Difference between revisions of "2017 AMC 10A Problems/Problem 14"

Latest revision as of 14:54, 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 $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?

$\textbf{(A) } 9\%\qquad \textbf{(B) } 19\%\qquad \textbf{(C) } 22\%\qquad \textbf{(D) } 23\%\qquad \textbf{(E) } 25\%$

Solution

Let $m$ = cost of movie ticket
Let $s$ = cost of soda

We can create two equations:

$m = \frac{1}{5}(A - s)$ $s = \frac{1}{20}(A - m)$

Substituting we get:

$m = \frac{1}{5}(A - \frac{1}{20}(A - m))$ which yields:
$m = \frac{19}{99}A$

Now we can find s and we get:

$s = \frac{4}{99}A$

Since we want to find what fraction of $A$ did Roger pay for his movie ticket and soda, we add $m$ and $s$ to get:

$\frac{19}{99}A + \frac{4}{99}A \implies \boxed{\textbf{(D)}\ 23\%}$

Solution 2

We have two equations from the problem: $5M=A-S$ and $20S=A-M$ If we replace $A$ with $100$ we get a system of equations, and the sum of the values of $M$ and $S$ is the percentage of $A$ . Solving, we get $S=\frac{400}{99}$ and $M=\frac{1900}{99}$ . Adding, we get $\frac{2300}{99}$ , which is closest to $23$ which is $\boxed{\textbf{(D)}\ 23\%}$ .

-Harsha12345

Solution 4

Let $m$ be the price of a movie ticket and $s$ be the price of a soda.

Then,

$m=\frac{A-s}{5}$ and $s=\frac{A-m}{20}$ Then, we can turn this into $5m=A-s$ $20s=A-m$

Subtracting and getting rid of A, we have $20s-5m=-m+s \rightarrow 19s=4m$ . Assume WLOG that $s=4$ , $m=19$ , thus making a solution for this equation. Substituting this into the 1st equation, we get $A=99$ . Hence, $\frac{m+s}{A} = \frac{19+4}{99} \approx \boxed{\textbf{(D)}\ 23\%}$

~MrThinker

Video Solution

https://youtu.be/s4vnGlwwHHw

https://youtu.be/zY726PV6XU8

~savannahsolver

@@ 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?
-<math> \mathrm{\textbf{(A)} \ }9\%\qquad \mathrm{\textbf{(B)} \ } 19\%\qquad \mathrm{\textbf{(C)} \ } 22\%\qquad \mathrm{\textbf{(D)} \ } 23\%\qquad \mathrm{\textbf{(E)} \ }25\%</math>
+<math> \textbf{(A) } 9\%\qquad \textbf{(B) } 19\%\qquad \textbf{(C) } 22\%\qquad \textbf{(D) }  23\%\qquad \textbf{(E) } 25\%</math>
 ==Solution==
@@ 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>.
 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> which is <math>(D)</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
-==Solution 3==
+==Solution 4==
-WLOG let <math>A=20.</math>
+Let <math>m</math> be the price of a movie ticket and <math>s</math> be the price of a soda.
-Let <math>m</math> be the price of the movie ticket.
-Let <math>s</math> be the price of the soda.
-Thus,
-<cmath>\begin{align*}
-m &=\frac{1}{5}\left(20-s\right) \\
-s &= \frac{1}{20}\left(20-m\right)
-\end{align*}</cmath>
-Simplifying, we have
-<cmath>\begin{align*}
-m &= 20 - s \\
-s &= 20-m
-\end{align*}</cmath>
-Multiplying the first equation by <math>4</math> and adding them, we have
+Then,
-<cmath>m+s = \frac{100 - 4s - m}{20}</cmath>
-Finding <math>m</math> and <math>s</math> is straightforward from there.
+<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>
-~mathboy282
+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==
 https://youtu.be/s4vnGlwwHHw

2017 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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