4.5 - Rational Equations in Context

Math 3

Unit 4, Day 5

Rational Word Problems

1.) Factor the denominator

2.) Multiply for a common denominator.

3.) Eliminate denominators & solve for numerators.

4.) Check for extraneous solutions (can't divide by 0)

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​Recap of Solving Rational Equations (Monday)

Multiple Choice

Solve for x: $$\frac{5}{x+2}=\frac{1}{x-4}$$  

• x = 5

• x = 4

• x = 5.5

• x = -4

Multiple Choice

$$\frac{7x-49}{x^2+8x+12}=\frac{3}{x+6}+\frac{1}{x+2}$$  

• $$-2$$  

• $$5$$  

• $$6$$  

• $$\frac{61}{3}$$  

Multiple Choice

$$\frac{2n+16}{n-8}=\frac{1}{2n-16}-\frac{1}{2}$$  

• $$-\frac{23}{5}$$  

• $$7$$  

• $$-4$$  

• $$\frac{4}{3}$$  

Multiple Choice

$$\frac{n-2}{n+3}=\frac{1}{n^2+7n+12}+\frac{n+8}{n+4}$$  

• $$3,\ 4$$  

• $$3$$  

• $$3,\ 7$$  

• $$-\frac{11}{3}$$  

Multiple Choice

$$\frac{5}{k+7}=\frac{1}{k+3}+\frac{2k-2}{k^2+10k+21}$$  

• $$2$$  

• $$1$$  

• $$-5$$  

• $$5$$  

Poll

Ms. Vaughan and Mr. Cooper attempt the following problem:

"Jade and Katia are painting a bedroom. Jade can paint the room in 3 hours. Working together, the girls can paint the room in 2 hours. How long would it take Katia to paint the room, working alone?"

Whose equation (and solution) is correct to this problem? Why doesn't one of the solutions make sense?

• Mr. Cooper

• Ms. Vaughan

Word Problem #1: Combined Rates

Word Problem #1: Combined Rates.

Multiple Choice

Walter and Helen are asked to paint a house. Walter can paint the house by himself in 12 hours and Helen can paint the house by herself in 16 hours. How long would it take to paint the house if they worked together?

Which equation would show this situation?

• $$\frac{1}{12}+\frac{1}{16}=\frac{1}{x}$$

• $$\frac{x}{16}+x=\frac{x}{12}$$

• $$\frac{1}{16}+\frac{1}{x}=\frac{1}{12}$$

• $$16x+x=12$$

Multiple Choice

Walter and Helen are asked to paint a house. Walter can paint the house by himself in 12 hours and Helen can paint the house by herself in 16 hours. How long would it take to paint the house if they worked together? Enter your answer as a decimal rounded to the nearest tenths.

• 5.9

• 6.9

• 8.9

• 10.9

Multiple Choice

Brian, Mark, and Jeff are painting a house. Working together they can paint the house is 6 hours. Working alone Brain can paint the house in 15 hours and Jeff can paint the house in 20 hours. How long would it take Mark to paint the house working alone?

Which equation shows this scenario?

• $$\frac{x}{15}+\frac{x}{20}+\frac{x}{x}=\frac{\text{1}}{\text{6}}$$

• $$\frac{1}{15}+\frac{1}{20}+\frac{1}{x}=\frac{1}{6}$$

• $$\frac{1}{6}+\frac{1}{15}+\frac{1}{20}=\frac{1}{x}$$

• $$15x+20x+x=6$$

Multiple Choice

One pipe can fill a swimming pool in 10 hours, while another pipe can empty the pool in 15 hours. How long would it take to fill the pool if both pipes were accidentally left open?

Which equation correctly shows this situation?

• $$\frac{1}{10}+\frac{1}{15}=\frac{1}{x}$$

• $$\frac{1}{10}-\frac{1}{15}=\frac{1}{x}$$

• $$\frac{x}{10}+\frac{x}{15}=x$$

• $$10x-15x=x$$

Word Problem #2: Comparing Rates

Multiple Choice

Griffin and Allison are weeding a garden. Griffin can weed the garden twice as fast as Allison. Working together, they can weed the garden in 45 minutes. How fast would it take Allison to weed the garden alone?

Which equation correctly represents this situation?

• $$\frac{1}{x}+\frac{1}{2x}=\frac{1}{45}$$

• $$x+2x=45$$

• $$x+\frac{x}{2}=45$$

• $$\frac{1}{2}+\frac{1}{x}=\frac{1}{45}$$

Multiple Choice

A hose and a pipe are used to fill a swimming pool with water. The pipe can fill the pool 2 hours faster than the hose. When both are left running, the pool can be filled in 8 hours. How long would it take for the pool to be filled by the pipe running alone?

Which equation could be used to represent this scenario?

• $$\frac{1}{x}+\frac{1}{x+2}=\frac{1}{8}$$

• $$\frac{1}{x}+\frac{1}{2}=\frac{1}{8}$$

• $$\frac{1}{x}+\frac{1}{2x}=\frac{1}{8}$$

• $$x+\left(x+2\right)=8$$

Multiple Choice

Mr. Cooper reads the following problem:

"A tub is filled with a hose; however, there is a leak in the tub causing some water to drain out. The tub would take 5 minutes longer to drain through the leak than it would to be filled with the hose. The leaky tub can be filled in 3 minutes. How long does it take to fill the tub without the leak?"

And writes the following equation to model the situation $$\frac{1}{x-5}-\frac{1}{x}=\frac{1}{3}$$ . What does x represent in Mr. Cooper's Equation?

• The time to drain the tub though the leak.

• The time to fill the tub with the hose (w/o the leak)

• The time to fill the tub with the hose (w/ the leak)

Word Problem #3: Mixtures & %'s

Multiple Choice

Ms. Graham has 500 ml of a 10% hydrochloric acid solution. For a lab, she only need a 5% acid solution and will dilute her solution with water. Which equation would Ms. Graham use to find the correct amount of water to add?

• $$\frac{50+x}{500+x}=0.05$$

• $$\frac{50}{500+x}=0.05$$

• $$\frac{10+x}{500+x}=5$$

• $$\frac{10}{500+x}=5$$

Multiple Choice

Ms. Jones has 500 ml of a 10% hydrochloric acid solution. For a lab, she only need a 15% acid solution and will dilute her solution with water. Which equation would Ms. Jones use to find the correct amount of water to add?

• $$\frac{50+x}{500+x}=0.15$$

• $$\frac{50}{500+x}=0.15$$

• $$\frac{10+x}{500+x}=15$$

• $$\frac{50+x}{500}=0.05$$

Multiple Choice

Every day, Bella and Isla play pickleball after school. Currently, Bella has won 10 of their last 14 matches but wants to keep a winning record. Which equation could Bella use to find the number of consecutive losses she could take before having a losing record?

• $$\frac{10+x}{14+x}=\frac{1}{2}$$

• $$\frac{10+x}{14}=\frac{1}{2}$$

• $$\frac{10}{14+x}=\frac{1}{2}$$

• $$\frac{10}{14}x=\frac{1}{2}$$

Multiple Choice

Ms. Freiberg is trying to solve the following word problem:

Sammy has had 26 hits in 74 at-bats last baseball season. Sammy would like his batting average to be at least 0.300 (30%). How many consecutive hits would Sammy need in order to reach this goal?

She writes the following equation $$\frac{26+x}{74+x}=0.3$$

What does the x represent in this equation?

• The number of hits

• The number of at-bats

• Both the number of hits and at-bats

• Sammy's batting average

Multiple Choice

Ms. Freiberg is trying to solve the following word problem:

Sammy has had 26 hits in 74 at-bats last baseball season. Sammy would like his batting average to be at least 0.300 (30%). How many consecutive hits would Sammy need in order to reach this goal?

She writes the following equation $$\frac{26+x}{74+x}=0.3$$

What does the 26 represent in this equation?

• Sammy's current number of hits

• Sammy's current number of at-bats

• Sammy's current average

• Sammy's desired average

Multiple Choice

Ms. Freiberg and Ms. Woodcock are trying to solve the following word problem:

Sammy has had 26 hits in 74 at-bats last baseball season. Sammy would like his batting average to be at least 0.300 (30%). How many consecutive hits would Sammy need in order to reach this goal?

Ms. Freiberg writes the following equation $$\frac{26+x}{74+x}=0.3$$ , while Ms. Woodcock writes $$\frac{26+x}{74}=0.3$$

Whose equation is correct?

• Ms. Freiberg's; for each additional hit, Sammy has one additional at-bat so you'd need to add x to both 26 and 74

• Ms. Woodcock's; Sammy is getting additional hits, so you should add x to 26 since that represents hits in the equation.