Difference between revisions of "2017 AMC 10A Problems/Problem 14"
(→Video Solution) |
m (consistent formatting of answer choices) |
||
(8 intermediate revisions by 5 users not shown) | |||
Line 2: | Line 2: | ||
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was <math>A</math> dollars. The cost of his movie ticket was <math>20\%</math> of the difference between <math>A</math> and the cost of his soda, while the cost of his soda was <math>5\%</math> of the difference between <math>A</math> and the cost of his movie ticket. To the nearest whole percent, what fraction of <math>A</math> did Roger pay for his movie ticket and soda? | Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was <math>A</math> dollars. The cost of his movie ticket was <math>20\%</math> of the difference between <math>A</math> and the cost of his soda, while the cost of his soda was <math>5\%</math> of the difference between <math>A</math> and the cost of his movie ticket. To the nearest whole percent, what fraction of <math>A</math> did Roger pay for his movie ticket and soda? | ||
− | <math> | + | <math> \textbf{(A) } 9\%\qquad \textbf{(B) } 19\%\qquad \textbf{(C) } 22\%\qquad \textbf{(D) } 23\%\qquad \textbf{(E) } 25\%</math> |
==Solution== | ==Solution== | ||
Line 8: | Line 8: | ||
Let <math>s</math> = cost of soda | Let <math>s</math> = cost of soda | ||
− | We can create two equations: | + | We can create two equations: |
− | < | + | <cmath>m = \frac{1}{5}(A - s)</cmath> |
− | < | + | <cmath>s = \frac{1}{20}(A - m)</cmath> |
Substituting we get: <br> | Substituting we get: <br> | ||
− | < | + | <cmath>m = \frac{1}{5}(A - \frac{1}{20}(A - m))</cmath> |
which yields:<br> | which yields:<br> | ||
− | < | + | <cmath>m = \frac{19}{99}A</cmath> |
Now we can find s and we get:<br> | Now we can find s and we get:<br> | ||
− | < | + | <cmath>s = \frac{4}{99}A</cmath> |
− | Since we want to find what fraction of <math>A</math> did Roger pay for his movie ticket and soda, we add <math>m</math> and <math>s</math> to get: | + | Since we want to find what fraction of <math>A</math> did Roger pay for his movie ticket and soda, we add <math>m</math> and <math>s</math> to get: |
− | < | + | <cmath>\frac{19}{99}A + \frac{4}{99}A \implies \boxed{\textbf{(D)}\ 23\%}</cmath> |
==Solution 2== | ==Solution 2== | ||
Line 33: | Line 33: | ||
If we replace <math>A</math> with <math>100</math> we get a system of equations, and the sum of the values of <math>M</math> and <math>S</math> is the percentage of <math>A</math>. | If we replace <math>A</math> with <math>100</math> we get a system of equations, and the sum of the values of <math>M</math> and <math>S</math> is the percentage of <math>A</math>. | ||
Solving, we get <math>S=\frac{400}{99}</math> and <math>M=\frac{1900}{99}</math>. | Solving, we get <math>S=\frac{400}{99}</math> and <math>M=\frac{1900}{99}</math>. | ||
− | Adding, we get <math>\frac{2300}{99}</math>, which is closest to <math>23</math>. | + | Adding, we get <math>\frac{2300}{99}</math>, which is closest to <math>23</math> which is <math>\boxed{\textbf{(D)}\ 23\%}</math>. |
-Harsha12345 | -Harsha12345 | ||
+ | ==Solution 4== | ||
+ | Let <math>m</math> be the price of a movie ticket and <math>s</math> be the price of a soda. | ||
+ | |||
+ | Then, | ||
+ | |||
+ | <cmath>m=\frac{A-s}{5}</cmath> | ||
+ | and | ||
+ | <cmath>s=\frac{A-m}{20}</cmath> | ||
+ | Then, we can turn this into | ||
+ | <cmath>5m=A-s</cmath> | ||
+ | <cmath>20s=A-m</cmath> | ||
+ | |||
+ | Subtracting and getting rid of A, we have <math>20s-5m=-m+s \rightarrow 19s=4m</math>. Assume WLOG that <math>s=4</math>, <math>m=19</math>, thus making a solution for this equation. Substituting this into the 1st equation, we get <math>A=99</math>. Hence, <math>\frac{m+s}{A} = \frac{19+4}{99} \approx \boxed{\textbf{(D)}\ 23\%}</math> | ||
+ | |||
+ | ~MrThinker | ||
==Video Solution== | ==Video Solution== | ||
https://youtu.be/s4vnGlwwHHw | https://youtu.be/s4vnGlwwHHw |
Latest revision as of 15:54, 4 July 2023
Problem
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was dollars. The cost of his movie ticket was
of the difference between
and the cost of his soda, while the cost of his soda was
of the difference between
and the cost of his movie ticket. To the nearest whole percent, what fraction of
did Roger pay for his movie ticket and soda?
Solution
Let = cost of movie ticket
Let = cost of soda
We can create two equations:
Substituting we get:
which yields:
Now we can find s and we get:
Since we want to find what fraction of did Roger pay for his movie ticket and soda, we add
and
to get:
Solution 2
We have two equations from the problem:
and
If we replace
with
we get a system of equations, and the sum of the values of
and
is the percentage of
.
Solving, we get
and
.
Adding, we get
, which is closest to
which is
.
-Harsha12345
Solution 4
Let be the price of a movie ticket and
be the price of a soda.
Then,
and
Then, we can turn this into
Subtracting and getting rid of A, we have . Assume WLOG that
,
, thus making a solution for this equation. Substituting this into the 1st equation, we get
. Hence,
~MrThinker
Video Solution
~savannahsolver
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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.