2016 AMC 8 Problems/Problem 10

Problem

Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if $2 * (5 * x)=1$ $\textbf{(A) }\frac{1}{10} \qquad\textbf{(B) }2\qquad\textbf{(C) }\frac{10}{3} \qquad\textbf{(D) }10\qquad \textbf{(E) }14$

Solution 1

Let us plug in $(5 * x)=1$ into $3a-b$ . Thus it would be $3(5)-x$ . Now we have $2*(15-x)=1$ . Plugging $2*(15-x)$ into $3a-b$ , we have $6-15+x=1$ . Solving for $x$ we have $-9+x=1$ $x=\boxed{\textbf{(D)} \, 10}$

Solution 2

Let us set a variable $y$ equal to $5 * x$ . Solving for y in the equation $3(2)-y=1$ , we see that y is equal to five. By substitution, we see that $5 * x$ = 5. Solving for x in the equation $5(3)-x = 5$ we get $x=\boxed{\textbf{(D)} \, 10}$

Video Solution (CREATIVE THINKING!!!)

https://youtu.be/0FhGMy0mCVU

~Education, the Study of Everything

Video Solution

https://youtu.be/ysYNhG_uG-I

~savannahsolver

Video Solution by OmegaLearn

https://youtu.be/TkZvMa30Juo?t=638

~pi_is_3.14

2016 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search