Rational Expressions - Dimensional Analysis

 

Objective: Use dimensional analysis to perform single unit, dual unit, square unit, and cubed unit conversions.

One application of rational expressions deals with converting units. When we convert units of measure we can do so by multiplying several fractions together in a process known as dimensional analysis. The trick will be to decide what fractions to multiply. When multiplying, if we multiply by 1, the value of the expression does not change. One written as a fraction can look like many different things as long as the numerator and denominator are identical in value. Notice the numerator and denominator are not identical in appearance, but rather identical in value. Below are several fractions, each equal to one where numerator and denominator are identical in value.

\dfrac{1}{1}=\dfrac{4}{4}=\dfrac{\dfrac{1}{2}}{\dfrac{2}{4}}=\dfrac{100 \, cm }{1  \, m}=\dfrac{1  \, lb }{16  \, oz }=\dfrac{1  \, hr }{60  \, min }=\dfrac{60  \, min }{1  \, hr }

The last few fractions that include units are called conversion factors. We can make a conversion factor out of any two measurements that represent the same distance. For example, \text{1 \, mile } =\text{5280 \, feet} . We could then make a conversion factor \dfrac{1 mi }{5280 ft } because both values are the same, the fraction is still equal to one. Similarly we could make a conversion factor \dfrac{5280 \,  ft }{1 \,  mi }. The trick for conversions will be to use the correct fractions.

The idea behind dimensional analysis is we will multiply by a fraction in such a way that the units we don't want will divide out of the problem. We found out when multiplying rational expressions that if a variable appears in the numerator and denominator we can divide it out of the expression. It is the same with units. Consider the following conversion.

 

Example 1.

17.37 \text { miles to feet } Write 17.37 miles as a fraction, put it over 1
\left(\dfrac{17.37 m i}{1}\right)\left(\dfrac{? ? f t}{? ? m i}\right) To divide out the miles we need miles in the denominator
\left(\dfrac{17.37 \, m i}{1}\right)\left(\dfrac{? ? \, f t}{? ? \, m i}\right) We are converting to feet, so this will go in the numerator
\left(\dfrac{17.37 m i}{1}\right)\left(\dfrac{5280 f t}{1 m i}\right) Fill in the relationship described above, 1 mile =5280 feet
\left(\dfrac{17.37}{1}\right)\left(\dfrac{5280 \, f t}{1}\right) Divide out the miles and multiply across
91,713.6 \, f t Our Solution


In the previous example, we had to use the conversion factor \dfrac{5280  \, ft }{1  \, mi } so the miles would divide out. If we had used \dfrac{1  \, mi }{5280  \, ft } we would not have been able to divide out the miles. This is why when doing dimensional analysis it is very important to use units in the set-up of the problem, so we know how to correctly set up the conversion factor.

 

Example 2.

If 1 pound = 16 ounces, how many pounds 435 ounces?

\left(\dfrac{4350  \, z}{1}\right) Write 435 as a fraction, put it over 1
\left(\dfrac{435 \,  o z}{1}\right)\left(\dfrac{? ?  \, l b s}{? ?  \, o z}\right) To divide out oz,
put it in the denominator and lbs in numerator
\left(\dfrac{435  \, o z}{1}\right)\left(\dfrac{1  \, l b s}{16  \, o z}\right) To divide out oz,
Fill in the given relationship, 1 pound =16 ounces
\left(\dfrac{435}{1}\right)\left(\dfrac{1 \,  l b s}{16}\right)=\dfrac{435 \, l b s}{16} Divide out oz, multiply across. Divide result
27.1875 \, \mathrm{lbs} Our Solution


The same process can be used to convert problems with several units in them. Consider the following example.

 

Example 3.

A student averaged 45 miles per hour on a trip. What was the student's speed in feet per second?

\left(\dfrac{45 \, mi}\right) "per" is the fraction bar, put hr in denominator
\left(\dfrac{5280 \, ft}{1 \, mi}\right)\left(\dfrac{1 \, hr}{3600 \, \mathrm{sec}}\right) To clear mi they must go in denominator and become \mathrm{ft}
\left(\dfrac{45 \, mi}\right)\left(\dfrac{5280 \, ft}{1 \, mi}\right)\left(\dfrac{1 \, h r}{3600 \, \mathrm{sec}}\right) To clear hr they must go in numerator and become sec
\left(\dfrac{45}{1}\right)\left(\dfrac{5280  \, f t}{1}\right)\left(\dfrac{1}{3600 \, \mathrm{sec}}\right) Divide out mi and hr. Multiply across
\dfrac{237600 \, ft}{3600 \, sec} Divide numbers
66 \, \mathrm{ft} per sec Our Solution


If the units are two-dimensional (such as square inches - in { }^{2} ) or three-dimensional (such as cubic feet - ft ^{3} ) we will need to put the same exponent on the conversion factor. So if we are converting square inches \left(\right. in \left.^{2}\right) to square ft \left( ft ^{2}\right), the conversion factor would be squared, \left(\dfrac{1  \, ft }{12  \, in }\right)^{2}. Similarly if the units are cubed, we will cube the convesion factor.

 

Example 4.

Convert 8 cubic feet to y d^{3} Write 8 \, \mathrm{ft}^{3} as fraction, put it over 1
\left(\dfrac{8  \, f t^{3}}{1}\right) To clear \mathrm{ft}, put them in denominator, yard in numerator
\left(\dfrac{8  \, f t^{3}}{1}\right)\left(\dfrac{? ?  \,  y d}{? ?  \, f t}\right)^{3} Because the units are cubed,
we cube the conversion factor
\left(\dfrac{8  \, f t^{3}}{1}\right)\left(\dfrac{1  \,  y d}{3 \,  f t}\right)^{3} Evaluate exponent, cubing all numbers and units
\left(\dfrac{8  \, f t^{3}}{1}\right)\left(\dfrac{1  \, y d^{3}}{27  \, f t^{3}}\right) Divide out f t^{3}
\left(\dfrac{8}{1}\right)\left(\dfrac{1  \, y^{3}}{27}\right)=\dfrac{8 y^{3}}{27} Multiply across and divide
0.296296  \, y d^{3} Our Solution

 

When calculating area or volume, be sure to use the units and multiply them as well.

 

Example 5.

A room is 10 ft by 12 ft. How many square yards are in the room?

A=l w=(10 \, f t)(12 \, f t)=120 f t^{2} Multiply length by width, also multiply units
\left(\dfrac{120 \, f t^{2}}{1}\right) Write area as a fraction, put it over 1
\left(\dfrac{120 \, f t^{2}}{1}\right)\left(\dfrac{? ? \, y d}{? ? \, f t}\right)^{2} Put f t in denominator to clear,
square conversion factor
\left(\dfrac{120 \, f t^{2}}{1}\right)\left(\dfrac{1 \, y d}{3 \, f t}\right)^{2} Evaluate exponent, squaring all numbers and units
\left(\dfrac{120 \, f t^{2}}{1}\right)\left(\dfrac{1 \, y d^{2}}{9 \, f t^{2}}\right) Divide out f t^{2}
\left(\dfrac{120}{1}\right)\left(\dfrac{1 \, y d^{2}}{9}\right)=\dfrac{120  \, y d^{2}}{9} Multiply across and divide
13.33 \, y d^{2} Our solution


To focus on the process of conversions, a conversion sheet has been included at the end of this lesson which includes several conversion factors for length, volume, mass, and time in both English and Metric units.

The process of dimensional analysis can be used to convert other types of units as well. If we can identify relationships that represent the same value we can make them into a conversion factor.

 

Example 6.

A child is prescribed a dosage of 12  \, mg of a certain drug 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 \text{3 prescriptions (original +2 refills)}?

\text { Convert 3 Rx}  \text { to doses } Identify what the problem is asking
1 R x=60 \, \mathrm{tab}, 1 \, \mathrm{tab}=4 \, \mathrm{mg}, 1 \, \text { dose }=12 \, \mathrm{mg} Identify given conversion factors
\left(\dfrac{3 R x}{1}\right) Write 3 R x as fraction, put over 1
\left(\dfrac{3 R x}{1}\right)\left(\dfrac{60 t a b}{1 R x}\right) Convert \mathrm{Rx} to tab, put \mathrm{Rx} in denominator
\left(\dfrac{3 R x}{1}\right)\left(\dfrac{60 \mathrm{tab}}{1 R x}\right)\left(\dfrac{4 m g}{1 t a b}\right) Convert tab to mg, put tab in denominator
\left(\dfrac{3 \mathrm{Rx}}{1}\right)\left(\dfrac{60 \mathrm{tab}}{1 \mathrm{Rx}}\right)\left(\dfrac{4 \mathrm{mg}}{1 \mathrm{tab}}\right)\left(\dfrac{1 \text { dose }}{12 \mathrm{mg}}\right) Convert mg to dose, put mg in denominator
\left(\dfrac{3}{1}\right)\left(\dfrac{60}{1}\right)\left(\dfrac{4}{1}\right)\left(\dfrac{1 \text { dose }}{12}\right) Divide out R x, tab, and mg, multiply across
\dfrac{720 \text { dose }}{12} Divide
60 \, \text { doses} Our Solution


World View Note: Only three countries in the world still use the English system commercially: Liberia (Western Africa), Myanmar (between India and Vietnam), and the USA.

 

Source: Tyler Wallace, http://www.wallace.ccfaculty.org/book/7.8%20Dimentional%20Analysis.pdf
Creative Commons License This work is licensed under a Creative Commons Attribution 3.0 License.
Last modified: Monday, 21 March 2022, 11:22 AM