Difference between revisions of "2017 AMC 10B Problems/Problem 25"

Revision as of 08:48, 11 October 2017

Problem

Last year Isabella took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100$ , inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95$ . What was her score on the sixth test?

$\textbf{(A)}\ 92\qquad\textbf{(B)}\ 94\qquad\textbf{(C)}\ 96\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ 100$

Solution 1

Let the sum of the scores of Isabella's first $6$ tests be $S$ . Since the mean of her first $7$ scores is an integer, then $S + 95 \equiv 0 \text{ (mod 7)}$ , or $S \equiv 3 \text{ (mod 7)}$ . Also, $S \equiv 0 \text{ (mod 6)}$ , so by CRT, $S \equiv 24 \text{ (mod 42)}$ . We also know that $91 \cdot 6 \leq S \leq 100 \cdot 6$ , so by inspection, $S = 570$ . However, we also have that the mean of the first $5$ integers must be an integer, so the sum of the first $5$ test scores must be an multiple of $5$ , which implies that the $6$ th test score is $\boxed{\textbf{(E) } 100}$ .

Solution 1.1

First, we find the largest sum of scores which is $100+99+98+97+96+95+94$ which equals $7(98)$ . Then we find the smallest sum of scores which is $91+92+93+94+95+96+97$ which is $7(94)$ . So the possible sums for the 7 test scores so that they provide an integer average are $7(98), 7(97), 7(96), 7(95)$ and $7(94)$ which are $686, 679, 672, 665,$ and $658$ respectively. Now in order to get the sum of the first 6 tests, we negate $95$ from each sum producing $591, 584, 577, 570,$ and $563$ . Notice only $570$ is divisible by $6$ so, therefore, the sum of the first $6$ tests is $570$ . We need to find her score on the $6th$ test so what number minus $570$ will give us a number divisible by $5$ . Since $95$ is the $7th$ test score and all test scores are distinct that only leaves $\boxed{\textbf{(E) } 100}$ .

Solution 2 (Cheap Solution)

By inspection, the sequences $91,93,92,96,98,100,95$ and $93,91,92,96,98,100,95$ work, so the answer is $\boxed{\textbf{(E) } 100}$ . Note: A method of finding this "cheap" solution is to create a "mod chart", basically list out the residues of 91-100 modulo 1-7 and then finding the two sequences should be made substantially easier.

@@ Line 8: / Line 8: @@
 ==Solution 1.1==
-First, we find the largest sum of scores which is <math>100+99+98+97+96+95+94</math> which equals <math>7(98)</math>. Then we find the smallest sum of scores which is <math>91+92+93+94+95+96+97</math> which is <math>7(94). So the possible sums for the 7 test scores so that they provide an integer average are </math>7(98), 7(97), 7(96), 7(95)<math> and </math>7(94)<math> which are </math>686, 679, 672, 665,<math> and </math>658<math> respectively.  Now in order to get the sum of the first 6 tests, we negate </math>95<math> from each sum producing </math>591, 584, 577, 570,<math> and </math>563<math>. Notice only </math>570<math> is divisible by </math>6<math> so, therefore, the sum of the first </math>6<math> tests is </math>570<math>. We need to find her score on the </math>6th<math> test so what number minus </math>570<math> will give us a number divisible by </math>5<math>. Since </math>95<math> is the </math>7th<math> test score and all test scores are distinct that only leaves </math>\boxed{\textbf{(E) } 100}<math>.
+First, we find the largest sum of scores which is <math>100+99+98+97+96+95+94</math> which equals <math>7(98)</math>. Then we find the smallest sum of scores which is <math>91+92+93+94+95+96+97</math> which is <math>7(94)</math>. So the possible sums for the 7 test scores so that they provide an integer average are <math>7(98), 7(97), 7(96), 7(95)</math> and <math>7(94)</math> which are <math>686, 679, 672, 665,</math> and <math>658</math> respectively.  Now in order to get the sum of the first 6 tests, we negate <math>95</math> from each sum producing <math>591, 584, 577, 570,</math> and <math>563</math>. Notice only <math>570</math> is divisible by <math>6</math> so, therefore, the sum of the first <math>6</math> tests is <math>570</math>. We need to find her score on the <math>6th</math> test so what number minus <math>570</math> will give us a number divisible by <math>5</math>. Since <math>95</math> is the <math>7th</math> test score and all test scores are distinct that only leaves <math>\boxed{\textbf{(E) } 100}</math>.
 ==Solution 2 (Cheap Solution)==
-By inspection, the sequences </math>91,93,92,96,98,100,95<math> and </math>93,91,92,96,98,100,95<math> work, so the answer is </math>\boxed{\textbf{(E) } 100}$.
+By inspection, the sequences <math>91,93,92,96,98,100,95</math> and <math>93,91,92,96,98,100,95</math> work, so the answer is <math>\boxed{\textbf{(E) } 100}</math>.
 Note: A method of finding this "cheap" solution is to create a "mod chart", basically list out the residues of 91-100 modulo 1-7 and then finding the two sequences should be made substantially easier.

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
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