Looks like a perfect opportunity to use weighted averages
Weighted average of groups combined = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...

Let x = the number of liters of MILK needed in the mixture
So, the PROPORTION of milk in the mixture = x/60 [since we have a total of 60 liters in the final mixture]

Since we have a total of 60 liters in the mixture, 60 - x = the number of liters of CREAM needed in the mixture
So, the PROPORTION of cream in the mixture = (60 - x)/60

We want the resulting mixture to contain 50% butterfat

Plug all of these values into the equation to get: 50 = (x/60)(5) + ((60 - x)/60)(75)
Multiply both sides by 60 to get: 3000 = (5)(x) + (60 - x)(75)
Expand: 3000 = 5x + 4500 - 75x
Simplify: 3000 = 4500 - 70x
Subtract 4500 from both sides: -1500 = -70x
Solve: x = (-1500)/(-70) = (1500)/(70) = 150/7

If you'd rather not divide 150 by 7, you can quickly evaluate this by first recognizing that 140/7 = 20
So, 150/7 must be a little bit bigger than 20

Answer: B

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B?
Milk =x
Cream = 60-x

5x/100 + 75(60-x)/100 = 30

Solving: x = 150/7 = 21.xx
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Use options.
Milk content will have 5% butter fat.
Total content is 60
So, 60-milk = cream

Now,
A) 20 is milk
40 will be cream.
=> 20*5/100= 1litre of butterfat
=> 40*75/100 = 30 litre of butterfat
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Total 31 litre of butterfat (greater than 50% of 60 litre of mixture)

(So it has to be slightier bigger than 20litre of milk)

B) 21.43 is milk
60-21.43 = 38.57 is creame
=> 21.43*5/100= 1.0715 litre of butterfat.
=> 38.57*75/100= 28.9275 litre of butterfat.
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Total butterfat is 1.0715+28.9275=30 ( desired result, 50% of 60 litre of mixture)

We can stop here, because we already got answer but if you want to check, you can check rest with the similar approach.

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Thank you BrentGMATPrepNow for your explanation! I have some questions...
1. Shouldn't milk (marked in bold above) be cream? If not, why?
2. I am confused why you use 50 and not 30(L) in the equation (I understand you get the 50 from the 50% ratio). Does it has to do with the fact that we are only using proportions and percentages in the equation? If that is true, why are we not using 1/2 instead of 50?

Please let me know!

1. Thanks for catching that. Yes, it should be cream. I have edited my response accordingly.

2. I'm not sure where the 30 comes in.
You can think of this as combining two groups: Milk and Cream.
Each group has a certain percentage of butterfat. Milk is 5% butterfat, and cream is 75% butterfat,
We want the combined mixture to contain 60 liters, and we want that mixture to be 50% butterfat (that's where I got the 50 from)

Does that help?
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That makes sense! FYI: The 30 comes from the 50% ratio of butterfat in the 60L mixture (which is 30L). I got confused and mixed two formulas. I just have to get used to the weighted average formula! Again thanks!

Solution

Given
In this question, we are given that
• Milk and cream are combined to make a 60-liter mixture that is 50 percent butterfat
• Milk is 5 percent butterfat and the cream is 75 percent butterfat

To find
We need to determine
• The number of litres of milk are in the mixture

Approach and Working out
Let x be the number of litres of milk in the mixture, and y be the number of litres of cream

• \((\frac{5}{100})x + (\frac{75}{100})y = 30\)

o 5x +75y = 3000
o x + 15y = 600
o And, we know x + y = 60


• Thus, 14x = 300

o Implies, \(x = \frac{150}{7} = 21.43\)

Thus, option B is the correct answer.

Correct Answer: Option B
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We are told that milk and cream are combined to make a 60-liter mixture that is 50% butterfat.

Milk is 5% butterfat and cream is 75% butterfat. Approximately how many liters of milk are in the mixture?

If butterfat and cream were in the mixture were in equal ratio, the mixture would be 35% butterfat. However, we're told that the 6-liter mixture is 50% butterfat. This must mean that the mixture has more cream than milk. We can eliminate answers C, D, and E right away.

Now, we have to choose between A and B.

Lets take a look at the figure we're given:

Milk / Butterfat = 50-5 / 75-50 = 45/25
45/25 = 9/5

The ratio of cream to butterfat is 9 : 5.

60/14 = 4.2
4.2 * 5 = milk is approximately 21 liters.

Answer is B.
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GMATNinja, Bunuel, can you please advice how to solve this problem by using a weighted average concept?

When X liter of milk ( 5 % butterfat ) is mixed with Y liter of cream ( 75 % butterfat ), we will get a 60 - liter mixture of milk and cream ( 50 % butterfat).

This is an application of the weighted average.

50% of (X + Y ) = 5 % of X + 75 % Y

50X + 50Y = 5X + 75Y

50X - 5X = 75Y - 50Y

X(50-5) = Y(75-50)

X/Y = 75-50/50-5 = 25/45 = 5/9

Instead of solving using the weighted average formula, we can directly get the final result by using the alligation method.

The alligation method is just a shortcut to bypass above mentioned steps and to reach the final result in a single calculation.

Kindly refer to the below figure for further reference. Write the final average value in between both the averages and take the diagonal difference. when you take the diagonal difference, always calculate the greater value - the smaller value.

X/Y = the value below quantity X in the figure/ Value below quantity Y in the figure.





Using the above mentioned alligation method, we can say that \(\frac{ x}{y} = \frac{25}{45} = \frac{5}{9}
\)
X = \(\frac{5}{14 }* 60 = \frac{150}{7} \)= 21.43

Option B is the right answer.

Thanks,
Clifin J Francis,
GMAT QUANT SME
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