2017 AMC 10B Problems/Problem 25
Contents
Problem
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?
Solution 1
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 the CRT,
. We also know that
, so by inspection,
. However, we also have that the mean of the first
test scores 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
.
Solution 2
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
.
Solution 3
Since all of the scores are from , we can subtract 90 from all of the scores. Basically, we're looking at their 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
.
Solution 4 (Working Backwards)
I am going to work backwards to solve this problem. In the test,
(mod n) = 0. In the following table, the tests already taken are in bold, the latest test is underlined. We will work from the row of (mod 7), (mod 6), and (mod 5) to determine the test order by trial and error.
In the test, the test score could be 91 or 97.
As you can see, the test score of the test is
Video Solution
https://youtu.be/YFz4bctJYVE - Happytwin
See Also
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.