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AMC-8 USA Math Olympiad

Ratio and proportion AMC 8, 2010 Problem 2

Try this beautiful problem from AMC 8. It involves the concept of a variable, binary operation and ratio-proportion. We provide sequential hints so that you can try the problem.

What are we learning ?

Competency in Focus: Ratio and proportion.

This problem from American Mathematics contest (AMC 8, 2010) is based on the concept of ratio and proportion and binary composition .

First look at the knowledge graph.

Next understand the problem

If $a @ b = \frac{a\times b}{a+b}$ for $a,b$ positive integers, then what is $5 @10$? $\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50$
Source of the problem
American Mathematical Contest 2010, AMC 8  Problem 2
Key Competency

Ratio and Proportion

Difficulty Level
2/10

Start with hints 

Do you really need a hint? Try it first!
Take a look at the knowledge graph first
Substitute $a \textbf{ by } 5 \textbf{ and } b \text{ by } 10$   
$5@10=\frac{5\times10}{5+10}=\frac{50}{15}=\frac{10}{3}$

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