The amount of water used to make pancakes varies directly with the amount of pancake mix. If $m$ represents the amount of mix, and w is the amount of water, given that nine cups of mix requires three cups of water, the function is

Step 1 ：This is a problem of direct variation. In a direct variation, as one quantity increases, so does the other. The formula for direct variation is \(y = kx\), where \(k\) is the constant of variation. In this case, \(m\) represents the amount of mix (\(x\)), and \(w\) represents the amount of water (\(y\)). We are given that nine cups of mix requires three cups of water, so we can use these values to find the constant of variation (\(k\)).

Step 2 ：Given that \(m = 9\) and \(w = 3\), we can substitute these values into the formula to find \(k\).

Step 3 ：\(k = \frac{w}{m} = \frac{3}{9} = 0.3333333333333333\)

Step 4 ：The constant of variation (\(k\)) is approximately 0.33. This means that for every one cup of mix, we need approximately 0.33 cups of water. Therefore, the function that represents this relationship is \(w = 0.33m\).

Step 5 ：Final Answer: The function is \(\boxed{w = 0.33m}\).