SOLUTION: Bobo's average for six mathematics tests is 97%. He made the same grade on each of the first five tests. His score on the last test was 12 points lower than the first five scores.

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Question 1204325: Bobo's average for six mathematics tests is 97%. He made the same grade on each of the first five tests. His score on the last test was 12 points lower than the first five scores. What was Bobo's score on the last test?
Found 3 solutions by ikleyn, greenestamps, math_tutor2020:
Answer by ikleyn(52797)   (Show Source):
You can put this solution on YOUR website!
.

x scores for the last, 6th test; (x+12) scores for each of the first five tests. An equation for x = 97. Solve by simplifying, step by step 5*(x+12) + x = 6*97 5x + 60 + x = 582 6x = 582 - 60 6x = 522 x = 522/6 = 87. ANSWER. His score on the last, 6-th test, is 87.

Solved.

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It is a typical problem on average.

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    - Miscellaneous problems on average values
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Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!

The response from tutor @ikleyn shows a good typical formal algebraic solution.

For problems like this involving averages of numbers that are all relatively close together, here is a VERY different way to solve the problem.

The score on the last test was 12 points lower than on each of the other tests.

"Distribute" those 12 points less over the 6 total tests. That's 2 points per test. That means the scores on his other 5 tests were each 97+2 = 99, and the score on his last test was 97-5(2) = 87.

ANSWER: 87

And for a more standard formal algebraic solution somewhat different than from the other tutor....

let x = his score on each of the first 5 tests

Then his score on the last test was x-12; and the average for the 6 tests was 97:

ANSWER: his score on the last test was x-12 = 99-12 = 87

Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!

Answer: 87

Explanation

The other tutors have great approaches.
I'll use a method that doesn't involve algebra.

Let's assume the first 5 tests had scores of 97 since this is the average we want.
The 6th test would have a score of 97-12 = 85.

The 6 scores are: 97,97,97,97,97,85

To get the average we add up the values and divide by the sample size (the number of items in the set).
Add the scores: 97+97+97+97+97+85 = 570
Divide by 6 to get: 570/6 = 95

The average we get (95) is smaller than the average we want (97).

Let's bump each grade up by 1 point and repeat the process
scores = 98,98,98,98,98,86
average = (98+98+98+98+98+86)/6 = 576/6 = 96
We're getting closer to the goal.

Let's bump each grade up by 1 point and repeat the process
scores = 99,99,99,99,99,87
average = (99+99+99+99+99+87)/6 = 582/6 = 97
We finally land on the average we want.

Therefore, Bobo's score on the 6th test was 87.

This trial-and-error process didn't take too long.
However, we might not be so lucky in other problems.
That's why the algebraic approach is often more efficient.

If you are familiar and confident with algebraic notation, then check out this lesson
https://www.algebra.com/algebra/homework/Probability-and-statistics/statistics-transformations1.lesson
In that lesson I prove that adding a certain value k to each item in a set would increase the mean by k.
This helps explain why the mean increased by 1 each time we bumped the grades up by 1.