Oil, vinegar, and water are mixed in a 3 to 2 to 1 ratio to make salad dressing.
If Larry has 8 cups of oil, 7 cups of vinegar, and access to any amount of water, what is the maximum number of cups of salad dressing he can make with the ingredients he has available, if fractional cup measurements are possible?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16
My inference is that one cup of salad dressing is 3 oil, 2 vinegar, and one cup of water is one cup of salad dressing.
The second ingredient is 2/(3) of the first ingredient. 3:2:1 will make one serving. Basically the cups of water within that ratio of mixture equals servings of salad dressing?
(2/3)*7 approximately equals 4.
After you use 8 cups of oil, you'll use 4 cups of vinegar, zero oil left, and we don't care about the water, $<3$ cups of vinegar left. You can't make anything after that step besides about two salad dressing cups?
There are two problems with your attempt.
The first is that you have taken $\frac{2}{3}$ of the vinegar when you really want $\frac{2}{3}$ of the oil.
The second is that you have ignored the part of the problem that states that you are allowed to use fractional cups. You took rounded $4\frac{2}{3}$ down to $4$.
To solve this we must use our ratios and consider the ingredients for which we only have a finite supply. We have eight cups of oil and seven cups of vinegar but we have an unlimited supply of water so water won't cause us any limitations.
If we use all $8$ cups of oil then the amount of vinegar is given by the ratio $3:2$. This means that $$\frac{8}{3}=\frac{v}{2}$$ where $v$ is the amount of vinegar. In words, we are saying that eight is to three as the amount of vinegar is to two.
Solving this gives $$v=\frac{16}{3}=5\frac{1}{3}\text{ cups of vinegar}$$
To get the amount of water, we can use $$\frac{8}{3}=\frac{w}{1}\implies w=\frac{8}{3}$$ where $w$ is the amount of water. Finally, we add $$8+\frac{16}{3}+\frac{8}{3}=8+8=16$$.
Since $16$ is the largest of your choices you can simply choose $E$ but, in general, we want to see if we can do better.
We arrived at our solution by using all of the oil. What if we try to use all of the vinegar? Again, we use our ratios to determine that $$\frac{o}{3}=\frac{7}{2}\implies o=\frac{21}{2}=10\frac{1}{2}\text{ cups of oil}$$
But we only have $8$ cups of oil so $16$ cups of dressing is the best we can do.