2017 AMC 10B Problems/Problem 25
Last year Isabella took math tests and received different scores, each an integer between and , inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was . What was her score on the sixth test?
Let the sum of the scores of Isabella's first tests be . Since the mean of her first scores is an integer, then , or . Also, , so by CRT, . We also know that , so by inspection, . However, we also have that the mean of the first integers must be an integer, so the sum of the first test scores must be an multiple of , which implies that the th test score is .
First, we find the largest sum of scores which is which equals . Then we find the smallest sum of scores which is which is . So the possible sums for the 7 test scores so that they provide an integer average are and which are and respectively. Now in order to get the sum of the first 6 tests, we subtract from each sum producing and . Notice only is divisible by so, therefore, the sum of the first tests is . We need to find her score on the test so we have to find which number will give us a number divisible by when subtracted from Since is the test score and all test scores are distinct that only leaves .
Since all of the scores are from , we can 'subtract' 90 off from all of the scores. Basically, we're looking at the units digits except for 100; we're looking at 10 in this case. Since the last score was a 95, the sum of the scores from the first six tests must be and . Trying out a few cases, the only solution possible is 30 (this is from adding numbers 1-10). The sixth test score must be because . The only possible test scores are and , and is already used, so the answer is .
https://youtu.be/YFz4bctJYVE - Happytwin
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