was to convert $120$ minutes to hours, $120$ minutes would be multiplied by this conversion factor.

Given Quantity | Conversion | Result |
---|---|---|

$120$ $hours$ | $120minutes ⋅60minutes 1hour $ | $2$ $hours$ |

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 |
---|---|---|

$3$ $pounds$ | $3pounds ⋅1pound 16ounces $ | $48$ $ounces$ |

$160$ $ounces$ | $160ounces ⋅16ounces 1pound $ | $10$ $pounds$ |

$1$ $mile$ | $1mile ⋅1mile 1760yards $ | $1760$ $yards$ |

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.

Discussion

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

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?

{"type":"choice","form":{"alts":["Yes","No"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

a Write a conversion factor using the fact that $1$ pound is $16$ ounces.

b Compare the answer from Part A with $6$ pounds.

a A conversion factor is needed to convert the given quantity from ounces to pounds. Recall how pounds and ounces are related.

$1pound is16ounces.or1lb=16oz $

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→Pounds16oz1lb $

Now convert the given quantity in ounces to pounds by multiplying $84$ ounces by the conversion factor.
$84oz⋅16oz1lb $

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$16oz84oz⋅1lb $

Cross out common units

$16oz 84oz ⋅1lb $

Cancel out common units

$1684⋅1lb $

MultByOne

$a⋅1=a$

$1684lb $

MovePartNumRight

$ca⋅b =ca ⋅b$

$1684 lb$

CalcQuot

Calculate quotient

$5.25lb$

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.

$5.25lb<6lb $

This means that Ignacio's team is eligible to participate in the event.
Discussion

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.
### Metric Units of Length

### Metric Units of Capacity

### Metric Units of 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. 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 |

Example

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?

a Write a conversion factor using the fact that $1$ meter is $100$ centimeters.

b Compare the answer from Part A with $40$ centimeters.

a The given length is in meters. To convert from meters to centimeters, use the fact that $1$ meter is $100$ centimeters.

$1meter is100centimeters.or1m=100cm $

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→Centimeters1m100cm $

Now convert the given quantity from meters to centimeters by multiplying it by the conversion factor.
$0.35m⋅1m100cm $

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$1m0.35m⋅100cm $

Cross out common units

$1m0.35m⋅100cm $

Cancel out common units

$10.35⋅100cm $

DivByOne

$1a =a$

$0.35⋅100cm$

Multiply

Multiply

$35cm$

b The car has a length of $35$ centimeters. This is less than $40$ centimeters.

$35cm<40cm $

It follows that Photon meets this criteria.
Discussion

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

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 |

{"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"ordermatters":true,"numinput":4,"listEditable":false,"hideNoSolution":true},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.915143125em;vertical-align:-0.19444em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">A<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">B<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">C<\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">D<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["2.39","38.1","5.5","13.78"]}}

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.

$1kilogram is about2.2pounds.or1kg≈2.2lb $

The required conversion factor will be obtained using this information. $Conversion FactorPounds→Kilograms2.2lb1kg $

After determining the factor, multiply the weight of Hyperion and the conversion factor.
$5.25lb⋅2.2lb1kg $

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$2.2lb5.25lb⋅1kg $

Cross out common units

$2.2lb 5.25lb ⋅1kg $

Cancel out common units

$2.25.25⋅1kg $

MultByOne

$a⋅1=a$

$2.25.25kg $

MovePartNumRight

$ca⋅b =ca ⋅b$

$2.25.25 kg$

CalcQuot

Calculate quotient

$2.386363…kg$

RoundDec

Round to $2$ decimal place(s)

$≈2.39$

$Conversion FactorKilograms→Pounds1kg2.2lb $

Use this factor to find the equivalent weight in pounds.
$2.5kg⋅1kg2.2lb $

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$1kg2.5kg⋅2.2lb $

Cross out common units

$1kg 2.5kg ⋅2.2lb $

Cancel out common units

$12.5⋅2.2lb $

DivByOne

$1a =a$

$2.5⋅2.2lb$

Multiply

Multiply

$5.5lb$

$1inch is2.54centimeters.or1in.=2.54cm $

$Conversion FactorInches→Centimeters1in.2.54cm $

Multiply the length of Hyperion by this conversion factor to get the equivalent length in centimeters.
$15in.⋅1in.2.54cm $

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$1in.15in.⋅2.54cm $

Cross out common units

$1in.15in.⋅2.54cm $

Cancel out common units

$115⋅2.54cm $

DivByOne

$1a =a$

$15⋅2.54cm$

Multiply

Multiply

$38.1cm$

$Conversion FactorCentimeters→Inches2.54cm1in. $

Multiply $35cm$ by $2.54cm1in. .$
$35cm⋅2.54cm1in. $

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$2.54cm35cm⋅1in. $

Cross out common units

$2.54cm 35cm ⋅1in. $

Cancel out common units

$2.5435⋅1in. $

MultByOne

$a⋅1=a$

$2.5435in. $

MovePartNumRight

$ca⋅b =ca ⋅b$

$2.5435 in.$