Algebra 1 solving inequalities

Question

In order for Brady to earn a B in his biology course. His test scores must average at least 80%. On the first 5 tests he has a average of 77%. There is one test remain in the course. What is the minimum score Brady needs to earn on the last test to receive a b in the class

Answer 1

The average of a set of "n" numbers is calculated by summing together the individual terms and dividing by the total number of terms "n". Now in this problem, what is "n"? How many tests is Brady taking in total? Well, he has taken 5 thus far, and has 1 more to go so that would be 6. Now, I'm going to call each individual test attempt T₁, T₂, T₃, T₄, T₅, T₆. We can write the average of his scores as follows:

Avg_total = (T₁+T₂+T₃+T₄+T₅+T₆)/6 = 80% (from the problem statement)

Now the issue is, we know Brady has taken 5 tests, but we don't know the individual results of each test. We only know the average of the first 5. How can we write his average thus far in an expression similar to above?

Avg_taken = (T₁+T₂+T₃+T₄+T₅)/5 = 77% (from the problem statement)

Next, we can perform a substitution to help us visualize the solution. Let x = (T₁+T₂+T₃+T₄+T₅). Now our first and second equations are rewritten as follows:

(x + T₆)/6 = 80

(x)/5 = 77

Simply solve for x in the second equation and substitute into the first.

x = 5*(77) = 385

(385 + T₆)/6 = 80, (multiply both sides by 6)

385 + T₆ = 480, (subtract 385)

T₆ = 95

So he must score 95% or better on his last exam to secure an average of 80%.

As a logical follow up, if Brady managed to score perfect on his last exam (assuming no extra credit), the highest he could possibly average in the course is (385 + 100)/6 = 80.8% ≈ 81%.