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Try this beautiful problem from Number system: largest possible value

## Largest Possible Value – AMC-10A, 2004- Problem 15

Given that $-4 \leq x \leq -2$ and $2 \leq y \leq 4$, what is the largest possible value of $\frac{x+y}{2}$

• $\frac {-1}{2}$
• $\frac{1}{6}$
• $\frac{1}{2}$
• $\frac{1}{4}$
• $\frac{1}{9}$

### Key Concepts

Number system

Inequality

divisibility

But try the problem first…

Answer: $\frac{1}{2}$

Source

AMC-10A (2003) Problem 15

Pre College Mathematics

## Try with Hints

First hint

The given expression is $\frac{x+y}{x}=1+\frac{y}{x}$

Now $-4 \leq x \leq -2$ and $2 \leq y \leq 4$ so we can say that $\frac{y}{x} \leq 0$

can you finish the problem……..

Second Hint

Therefore, the expression $1+\frac{y}x$ will be maximized when $\frac{y}{x}$ is minimized, which occurs when $|x|$ is the largest and $|y|$ is the smallest.

can you finish the problem……..

Final Step

Therefore in the region $(-4,2)$ , $\frac{x+y}{x}=1-\frac{1}{2}=\frac{1}{2}$