2015 AMC 12A Problems/Problem 10: Difference between revisions

Latest revision as of 14:47, 28 September 2025

Problem

Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$ . What is $x$ ?

$\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26$

Solution 1

Use SFFT to get $(x+1)(y+1)=81$ . The terms $(x+1)$ and $(y+1)$ must be factors of $81$ , which include $1, 3, 9, 27, 81$ . Because $x > y$ , $x+1$ is equal to $27$ or $81$ . But if $x+1=81$ , then $y=0$ and so $x=\boxed{\textbf{(E)}\ 26}$ .

Solution 2

Plug in values of $x$ and solve for $y$ , noting $x > y$ and that $y$ is an integer.

$x = 8$ :

$8 + y + 8y = 80$

$y = 8$ (Does not work because $x > y$ )

$x = 10$ :

$10 + y + 10y = 80$

$y = 70/11$ (Does not work because $y$ is not an integer)

$x = 15$ :

$15 + y + 15y = 80$

$y = 65/16$ (Does not work because $y$ is not an integer)

$x = 18$ :

$18 + y + 18y = 80$

$y = 62/19$ (Does not work because $y$ is not an integer)

Thus $x = 26$ , $\boxed{\textbf{(E)}\ 26}$ .

~ Solution by CYB3RFLARE7408

Video Solution by OmegaLearn

https://youtu.be/ba6w1OhXqOQ?t=4512

~ pi_is_3.14

2015 AMC 12A (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 12 Problems and Solutions

@@ Line 5: / Line 5: @@
 <math> \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 26</math>
-==Solution==
+==Solution 1==
 Use [[SFFT]] to get <math>(x+1)(y+1)=81</math>. The terms <math>(x+1)</math> and <math>(y+1)</math> must be factors of <math>81</math>, which include <math>1, 3, 9, 27, 81</math>. Because <math>x > y</math>, <math>x+1</math> is equal to <math>27</math> or <math>81</math>. But if <math>x+1=81</math>, then <math>y=0</math> and so <math>x=\boxed{\textbf{(E)}\ 26}</math>.
+==Solution 2==
+Plug in values of <math>x</math> and solve for <math>y</math>, noting <math>x > y</math> and that <math>y</math> is an integer.
+<math>x = 8</math>:
+<math>
++ y + 8y = 80
+</math>
+<math>y = 8</math> (Does not work because <math>x > y</math>)
+<math>x = 10</math>:
+<math>
++ y + 10y = 80
+</math>
+<math>y = 70/11</math> (Does not work because <math>y</math> is not an integer)
+<math>x = 15</math>:
+<math>
++ y + 15y = 80
+</math>
+<math>y = 65/16</math> (Does not work because <math>y</math> is not an integer)
+<math>x = 18</math>:
+<math>
++ y + 18y = 80
+</math>
+<math>y = 62/19</math> (Does not work because <math>y</math> is not an integer)
+Thus <math>x = 26</math>, <math>\boxed{\textbf{(E)}\ 26}</math>.
+~ Solution by CYB3RFLARE7408
+== Video Solution by OmegaLearn ==
+https://youtu.be/ba6w1OhXqOQ?t=4512
+~ pi_is_3.14
 == See Also ==
 {{AMC12 box|year=2015|ab=A|num-b=9|num-a=11}}

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