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Study Guides > ALGEBRA / TRIG I

Making Unit Conversions in the U.S. System of Measurement

Learning Outcomes

  • Make unit conversions in the U.S. system
  • Use mixed units of measurement in the U.S. system
There are two systems of measurement commonly used around the world. Most countries use the metric system. The United States uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first. The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, or hours. The equivalencies among the basic units of the U.S. system of measurement are listed in the table below. The table also shows, in parentheses, the common abbreviations for each measurement.
U.S. System Units
Length Volume
11 foot (ft) = 1212 inches (in) 11 yard (yd) = 33 feet (ft) 11 mile (mi) = 52805280 feet (ft) 33 teaspoons (t) = 11 tablespoon (T) 1616 Tablespoons (T) = 11 cup (C) 11 cup (C) = 88 fluid ounces (fl oz) 11 pint (pt) = 22 cups (C) 11 quart (qt) = 22 pints (pt) 11 gallon (gal) = 44 quarts (qt)
Weight Time
11 pound (lb) = 1616 ounces (oz) 11 ton = 20002000 pounds (lb) 11 minute (min) = 6060 seconds (s) 11 hour (h) = 6060 minutes (min) 11 day = 2424 hours (h) 11 week (wk) = 77 days 11 year (yr) = 365365 days
In many real-life applications, we need to convert between units of measurement. We will use the identity property of multiplication to do these conversions. We’ll restate the Identity Property of Multiplication here for easy reference. For any real number a,a1=a[/latex]  [latex]1a=a\text{For any real number }a,a\cdot 1=a[/latex]    [latex]1\cdot a=a To use the identity property of multiplication, we write 11 in a form that will help us convert the units. For example, suppose we want to convert inches to feet. We know that 11 foot is equal to 1212 inches, so we can write 11 as the fraction 1 ft12 in\Large\frac{\text{1 ft}}{\text{12 in}}. When we multiply by this fraction, we do not change the value but just change the units. But 12 in1 ft\Large\frac{\text{12 in}}{\text{1 ft}} also equals 11. How do we decide whether to multiply by 1 ft12 in\Large\frac{\text{1 ft}}{\text{12 in}} or 12 in1 ft?\Large\frac{\text{12 in}}{\text{1 ft}}? We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert 6060 inches to feet. If we choose the fraction that has inches in the denominator, we can eliminate the inches. 60)in1 ft12)in=5 ft60\overline{)\text{in}}\cdot\Large\frac{\text{1 ft}}{12\overline{)\text{in}}}\normalsize=\text{5 ft} On the other hand, if we wanted to convert 55 feet to inches, we would choose the fraction that has feet in the denominator. 5 ft12 in1)ft=60 in\text{5 ft}\cdot\Large\frac{\text{12 in}}{1\overline{)\text{ft}}}\normalsize=\text{60 in} We treat the unit words like factors and ‘divide out’ common units like we do common factors.

Make unit conversions.

  1. Multiply the measurement to be converted by 11; write 11 as a fraction relating the units given and the units needed.
  2. Multiply.
  3. Simplify the fraction, performing the indicated operations and removing the common units.
 

example

Mary Anne is 6666 inches tall. What is her height in feet? Solution
Convert 6666 inches into feet.
Multiply the measurement to be converted by 11. 6666 inches 1\cdot 1
Write 11 as a fraction relating the units given and the units needed. 66 inches1 foot12 inches\text{66 inches}\cdot \Large\frac{\text{1 foot}}{\text{12 inches}}
Multiply. 66 inches1 foot12 inches\Large\frac{\text{66 inches}\cdot \text{1 foot}}{\text{12 inches}}
Simplify the fraction. 66)inches1 foot12)inches\Large\frac{66\overline{)\text{inches}}\cdot \text{1 foot}}{12\overline{)\text{inches}}}
66 feet12\Large\frac{\text{66 feet}}{12}
5.5 feet\text{5.5 feet}
Notice that when we simplified the fraction, we first divided out the inches. Mary Anne is 5.55.5 feet tall.
 

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[ohm_question]146826[/ohm_question] [ohm_question]146830[/ohm_question]
In the following video we show more examples of converting between units of length in the U.S. system of measurement. https://youtu.be/atN7JU1BaIY When we use the Identity Property of Multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.

example

Ndula, an elephant at the San Diego Safari Park, weighs almost 3.23.2 tons. Convert her weight to pounds. (credit: Guldo Da Rozze, Flickr) A photograph of an adult elephant.

Answer: Solution We will convert 3.23.2 tons into pounds, using the equivalencies in the table from earlier. We will use the Identity Property of Multiplication, writing 11 as the fraction 2000 pounds1 ton\Large\frac{\text{2000 pounds}}{\text{1 ton}}.

3.2 tons\text{3.2 tons}
Multiply the measurement to be converted by 11. 3.2 tons1\text{3.2 tons}\cdot 1
Write 11 as a fraction relating tons and pounds. 3.2 tons2000 lbs1 ton\text{3.2 tons}\cdot\Large\frac{\text{2000 lbs}}{\text{1 ton}}
Simplify. 3.2)tons2000 lbs1)ton\Large\frac{3.2\overline{)\text{tons}}\cdot \text{2000 lbs}}{1\overline{)\text{ton}}}
Multiply. 6400 lbs\text{6400 lbs}
Ndula weighs almost 6,4006,400 pounds.

 

try it

[ohm_question]146831[/ohm_question] [ohm_question]146833[/ohm_question]
In  the next video we show more examples of how to convert between units of weight using the U. S. system of measurement. https://youtu.be/_wxrSBEhLpI Sometimes to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

example

Juliet is going with her family to their summer home. She will be away for 99 weeks. Convert the time to minutes.

Answer: Solution To convert weeks into minutes, we will convert weeks to days, days to hours, and then hours to minutes. To do this, we will multiply by conversion factors of 11.

9 weeks\text{9 weeks}
Write 1 as 7days1week,24hours1day,60minutes1hour\Large\frac{7\text{days}}{1\text{week}}\normalsize ,\Large\frac{24\text{hours}}{1\text{day}}\normalsize ,\Large\frac{60\text{minutes}}{1\text{hour}} . 9wk17days1wk24hr1day60min1hr\Large\frac{9 wk}{1}\normalsize\cdot\Large\frac{7 days}{1 wk}\normalsize\cdot\Large\frac{24 hr}{1 day}\normalsize\cdot\Large\frac{60 min}{1 hr}
Cancel common units. 9wk17days1wk24hr1day60min1hr\Large\frac{9 wk}{1}\normalsize\cdot\Large\frac{7\color{blue}{days}}{1 wk}\normalsize\cdot\Large\frac{24\color{red}{hr}}{1\color{blue}{day}}\normalsize\cdot\Large\frac{60 min}{1\color{red}{hr}}
Multiply. 972460min1111=90,720min\Large\frac{9\cdot 7\cdot 24\cdot 60\text{min}}{1\cdot 1\cdot 1\cdot 1}\normalsize =90,720\text{min}
Juliet will be away for 90,72090,720 minutes.

 

try it

The distance between Earth and the moon is about 250,000250,000 miles. Convert this length to yards.

Answer: Solution To convert miles into yards, we will convert miles to feet and then feet to yards. To do this, we will multiply by conversion factors of 11.

250,000 miles\text{250,000 miles}
Write 1 as 5,280feet1mile,1yard3feet\Large\frac{5,280\text{feet}}{1\text{mile}}\normalsize ,\Large\frac{1\text{yard}}{3\text{feet}}\normalsize. 250,000mi15,280feet1mi1yard3feet\Large\frac{250,000 mi}{1}\normalsize\cdot\Large\frac{5,280 feet}{1 mi}\normalsize\cdot\Large\frac{1 yard}{3 feet}
Cancel common units. 250,000mi15,280feet1mi1yard3feet\Large\frac{250,000\color{red}{mi}}{1}\normalsize\cdot\Large\frac{5,280\color{blue}{feet}}{1\color{red}{ mi}}\normalsize\cdot\Large\frac{1 yard}{3\color{blue}{feet}}
Multiply. 250,0005,2801yard113=44,000,000yard\Large\frac{250,000\cdot 5,280\cdot 1\text{yard}}{1\cdot 1\cdot 3}\normalsize =44,000,000\text{yard}
The distance between the earth and the moon is 44,000,00044,000,000 yards.

A team of astronauts spends 1515 weeks in space. Convert the time to minutes.

Answer: Solution To convert weeks into minutes, we will convert weeks to days, days to hours, and then hours to minutes. To do this, we will multiply by conversion factors of 11.

15 weeks\text{15 weeks}
Write 1 as 7days1week,24hours1day,60minutes1hour\Large\frac{7\text{days}}{1\text{week}}\normalsize ,\Large\frac{24\text{hours}}{1\text{day}}\normalsize ,\Large\frac{60\text{minutes}}{1\text{hour}} . 15wk17days1wk24hr1day60min1hr\Large\frac{15 wk}{1}\normalsize\cdot\Large\frac{7 days}{1 wk}\normalsize\cdot\Large\frac{24 hr}{1 day}\normalsize\cdot\Large\frac{60 min}{1 hr}
Cancel common units. 15wk17days1wk24hr1day60min1hr\Large\frac{15 \color{green}{wk}}{1}\normalsize\cdot\Large\frac{7\color{blue}{days}}{1 \color{green}{wk}}\normalsize\cdot\Large\frac{24\color{red}{hr}}{1\color{blue}{day}}\normalsize\cdot\Large\frac{60 min}{1\color{red}{hr}}
Multiply. 1572460min1111=151,200min\Large\frac{15\cdot 7\cdot 24\cdot 60\text{min}}{1\cdot 1\cdot 1\cdot 1}\normalsize =151,200\text{min}
The astronauts will be in space for 151,200151,200 minutes.

 

example

How many fluid ounces are in 11 gallon of milk? (credit: www.bluewaikiki.com, Flickr) A photograph of a milk display in a grocery store.

Answer: Solution Use conversion factors to get the right units: convert gallons to quarts, quarts to pints, pints to cups, and cups to fluid ounces.

11 gallon
Multiply the measurement to be converted by 1. 1 gal14 qt1 gal2 pt1 qt2 C1 pt8 fl oz1 C\Large\frac{\text{1 gal}}{1}\normalsize\cdot\Large\frac{\text{4 qt}}{\text{1 gal}}\normalsize\cdot\Large\frac{\text{2 pt}}{\text{1 qt}}\normalsize\cdot\Large\frac{\text{2 C}}{\text{1 pt}}\normalsize\cdot\Large\frac{\text{8 fl oz}}{\text{1 C}}
Simplify. 1)gal14)qt1)gal2)pt1)qt2)C1)pt8 fl oz1)C\Large\frac{1\overline{)\text{gal}}}{1}\normalsize\cdot\Large\frac{4\overline{)\text{qt}}}{1\overline{)\text{gal}}}\normalsize\cdot\Large\frac{2\overline{)\text{pt}}}{1\overline{)\text{qt}}}\normalsize\cdot\Large\frac{2\overline{)\text{C}}}{1\overline{)\text{pt}}}\normalsize\cdot\Large\frac{\text{8 fl oz}}{1\overline{)\text{C}}}
Multiply. 14228 fl oz11111\Large\frac{1\cdot 4\cdot 2\cdot 2\cdot \text{8 fl oz}}{1\cdot 1\cdot 1\cdot 1\cdot 1}
Simplify. 128128 fluid ounces
There are 128128 fluid ounces in a gallon.

 

try it

[ohm_question]146837[/ohm_question] How many teaspoons are in 11 cup? [ohm_question]146838[/ohm_question]

Use Mixed Units of Measurement in the U.S. System

Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units.

example

Charlie bought three steaks for a barbecue. Their weights were 1414 ounces, 11 pound 22 ounces, and 11 pound 66 ounces. How many total pounds of steak did he buy? (credit: Helen Penjam, Flickr) A photograph of meat being cooked on a charcoal grill.

Answer: Solution We will add the weights of the steaks to find the total weight of the steaks.

Add the ounces. Then add the pounds. .
Convert 2222 ounces to pounds and ounces.
Add the pounds. 22 pounds +1+1 pound, 66 ounces 33 pounds, 66 ounces
Charlie bought 33 pounds 66 ounces of steak.

 

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[ohm_question]146961[/ohm_question] [ohm_question]146962[/ohm_question]
 

example

Anthony bought four planks of wood that were each 66 feet 44 inches long. If the four planks are placed end-to-end, what is the total length of the wood? The image shows 4 planks of wood placed end-to-end horizontally. Each plank is labeled 6 feet 4 inches. A line starts at the left of the first plank and runs horizontally to the right of the fourth plank. The line is labeled with the letter l to represent length.

Answer: Solution We will multiply the length of one plank by 44 to find the total length.

Multiply the inches and then the feet. .
Convert 1616 inches to feet. 2424 feet +1+1 foot 44 inches
Add the feet. 2525 feet 44 inches
Anthony bought 2525 feet 44 inches of wood.

 

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[ohm_question]146965[/ohm_question] [ohm_question]146970[/ohm_question]

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