1.6 Unit Conversion Word Problems
One application of rational expressions deals with converting units. Units of measure can be converted by multiplying several fractions together in a process known as dimensional analysis.
The trick is to decide what fractions to multiply. If an expression is multiplied by 1, its value does not change. The number 1 can be written as a fraction in many different ways, so long as the numerator and denominator are identical in value. Note that the numerator and denominator need not be identical in appearance, but rather only identical in value. Below are several fractions, each equal to 1, where the numerator and the denominator are identical in value. This is why, when doing dimensional analysis, it is very important to use units in the setup of the problem, so as to ensure that the conversion factor is set up correctly.
Example 1.6.1
If 1 pound = 16 ounces, how many pounds are in 435 ounces?
[latex]\begin{array}{rrll} 435\text{ oz}&=&435\text{ \cancel{oz}}\times \dfrac{1\text{ lb}}{16\text{ \cancel{oz}}} \hspace{0.2in}& \text{This operation cancels the oz and leaves the lbs} \\ \\ &=&\dfrac{435\text{ lb}}{16} \hspace{0.2in}& \text{Which reduces to } \\ \\ &=&27\dfrac{3}{16}\text{ lb} \hspace{0.2in}& \text{Solution} \end{array}[/latex]
The same process can be used to convert problems with several units in them. Consider the following example.
Example 1.6.2
A student averaged 45 miles per hour on a trip. What was the student’s speed in feet per second?
[latex]\begin{array}{rrll} 45 \text{ mi/h}&=&\dfrac{\text{45 \cancel{mi}}}{\text{\cancel{hr}}}\times \dfrac{5280 \text{ ft}}{1\text{ \cancel{mi}}}\times \dfrac{1\text{ \cancel{hr}}}{3600\text{ s}}\hspace{0.2in}&\text{This will cancel the miles and hours} \\ \\ &=&45\times \dfrac{5280}{1}\times \dfrac{1}{3600} \text{ ft/s}\hspace{0.2in}&\text{This reduces to} \\ \\ &=&66\text{ ft/s}\hspace{0.2in}&\text{Solution} \end{array}[/latex]
Example 1.6.3
Convert 8 ft3 to yd3.
[latex]\begin{array}{rrll} 8\text{ ft}^3&=&8\text{ ft}^3 \times \dfrac{(1\text{ yd})^3}{(3\text{ ft})^3}&\text{Cube the parentheses} \\ \\ &=&8\text{ }\cancel{\text{ft}^3}\times \dfrac{1\text{ yd}^3}{27\text{ }\cancel{\text{ft}^3}}&\text{This will cancel the ft}^3\text{ and replace them with yd}^3 \\ \\ &=&8\times \dfrac{1\text{ yd}^3}{27}&\text{Which reduces to} \\ \\ &=&\dfrac{8}{27}\text{ yd}^3\text{ or }0.296\text{ yd}^3&\text{Solution} \end{array}[/latex]
Example 1.6.4
A room is 10 ft by 12 ft. How many square yards are in the room? The area of the room is 120 ft2 (area = length × width).
Converting the area yields:
[latex]\begin{array}{rrll} 120\text{ ft}^2&=&120\text{ }\cancel{\text{ft}^2}\times \dfrac{(1\text{ yd})^2}{(3\text{ }\cancel{\text{ft}})^2}&\text{Cancel ft}^2\text{ and replace with yd}^2 \\ \\ &=&\dfrac{120\text{ yd}^2}{9}&\text{This reduces to} \\ \\ &=&13\dfrac{1}{3}\text{ yd}^2&\text{Solution} \\ \\ \end{array}[/latex]
The process of dimensional analysis can be used to convert other types of units as well. Once relationships that represent the same value have been identified, a conversion factor can be determined.
Example 1.6.5
A child is prescribed a dosage of 12 mg of a certain drug per day and is allowed to refill his prescription twice. If there are 60 tablets in a prescription, and each tablet has 4 mg, how many doses are in the 3 prescriptions (original + 2 refills)?
[latex]\begin{array}{rrll} 3\text{ prescriptions}&=&3\text{ \cancel{pres.}}\times \dfrac{60\text{ \cancel{tablets}}}{1\text{ \cancel{pres.}}}\times \dfrac{4\text{ \cancel{mg}}}{1\text{ \cancel{tablet}}}\times \dfrac{1\text{ dosage}}{12\text{ \cancel{mg}}}&\text{This cancels all unwanted units} \\ \\ &=&\dfrac{3\times 60\times 4\times 1}{1\times 1\times 12}\text{ or }\dfrac{720}{12}\text{ dosages}&\text{Which reduces to} \\ \\ &=&60\text{ daily dosages}&\text{Solution} \\ \\ \end{array}[/latex]
Metric and Imperial (U.S.) Conversions
Distance
\[\begin{array}{rrlrrl}
12\text{ in}&=&1\text{ ft}\hspace{1in}&10\text{ mm}&=&1\text{ cm} \\
3\text{ ft}&=&1\text{ yd}&100\text{ cm}&=&1\text{ m} \\
1760\text{ yds}&=&1\text{ mi}&1000\text{ m}&=&1\text{ km} \\
5280\text{ ft}&=&1\text{ mi}&&&
\end{array}\]
Imperial to metric conversions:
\[\begin{array}{rrl}
1\text{ inch}&=&2.54\text{ cm} \\
1\text{ ft}&=&0.3048\text{ m} \\
1\text{ mile}&=&1.61\text{ km}
\end{array}\]
Area
\[\begin{array}{rrlrrl}
144\text{ in}^2&=&1\text{ ft}^2\hspace{1in}&10,000\text{ cm}^2&=&1\text{ m}^2 \\
43,560\text{ ft}^2&=&1\text{ acre}&10,000\text{ m}^2&=&1\text{ hectare} \\
640\text{ acres}&=&1\text{ mi}^2&100\text{ hectares}&=&1\text{ km}^2
\end{array}\]
Imperial to metric conversions:
\[\begin{array}{rrl}
1\text{ in}^2&=&6.45\text{ cm}^2 \\
1\text{ ft}^2&=&0.092903\text{ m}^2 \\
1\text{ mi}^2&=&2.59\text{ km}^2
\end{array}\]
Volume
\[\begin{array}{rrlrrl}
57.75\text{ in}^3&=&1\text{ qt}\hspace{1in}&1\text{ cm}^3&=&1\text{ ml} \\
4\text{ qt}&=&1\text{ gal}&1000\text{ ml}&=&1\text{ litre} \\
42\text{ gal (petroleum)}&=&1\text{ barrel}&1000\text{ litres}&=&1\text{ m}^3
\end{array}\]
Imperial to metric conversions:
\[\begin{array}{rrl}
16.39\text{ cm}^3&=&1\text{ in}^3 \\
1\text{ ft}^3&=&0.0283168\text{ m}^3 \\
3.79\text{ litres}&=&1\text{ gal}
\end{array}\]
Mass
\[\begin{array}{rrlrrl}
437.5\text{ grains}&=&1\text{ oz}\hspace{1in}&1000\text{ mg}&=&1\text{ g} \\
16\text{ oz}&=&1\text{ lb}&1000\text{ g}&=&1\text{ kg} \\
2000\text{ lb}&=&1\text{ short ton}&1000\text{ kg}&=&1\text{ metric ton}
\end{array}\]
Imperial to metric conversions:
\[\begin{array}{rrl}
453\text{ g}&=&1\text{ lb} \\
2.2\text{ lb}&=&1\text{ kg}
\end{array}\]
Temperature
Fahrenheit to Celsius conversions:
\[\begin{array}{rrl}
^{\circ}\text{C} &= &\dfrac{5}{9} (^{\circ}\text{F} – 32) \\ \\
^{\circ}\text{F}& =& \dfrac{9}{5}(^{\circ}\text{C} + 32)
\end{array}\]
Questions
For questions 1 to 18, use dimensional analysis to perform the indicated conversions.
- 7 miles to yards
- 234 oz to tons
- 11.2 mg to grams
- 1.35 km to centimetres
- 9,800,000 mm to miles
- 4.5 ft2 to square yards
- 435,000 m2 to square kilometres
- 8 km2 to square feet
- 0.0065 km3 to cubic metres
- 14.62 in3 to square centimetres
- 5500 cm3 to cubic yards
- 3.5 mph (miles per hour) to feet per second
- 185 yd per min. to miles per hour
- 153 ft/s (feet per second) to miles per hour
- 248 mph to metres per second
- 186,000 mph to kilometres per year
- 7.50 tons/yd2 to pounds per square inch
- 16 ft/s2 to kilometres per hour squared
For questions 19 to 27, solve each conversion word problem.
- On a recent trip, Jan travelled 260 miles using 8 gallons of gas. What was the car’s miles per gallon for this trip? Kilometres per litre?
- A certain laser printer can print 12 pages per minute. Determine this printer’s output in pages per day.
- An average human heart beats 60 times per minute. If the average person lives to the age of 86, how many times does the average heart beat in a lifetime?
- Blood sugar levels are measured in milligrams of glucose per decilitre of blood volume. If a person’s blood sugar level measured 128 mg/dL, what is this in grams per litre?
- You are buying carpet to cover a room that measures 38 ft by 40 ft. The carpet cost $18 per square yard. How much will the carpet cost?
- A cargo container is 50 ft long, 10 ft wide, and 8 ft tall. Find its volume in cubic yards and cubic metres.
- A local zoning ordinance says that a house’s “footprint” (area of its ground floor) cannot occupy more than ¼ of the lot it is built on. Suppose you own a [latex]\frac{1}{3}[/latex]-acre lot (1 acre = 43,560 ft2). What is the maximum allowed footprint for your house in square feet? In square metres?
- A car travels 23 km in 15 minutes. How fast is it going in kilometres per hour? In metres per second?
- The largest single rough diamond ever found, the Cullinan Diamond, weighed 3106 carats. One carat is equivalent to the mass of 0.20 grams. What is the mass of this diamond in milligrams? Weight in pounds?
Long Descriptions
Celsius to Fahrenheit conversion scale long description: Scale showing conversions between Celsius and Fahrenheit. The following table summarizes the data:
| Celsius | Fahrenheit |
|---|---|
| −40°C | −40°F |
| −30°C | −22°F |
| −20°C | −4°F |
| −10°C | 14°F |
| 0°C | 32°F |
| 10°C | 50°F |
| 20°C | 68°F |
| 30°C | 86°F |
| 40°C | 104°F |
| 50°C | 122°F |
| 60°C | 140°F |
| 70°C | 158°F |
| 80°C | 176°F |
| 90°C | 194°F |
| 100°C | 212°F |
