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# Largest possible value | AMC-10A, 2004 | Problem 15

Try this beautiful problem from Number Theory based on largest possible value from AMC-10A, 2004. You may use sequential hints to solve the problem.

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

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