Oil, vinegar, and water are mixed in a 3 to 2 to 1 ratio to make salad

Excellent opportunity for the k technique:

\(\left\{ \matrix{\\
\,{\rm{oil}} = 3k\,\,\,\,\,\, \to \,\,\,\,\,\,3k \le 8\,\,\, \Rightarrow \,\,\,k \le 2{2 \over 3} \hfill \cr \\
\,{\rm{vin}} = 2k\,\,\,\,\,\, \to \,\,\,\,\,\,2k \le 7\,\,\, \Rightarrow \,\,\,k \le 3{1 \over 2} \hfill \cr \\
\,{\rm{wat}} = k \hfill \cr} \right.\)

\(?\,\,:\,\,\left( {3k + 2k + k} \right)\,\,{\rm{where}}\,\,k = 2{2 \over 3}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 6\left( {2 + {2 \over 3}} \right) = 12 + 4 = 16\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

GIVEN: oil: vinegar : water = 3 : 2 : 1

Let's first try to use ALL 8 cups of oil
Take 3 : 2 : 1 and multiply all 3 parts by 8/3 to get the EQUIVALENT ratio 8 : 16/3 : 8/3
Simplify to get: 8 : 5 1/3 : 2 2/3
So, in this case, the dressing is comprised of 8 cups of oil, 5 1/3 cups of vinegar, and 2 2/3 cups of water
Notice that we have enough vinegar to make this batch.

8 cups + 5 1/3 cups + 2 2/3 cups = 16 cups
So, the TOTAL volume = 16 cups

IMPORTANT: Notice that 16 is the biggest answer choice.
This means we need not explore what happens when we try to use ALL 7 cups on the vinegar, since the answer choices tell us that we will NOT get a value greater than 16

Answer: E

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This is how i solved

Notice the original ratio is 3: 2: 1 for Oil Vinegar and water.
which effectively means Oil is Half of the mixture = 3/6

Now if we know the max quantity we have is 8 cups of oil which could be = at max half the quantity of the mixture if we use the entire 8 cups of oil.

Now the other 2 ie Vinegar and Water are 2/3 * of the other half and 1/3 of other half of the mixture= 8

Or vinegar needed for the mixture if we were to use the entire 8 cups of oil = 2/3*8 = 5.3 cups of vinegar and we are told we have 7 cups at our disposal so great we need only 5.3 cups of vinegar

and water would be 1/3 of 8 = 8/3 = 2.7 cups ( We know Water availability is infinite )

So max quantity of salad we can make is 8 + 5.3 + 2.7 = 16 ans choice e

O+V+W =6x
6X: 1 CUP
96 X (Maximum number in option ,i.e divided by six): how many cups
cross multiply
ANS: 16

can someone explain the question in the task in a simple way? I could not understand it and have not been able to solve this task.

Well, I used a different approach-

Oil : Vinegar : Water
3 2 1

if just add them then 3x+2x+1x = S(salad)
so x= S/6 --> now assume x = 1 cup
so you can make 6 salads from above ratio

now you have -
Oil : Vinegar : Water
8 7 6(assume as we can use as much as we want)

now just make me 6 salads
Oil : Vinegar : Water = 6 ( what I am left with (8-3)C (7-2)C (6-1)C = 5C 5C 5C ) --- C=Cups
3 2 1


now make me more 6 salads
Oil : Vinegar : Water = 6 salads ( what I am left with (5-3)C (5-2)C (5-1)C = 2C 3C 4C ) --- C=Cups

total salads I have made till now are 12 and I am left with 2cups of oil, 3 cups of vinegar and 4 cups of water.
now I cannot make 6 salads but still, i can use 2 cups to make some salads. So my salads will be less than 6 and somewhat between 4 to 6

if I had 3 cups then I would have made 18 salads but I have 1 fewer cups so choose the answer nearest to less than 18 which is 16.

(These questions depend upon the approach and understanding)

The ratio of Oil to Vinegar to Water is 3:2:1, which means 3/6:2/6:1/6
Further simplifying, 1/2:1/3;1/6
we have 8 cups of vinegar so try to make the ratio of oil=8
16/2=8 so 16 cups can be made. (E)

I don't understand the significance of the phrase "if fractional cup measurements are possible"

What is the point of the blue phrasing ?

I know all the answers are integers but perhaps, because of the blue -- we can have non-integers answers too, say like 15.5 cups of salad dressing ? Is that the point of the blue.

OR

is the point of the blue phrase - that we can perhaps use 5.5 cups / 7 cups of vinegar OR 4.5 cups / 8 cups of Oil ?


Bunuel zhanbo ScottTargetTestPrep

It is the latter. The blue phrasing tells you that it is possible to use fractional number of cups in your recepie, such as 5.5 cups of vinegar or 4.5 cups of oil. Notice that in the 16 cups of salad dressing we obtain, there is an integer number of cups of oil, but the amount of vinegar is 16/3 cups and the amount of water is 8/3 cups. If we were told that the number of cups of each ingredient in the salad dressing was required to be an integer, the answer that we obtained would no longer be the correct answer. The purpose of the blue phrasing is to clear any confusions about this issue.

As a matter of fact, it could be the case that the quantities were required to be integers even when the question does not say anything about it. For instance, consider the following variation of this question: we need to put together a team which should contain interns, assistants, and managers in a ratio of 3 to 2 to 1. We have only 8 interns and 7 assistants, but we can find as many managers as we want. What is the largest team that we can form according to these restrictions? You cannot solve this question exactly the same way as we solved the original question, because if you did, you would find that you need 16/3 assistants and 8/3 managers, neither of which makes any sense.

Given: Oil, vinegar, and water are mixed in a 3 to 2 to 1 ratio to make salad dressing.

Asked: 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?

In 1 cup of salad: -
Oil = 3/6 = 1/2 cup
Vinegar = 2/6 = 1/3 cup
Water = 1/6 cup

Maximum number of cups of salad = min (8/(1/2), 7/(1/3)) = min (16,21) = 16 cups

IMO E

I took a different approach. Is this a wrong way of getting there?

As we know that 3 out of the 6 units that make up the dressing are oil units, we now that 3/6 of the entire dressing is oil.
Half the dressing is oil.

As a result, since we have 8 cups of oil, those can total up to 16 cups of dressing.

Since the amount of vinegar available respective to oil is more than what is needed in the recipe (7/8 > 2/3, since we have 7 cups of vinegar and 8 of oil, and since 2 units of vinegar are needed for every 3 of oil), we can conclude that the wew'll have enough vinegar to prepare 16 cups of dressing.

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