Question

If \((2m+n) + (2n+m)=27\), find the maximum value of \((2m-3)\), assuming m and n are positive integers. 
 

Hint:

If an algebra problem seems to have no solution or an unbounded solution (no maximum/minimum), double-check for potential typos. A product might be a sum, or there might be an unstated but implied constraint, such as the variables being positive integers.

Answer 1

Correct Answer: 13

The equation \( (2m+n)+(2n+m)=27 \) just adds up to \( 3m+3n=27 \), so \( m+n=9 \). Since \( m \) and \( n \) must both be positive whole numbers, pushing \( m \) as high as possible means pushing \( n \) as low as possible, and the smallest positive integer \( n \) can be is 1. That forces \( m=8 \), the largest \( m \) can get. Plugging into \( 2m-3 \) gives \( 2(8)-3=13 \), so the maximum value of \( (2m-3) \) is 13.