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

## Check the Answer

But try the problem first…

Answer: \(\frac{1}{2}\)

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

## Other useful links

- https://www.cheenta.com/probability-in-divisibility-amc-10a-2003-problem-15/
- https://www.youtube.com/watch?v=pLAMlNUOdTs

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