Unlock Measurement Conversion Skills: Metric System and Unit Ratios

Fact	$1$ $foot$ is equal to $12$ $inches .$	$1$ $hour$ is equal to $60$ $minutes .$
Ratio	$\frac{1 ft}{1 2 in .}$	$\frac{1 h}{6 0 min}$

Fact

1

foot

is equal to

12

inches .

1

hour

is equal to

60

minutes .

Ratio

\frac{1 ft}{1 2 in .}

\frac{1 h}{6 0 min}

Given Quantity	Conversion	Result
$2$ $hours$	$2 hours \cdot \frac{6 0 minutes}{1 hour}$	$120$ $minutes$

Given Quantity

Conversion

Result

2

hours

2 hours \cdot \frac{6 0 minutes}{1 hour}

120

minutes

Given Quantity	Conversion	Result
$120$ $hours$	$120 minutes \cdot \frac{1 hour}{6 0 minutes}$	$2$ $hours$

Given Quantity

Conversion

Result

120

hours

120 minutes \cdot \frac{1 hour}{6 0 minutes}

2

hours

Given Quantity	Conversion	Result
$3$ $pounds$	$3 pounds \cdot \frac{1 6 ounces}{1 pound}$	$48$ $ounces$
$160$ $ounces$	$160 ounces \cdot \frac{1 pound}{1 6 ounces}$	$10$ $pounds$
$1$ $mile$	$1 mile \cdot \frac{1 7 6 0 yards}{1 mile}$	$1760$ $yards$

Given Quantity

Conversion

Result

3

pounds

3 pounds \cdot \frac{1 6 ounces}{1 pound}

48

ounces

160

ounces

160 ounces \cdot \frac{1 pound}{1 6 ounces}

10

pounds

1

mile

1 mile \cdot \frac{1 7 6 0 yards}{1 mile}

1760

yards

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

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

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

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

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

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

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

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

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

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

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 \approx 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 Factor Pounds \to Kilograms \frac{1 kg}{2 . 2 lb}

After determining the factor, multiply the weight of Hyperion and the conversion factor.

5.25 lb \cdot \frac{1 kg}{2 . 2 lb}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

\frac{5 . 2 5 lb \cdot 1 kg}{2 . 2 lb}

Cross out common units

\frac{5 . 2 5 lb \cdot 1 kg}{2 . 2 lb}

Cancel out common units

\frac{5 . 2 5 \cdot 1 kg}{2 . 2}

MultByOne

$a \cdot 1 = a$

\frac{5 . 2 5 kg}{2 . 2}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

\frac{5 . 2 5}{2 . 2} kg

CalcQuot

Calculate quotient

2.386363 \dots kg

RoundDec

Round to $2$ decimal place(s)

\approx 2.39

Hyperion has a weight of about

2.39

kilograms. In other words, the value of A is

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 Factor Kilograms \to Pounds \frac{2 . 2 lb}{1 kg}

Use this factor to find the equivalent weight in pounds.

2.5 kg \cdot \frac{2 . 2 lb}{1 kg}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

\frac{2 . 5 kg \cdot 2 . 2 lb}{1 kg}

Cross out common units

\frac{2 . 5 kg \cdot 2 . 2 lb}{1 kg}

Cancel out common units

\frac{2 . 5 \cdot 2 . 2 lb}{1}

DivByOne

$\frac{a}{1} = a$

2.5 \cdot 2.2 lb

Multiply

5.5 lb

Photon has a weight of

5.5

pounds. The value of C is

5.5 .

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

\frac{2 . 5 4 cm}{1 in .}

will convert this length to centimeters.

Conversion Factor Inches \to Centimeters \frac{2 . 5 4 cm}{1 in .}

Multiply the length of Hyperion by this conversion factor to get the equivalent length in centimeters.

15 in . \cdot \frac{2 . 5 4 cm}{1 in .}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

\frac{1 5 in . \cdot 2 . 5 4 cm}{1 in .}

Cross out common units

\frac{1 5 in . \cdot 2 . 5 4 cm}{1 in .}

Cancel out common units

\frac{1 5 \cdot 2 . 5 4 cm}{1}

DivByOne

$\frac{a}{1} = a$

15 \cdot 2.54 cm

Multiply

38.1 cm

Hyperion has a length of

38.1

centimeters. The value of B is

38.1 .

Finding D

Finally, Photon's length will be converted to inches. This calls for multiplication by

\frac{1 in .}{2 . 5 4 cm} .

Conversion Factor Centimeters \to Inches \frac{1 in .}{2 . 5 4 cm}

Multiply

35 cm

\frac{1 in .}{2 . 5 4 cm} .

35 cm \cdot \frac{1 in .}{2 . 5 4 cm}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

\frac{3 5 cm \cdot 1 in .}{2 . 5 4 cm}

Cross out common units

\frac{3 5 cm \cdot 1 in .}{2 . 5 4 cm}

Cancel out common units

\frac{3 5 \cdot 1 in .}{2 . 5 4}

MultByOne

$a \cdot 1 = a$

\frac{3 5 in .}{2 . 5 4}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

\frac{3 5}{2 . 5 4} in .

CalcQuot

Calculate quotient

13.779527 \dots in .

RoundDec

Round to $2$ decimal place(s)

\approx 13.78 in .

Photon's length is about

13.78

inches, which means that D is

13.78 .

A = B = C = D = 2.39 38.1 5.5 13.78

Equivalent Quantities	Conversion Factor
$1 ft = 12 in .$	$\frac{1 2 in .}{1 ft}$
$1 min = 60 sec$	$\frac{1 min}{6 0 sec}$

Equivalent Quantities

Conversion Factor

1 ft = 12 in .

\frac{1 2 in .}{1 ft}

1 min = 60 sec

\frac{1 min}{6 0 sec}

Equivalent Quantities	Conversion Factor
$1 ft = 30.48 cm$	$\frac{3 0 . 4 8 cm}{1 ft}$
$1 min = 60 sec$	$\frac{1 min}{6 0 sec}$

Equivalent Quantities

Conversion Factor

1 ft = 30.48 cm

\frac{3 0 . 4 8 cm}{1 ft}

1 min = 60 sec

\frac{1 min}{6 0 sec}

	Hyperion	Photon
$Speed = \frac{Distance}{Time}$	$\frac{8 0 ft}{5 min}$	$\frac{2 4 0 ft}{1 6 min}$
Simplify	$16 \frac{ft}{min}$	$15 \frac{ft}{min}$

Hyperion

Photon

Speed = \frac{Distance}{Time}

\frac{8 0 ft}{5 min}

\frac{2 4 0 ft}{1 6 min}

Simplify

16 \frac{ft}{min}

15 \frac{ft}{min}

	Hyperion	Photon
$Speed$	$3.2 \frac{in .}{sec}$	$7.62 \frac{cm}{sec}$

Hyperion

Photon

Speed

3.2 \frac{in .}{sec}

7.62 \frac{cm}{sec}

	Given
Hyperion (Ignacio's)	$0.4$ miles in $2$ hours
Photon (Emily's)	$0.4$ kilometers per hour

Given

Hyperion (Ignacio's)

0.4

miles in

2

hours

Photon (Emily's)

0.4

kilometers per hour

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Catch-Up and Review

Different Measures: Comparing Average Speeds

Unit Ratio

Practice Rewriting Ratios

Converting the Map's Distance to the Actual Distance

Hint

Solution

Conversion Factor

Why

Customary System

Determining the Weight of the Robotic Car

Hint

Solution

Metric System

Metric Units of Length

Metric Units of Capacity

Metric Units of Weight

The Length of the Robot Car Designed by Emily’s Team

Hint

Solution

Converting Measures Between Systems

Filling Out the Competition Application

Hint

Solution

Finding Missing Equivalents of Weights

Finding A

Finding C

Finding Missing Equivalents of Lengths

Finding B

Finding D

Speed of the Robotic Cars on an 80-foot Track

Hint

Solution

Different Measures: Comparing Average Speeds

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