Given that the arithmetic mean of $m$ and $2n$ is 4, and the arithmetic mean of $2m$ and $n$ is 5, find the value of $m+n$.

1. Problem Description

Given that the arithmetic mean of $m$ and $2n$ is 4, and the arithmetic mean of $2m$ and $n$ is 5, find the value of $m+n$.

2. Solution Steps

The arithmetic mean of two numbers $a$ and $b$ is given by $\frac{a+b}{2}$.

We are given that the arithmetic mean of $m$ and $2n$ is 4, so

$\frac{m+2n}{2} = 4$.

Multiplying by 2, we get

$m+2n = 8$.

We are also given that the arithmetic mean of $2m$ and $n$ is 5, so

$\frac{2m+n}{2} = 5$.

Multiplying by 2, we get

$2m+n = 10$.

We now have a system of two linear equations in two variables:

$m+2n = 8$

$2m+n = 10$

We can solve this system of equations. Multiply the first equation by 2:

$2m+4n = 16$

Subtract the second equation from this new equation:

$(2m+4n) - (2m+n) = 16 - 10$

$3n = 6$

$n = 2$

Substitute $n=2$ into the first equation $m+2n=8$:

$m+2(2) = 8$

$m+4 = 8$

$m = 4$

Therefore, $m=4$ and $n=2$. Thus, $m+n = 4+2 = 6$.

3. Final Answer

$m+n = 6$