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Study Guides > MATH 0123

Applications with Rational Equations

Learning Outcomes

  • Solve a rational formula for a specified variable
  • Solve an application using a formula that must be solved for a specified variable.
  • Solve applications by defining and solving rational equations.

Rational Formulas

Rational formulas can be useful tools for representing real-life situations and for finding answers to real problems. You'll see later in this module that certain equations representing relationships called direct, inverse, and joint variation are examples of rational formulas that can model many real-life situations. When solving problems using rational formulas, after identifying the particular formula that represents the relationship between the known and unknown quantities, it is often helpful to then solve the formula for a specified variable. This is sometimes called solving a literal equation.

Example

The formula for finding the density of an object is D=mv D=\frac{m}{v}, where D is the density, m is the mass of the object, and v is the volume of the object. Rearrange the formula to solve for the mass (m) and then for the volume (v).

Answer: Start with the formula for density. D=mv D=\frac{m}{v} Multiply both side of the equation by v to isolate m. vD=mvv v\cdot D=\frac{m}{v}\cdot v Simplify and rewrite the equation, solving for m. vD=mvvvD=m1vD=m\begin{array}{l}v\cdot D=m\cdot \frac{v}{v}\\v\cdot D=m\cdot 1\\v\cdot D=m\end{array} To solve the equation D=mv D=\frac{m}{v} in terms of v, you will need do the same steps to this point and then divide both sides by D. vDD=mDDDv=mD1v=mDv=mD\begin{array}{r}\frac{v\cdot D}{D}=\frac{m}{D}\\\\\frac{D}{D}\cdot v=\frac{m}{D}\\\\1\cdot v=\frac{m}{D}\\\\v=\frac{m}{D}\end{array} Therefore, m=Dv m=D\cdot v and v=mD v=\frac{m}{D}.

Example

The formula for finding the volume of a cylinder is V=πr2hV=\pi{r^{2}}h, where V is the volume, r is the radius, and h is the height of the cylinder. Rearrange the formula to solve for the height (h).

Answer: Start with the formula for the volume of a cylinder. V=πr2h V=\pi{{r}^{2}}h Divide both sides by πr2 \pi {{r}^{2}} to isolate h. Vπr2=πr2hπr2 \frac{V}{\pi {{r}^{2}}}=\frac{\pi {{r}^{2}}h}{\pi {{r}^{2}}} Simplify. You find the height, h, is equal to Vπr2 \frac{V}{\pi {{r}^{2}}}. Vπr2=h \frac{V}{\pi {{r}^{2}}}=h Therefore, h=Vπr2 h=\frac{V}{\pi {{r}^{2}}}.

Watch the following video for more examples of solving for a particular variable in a formula, a literal equation. https://www.youtube.com/watch?v=ecEUUbRLDQs&feature=youtu.be

Work

A work problem is a useful real-world application involving literal equations. Let's say you'd like to calculate how long it will take different people working at different speeds to finish a task.  You may recall the formula that relates distance, rate and time, d=rtd=rt. A similar formula relates work performed to a work-rate and time spent working: W=rtW=rt. The amount of work done WW is the product of the rate of work rr and the time spent working tt. Using algebra, you can write the work formula 33 ways:

W=rtW=rt

Solved for time tt the formula is t=Wr t=\frac{W}{r} (divide both sides by r).

Solved for rate rr the formula is r=Wt r=\frac{W}{t}(divide both sides by t).

Rational equations can be used to solve a variety of problems that involve rates, times, and work. Using rational expressions and equations can help you answer questions about how to combine workers or machines to complete a job on schedule. Some work problems include multiple machines or people working on a project together for the same amount of time but at different rates. In this case, you can add their individual work rates together to get a total work rate.

Example

Myra takes 22 hours to plant 5050 flower bulbs. Francis takes 33 hours to plant 4545 flower bulbs. Working together, how long should it take them to plant 150150 bulbs?

Answer: Think about how many bulbs each person can plant in one hour. This is their planting rate. Myra: 50  bulbs2  hours \frac{50\,\,\text{bulbs}}{2\,\,\text{hours}} or 25  bulbs1  hour \frac{25\,\,\text{bulbs}}{1\,\,\text{hour}} Francis: 45  bulbs3  hours \frac{45\,\,\text{bulbs}}{3\,\,\text{hours}} or 15  bulbs1  hour \frac{15\,\,\text{bulbs}}{1\,\,\text{hour}} Combine their hourly rates to determine the rate they work together. Myra and Francis together: 25  bulbs1  hour+15  bulbs1  hour=40  bulbs1  hour \frac{25\,\,\text{bulbs}}{1\,\,\text{hour}}+\frac{15\,\,\text{bulbs}}{1\,\,\text{hour}}=\frac{40\,\,\text{bulbs}}{1\,\,\text{hour}} Use one of the work formulas to write a rational equation, for example, r=Wt r=\frac{W}{t}. You know r, the combined work rate, and you know W, the amount of work that must be done. What you do not know is how much time it will take to do the required work at the designated rate. 401=150t \frac{40}{1}=\frac{150}{t} Solve the equation by multiplying both sides by the common denominator and then isolating t. 1t401=150t1t40t=150t=15040=154t=334hours\begin{array}{c}1t\cdot\frac{40}{1} =\frac{150}{t}\cdot 1t\\\\40t=150\\\\t=\frac{150}{40}=\frac{15}{4}\\\\t=3\frac{3}{4}\text{hours}\end{array} It should take 33 hours 4545 minutes for Myra and Francis to plant 150150 bulbs together.

Other work problems take a different perspective. You can calculate how long it will take one person to do a job alone when you know how long it takes people working together to complete the job.

Example

Joe and John are planning to paint a house together. John thinks that if he worked alone, it would take him 33 times as long as it would take Joe to paint the entire house. Working together, they can complete the job in 2424 hours. How long would it take each of them, working alone, to complete the job?

Answer: Choose variables to represent the unknowns. Since it takes John 33 times as long as Joe to paint the house, his time is represented as 3x3x. Let xx = time it takes Joe to complete the job 3x3x = time it takes John to complete the job The work is painting 11 house or 11. Write an expression to represent each person’s rate using the formula r=Wt r=\frac{W}{t}. Joe’s rate: 1x \frac{1}{x} John’s rate: 13x \frac{1}{3x} Their combined rate is the sum of their individual rates. Use this rate to write a new equation using the formula W=rtW=rt. combined rate: 1x+13x \frac{1}{x}+\frac{1}{3x} The problem states that it takes them 2424 hours together to paint a house, so if you multiply their combined hourly rate (1x+13x) \left( \frac{1}{x}+\frac{1}{3x} \right) by 2424, you will get 11, which is the number of houses they can paint in 2424 hours. 1=(1x+13x)241=24x+243x \begin{array}{l}1=\left( \frac{1}{x}+\frac{1}{3x} \right)24\\\\1=\frac{24}{x}+\frac{24}{3x}\end{array} Now solve the equation for x. (Remember that x represents the number of hours it will take Joe to finish the job.)    1=3324x+243x   1=3243x+243x   1=723x+243x   1=72+243x   1=963x3x=96   x=32\begin{array}{l}\,\,\,1=\frac{3}{3}\cdot \frac{24}{x}+\frac{24}{3x}\\\\\,\,\,1=\frac{3\cdot 24}{3x}+\frac{24}{3x}\\\\\,\,\,1=\frac{72}{3x}+\frac{24}{3x}\\\\\,\,\,1=\frac{72+24}{3x}\\\\\,\,\,1=\frac{96}{3x}\\\\3x=96\\\\\,\,\,x=32\end{array} Check the solution in the original equation. 1=(1x+13x)241=[132+13(32)]241=2432+243(32)1=2432+24961=332432+24961=7296+2496\begin{array}{l}1=\left( \frac{1}{x}+\frac{1}{3x} \right)24\\\\1=\left[ \frac{\text{1}}{\text{32}}+\frac{1}{3\text{(32})} \right]24\\\\1=\frac{24}{\text{32}}+\frac{24}{3\text{(32})}\\\\1=\frac{24}{\text{32}}+\frac{24}{96}\\\\1=\frac{3}{3}\cdot \frac{24}{\text{32}}+\frac{24}{96}\\\\1=\frac{72}{96}+\frac{24}{96}\end{array} The solution checks. Since x=32x=32, it takes Joe 3232 hours to paint the house by himself. John’s time is 3x3x, so it would take him 9696 hours to do the same amount of work.

As shown above, many work problems can be represented by the equation ta+tb=1 \dfrac{t}{a}+\dfrac{t}{b}=1, where tt represents the quantity of time two people, A and BA \text{ and } B, complete the job working together, aa is the amount of time it takes person AA to do the job, and bb is the amount of time it takes person BB to do the job. The 11 on the right hand side represents 11 job, the total work done—in this case, the work was to paint a house. The key idea here is to figure out each worker’s individual rate of work. Then, once those rates are identified, add them together, multiply by the time t, set it equal to the amount of work done, and solve the rational equation. If person AA works at a rate of 11 job every aa hours, and person BB works at a rate of 11 job every bb hours, and if tt represents the total amount of time it takes to paint 11 house, we have W=(r1+r2)t1=(1a+1b)t1=ta+tb\begin{align}W&= \left(r_1 + r_2\right)t \\ 1 &= \left(\dfrac{1}{a}+\dfrac{1}{b}\right)t \\ 1 &= \dfrac{t}{a}+\dfrac{t}{b}\end{align} Watch the following video for an example of finding the total time given two people working together at known rates. https://youtu.be/SzSasnDF7Ms The following video shows an example of finding one person's work rate given a known combined work rate. https://www.youtube.com/watch?v=kbRSYb8UYqU&feature=youtu.be

Mixing

Mixtures are made of ratios of different substances that may include chemicals, foods, water, or gases. There are many different situations where mixtures may occur both in nature and as a means to produce a desired product or outcome. For example, chemical spills, manufacturing, and even biochemical reactions involve mixtures. Mixture problems become mathematically interesting when components of the mixture are added at different rates and concentrations. The next example shows how to define an equation that models the concentration (a ratio) of sugar to water in a large mixing tank over time.

Example

A large mixing tank currently contains 100100 gallons of water into which 55 pounds of sugar have been mixed. A tap will open pouring 1010 gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of 11 pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after 1212 minutes. Is that a greater concentration than at the beginning?

Answer:

Let t be the number of minutes since the tap opened. Since the water increases at 1010 gallons per minute, and the sugar increases at 11 pound per minute, these are constant rates of change. This tells us the amount of water in the tank is a linear equation, as is the amount of sugar in the tank. We can write an equation independently for each:

{water: W(t)=100+10t in gallonssugar: S(t)=5+1t in pounds\begin{cases}\text{water: }W\left(t\right)=100+10t\text{ in gallons}\\ \text{sugar: }S\left(t\right)=5+1t\text{ in pounds}\end{cases}

The concentration, C, will be the ratio of pounds of sugar to gallons of water

C(t)=5+t100+10tC\left(t\right)=\frac{5+t}{100+10t} The concentration after 1212 minutes is given by evaluating C(t)C\left(t\right) at t= 12t=\text{ }12. C(12)=5+12100+10(12)=17220C\left(12\right)=\frac{5+12}{100+10(12)}=\frac{17}{220} This means the concentration is 1717 pounds of sugar to 220220 gallons of water. At the beginning, the concentration is C(0)=5+0100+10(0)=5100=120C\left(0\right)=\frac{5+0}{100+10(0)}=\frac{5}{100}=\frac{1}{20} Since 172200.08>120=0.05\frac{17}{220}\approx 0.08>\frac{1}{20}=0.05, the concentration is greater after 1212 minutes than at the beginning.

The following video gives another example of how to use rational functions to model mixing. https://youtu.be/GD6H7BE_0EI

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