2019 AMC 8 Problems/Problem 7
Shauna takes five tests, each worth a maximum of points. Her scores on the first three tests are , , and . In order to average for all five tests, what is the lowest score she could earn on one of the other two tests?
We should notice that we can turn the information we are given into a linear equation and just solve for our set variables. I'll use the variables and for the scores on the last two tests. We can now cross multiply to get rid of the denominator. Now that we have this equation, we will assign as the lowest score of the two other tests, and so: Now we know that the lowest score on the two other tests is .
Right now, she scored and points, for a total of points. She wants her average to be for her tests, so she needs to score points in total. This means she needs to score a total of points in her next tests. Since the maximum score she can get on one of her tests is , the least possible score she can get is .
Note: You can verify that is the right answer because it is the lowest answer out of the 5. Since it is possible to get 48, we are guaranteed that that is the right answer.
We can compare each of the scores with the average of : , , , ;
So the last one has to be (since all the differences have to sum to ), which corresponds to .
We know that she scored and points on her first tests for a total of points and that she wants her average to be for her total tests. Therefore, she needs to score a total of points. In addition, one of the final tests needs to be the maximum of points, to make the final test score—the one that we are looking for—the lowest score possible for her to earn. We can now see here that the sum of has a units digit of and that the final test score must have a units digit ending with a . Now, needs to be added to a number that makes the sum divisible by . Among the answer choices of and , only has a units digit that works. (, giving us a units digit of .)
Video Solution 1
The Learning Royal : https://youtu.be/8njQzoztDGc
Video Solution 2
Solution detailing how to solve the problem: https://www.youtube.com/watch?v=mwHrUESo2_A&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=8
Video Solution 3
Video Solution 4
Video Solution by OmegaLearn
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