Difference between revisions of "2023 AMC 12A Problems/Problem 8"
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==Solution 2 (Variation on Solution 1)== | ==Solution 2 (Variation on Solution 1)== | ||
− | Suppose Maureen took n tests with an average of m | + | Suppose Maureen took <math>n</math> tests with an average of <math>m</math>. |
+ | If she takes another test, her new average is <math>\frac{(nm+11)}{(n+1)}=m+1</math> | ||
+ | Cross-multiplying: <math>nm+11=nm+n+m+1</math>, so <math>n+m=10</math>. | ||
− | + | If she takes <math>3</math> more tests, her new average is <math>\frac{(nm+33)}{(n+3)}=m+2</math> | |
+ | Cross-multiplying: <math>nm+33=nm+2n+3m+6</math>, so <math>2n+3m=27</math>. | ||
− | + | But <math>2n+3m</math> can also be written as <math>2(n+m)+m=20+m</math>. Therefore <math>m=27-20=\boxed{\textbf{(D) }7}</math> | |
− | + | ~Dilip ~megaboy6679 (latex) | |
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==Solution 3== | ==Solution 3== |
Revision as of 15:28, 10 November 2023
- The following problem is from both the 2023 AMC 10A #10 and 2023 AMC 12A #8, so both problems redirect to this page.
Contents
[hide]Problem
Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an on the next quiz, her mean will increase by . If she scores an on each of the next three quizzes, her mean will increase by . What is the mean of her quiz scores currently?
Solution 1
Let represent the amount of tests taken previously and the mean of the scores taken previously.
We can write the following equations:
Multiplying by and solving, we get:
Multiplying by and solving, we get:
Solving the system of equations for and , we find that and .
~walmartbrian ~Shontai ~andyluo ~megaboy6679
Solution 2 (Variation on Solution 1)
Suppose Maureen took tests with an average of . If she takes another test, her new average is Cross-multiplying: , so .
If she takes more tests, her new average is Cross-multiplying: , so .
But can also be written as . Therefore
~Dilip ~megaboy6679 (latex)
Solution 3
Let represent the sum of Maureen's test scores previously and be the number of scores taken previously.
So, and
We can use the first equation to write in terms of .
We then substitute this into the second equation:
From here, we solve for t, getting .
We substitute this to get .
Therefore, the solution to the problem is
~milquetoast
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/cMgngeSmFCY?si=MHL95YihFdxKROrU&t=2280 ~Math-X
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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 |
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.