# Maintaining an Average

## Problem

Samatha needs to maintain a B average in courses in her major. This term she needs an 80% average in IT. In the first three exams, Sam averaged 75%. What is the lowest score she can get on the fourth exam and still have any possibility at all of getting an 80% average?

There are eight exams through the semester, Sam’s average is based on exams only, and all exams are weighted equally.

## Solution

To get your arms around the question, think about what an average is: the sum of all the numbers in a set divided by the number of numbers.

Sam needs an 80% average, so the sum of all her scores divided by the number of exams adds up to 80. What do you know so far?

• There are eight exams, so the denominator will be 8.
• Sam’s average in the first three exams is 75. That means the sum of her first three scores divided by 3 is 75 — so the sum of her first three scores is 3 times 75, or 225.
• You are looking for the lowest score that can possibly work.

Sam can get away with a lower score on this test if she does better on later ones. The lowest score she can afford this time is the one that will get her just an 80% average if she gets 100 on each one of the last four exams. That’s the score you’re looking for.

You need a name for the number you’re looking for. Call it x.

Now you can write an equation for the average.

$80=\frac{225+x+100+100+100+100}{8}$

$80=\frac{625+x}{8}$

And solve for x.

$x=15$

Sam can get a 15% on the fourth test and still get her 80% average for the course. All she has to do is get 100% on every test after that.

To check, add up all the scores, including the three 75s for the first three exams; 15 for the fourth exam, and 100 for each of the last four exams, and divide by 8:

$\frac{75+75+75+15+100+100+100+100}{8}=80$

Check.

It’s a silly question, but I was trying to replicate an SAT question that was, well, equally silly in the same way. I don’t recommend that you make a habit of counting on your future self to get perfect exam scores.