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Studienführer > Prealgebra

Putting It Together: Multi-Step Linear Equations

Kristy, the college student we met earlier who was anxiously studying for her final exam, still isn't sure what grade she needs to earn to pass her math class, and the final is tomorrow! Let's put your new math skills to good use to help her figure it out. Here's what we already know:
  • Kristy's current grade in the class is 82.382.3%.
  • She wants to know what score she needs on the final to keep her B in the class (>80%).
  • She also wants to know what score she needs on the final to pass the class with a C (>70%).
Several students at their desks in a classroom. Exam time is coming.
Even though Kristy knows her current grade is 82.382.3%, that isn't enough information to determine what grade she needs on the final. She also needs to know how much the final exam can affect her grade. To figure this out, she consults her class syllabus, where she finds the following breakdown of points in the class:          
Homework 150 Points
Quizzes 100 Points
Midterm 100 Points
Final Exam 200 Points
TOTAL 550 Points
How can we put this information together in a way that will be useful? First, we know that Kristy has earned points for everything but the final exam. If we add up the points for homework, quizzes, and the midterm, we know there are 350350 points possible so far (150+100+100150+100+100). If Kristy's grade is currently 82.382.3%, this means she has earned 82.382.3% of the total 350350 points. We can multiply to find the number of points she has earned so far:

350×0.823=288.05350 \times 0.823 = 288.05 points

So we know that Kristy has 288.05288.05 points. The next question to answer is: how many total points does she need to achieve a B or a C?

Typically a B is 80% or .8, a C is 70% or .7

There are a total of 550550 points in the class. We can use this to figure out how many points Kristy would need to get a certain grade:

To get a B, she needs 550×0.8=440550 \times 0.8 = 440 points.

To get a C, she needs 550×0.7=385550 \times 0.7 = 385 points.

Now we can write a linear equation to determine the number of points she needs on the final to earn either a B or a C in the class.

current number of points + points needed on the final = total points needed

Again, Kristy has 288.05288.05 points. Let's call the number of points she needs on the final xx. We know that for a B, she needs 440440 points total. We can set up the equation and solve for xx to determine the number of points she needs on the final to earn a B.

288.05+x=440288.05+x=440

288.05+x−288.05=440−288.05288.05+x\color{red}{-288.05}=440\color{red}{-288.05}

x=151.95x=151.95

To state that grade as a percentage, we can divide it by the total number of points possible on the final exam: 200200. That means Kristy needs 151.95/200=75.975151.95/200=75.975%. In other words, a C on the final will be enough for her to get a B in the class. Now let's calculate how many points she can earn on the final to pass the class with a C. We'll use the same equation, but this time, we'll use 385385 for the total number of points.

288.05+x=385288.05 + x = 385

Solve again, and you'll find that x=96.95x = 96.95. If we divide that by 200200 to figure out the percentage, we learn that Kristy needs a score of 48.47548.475% on the final to get a C in the class. In other words, she can fail the final exam and still pass. Kristy can now take the final exam knowing exactly how well she needs to do to pass the class and keep her scholarship--and it turns out that she doesn't need to be as nervous as she thought!

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