If y varies directly as x and y is 180 when x is n and y is n when x is 5 what is the value of n?

To solve this problem, we begin with the understanding that if y varies directly as x, it can be represented mathematically as:

y = kx

where k is the constant of variation.

From the question, we have two scenarios:

1. When x = n, y = 180
2. When x = 5, y = n

From the first scenario, we can express k in terms of n:

180 = kn

So, we have:

k = rac{180}{n}

Now, we can substitute k into the second scenario:

n = k imes 5

By substituting our expression for k, we get:

n = rac{180}{n} imes 5

Multiplying both sides by n to eliminate the fraction leads to:

n^2 = 900

Taking the square root of both sides gives:

n = 30

Therefore, the value of n is 30.