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# Milk and cream are combined to make a 60-liter mixture that is 50 perc

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Math Expert
Joined: 02 Sep 2009
Posts: 65187
Milk and cream are combined to make a 60-liter mixture that is 50 perc  [#permalink]

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22 May 2020, 11:01
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35% (medium)

Question Stats:

70% (02:24) correct 30% (02:14) wrong based on 96 sessions

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Milk and cream are combined to make a 60-liter mixture that is 50 percent butterfat. If the milk is 5 percent butterfat and the cream is 75 percent butterfat, approximately how many liters of milk are in the mixture?

A. 20.00
B. 21.43
C. 30.00
D. 38.57
E. 42.75

PS20433

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Milk and cream are combined to make a 60-liter mixture that is 50 perc  [#permalink]

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Updated on: 23 May 2020, 04:58
Top Contributor
Bunuel wrote:
Milk and cream are combined to make a 60-liter mixture that is 50 percent butterfat. If the milk is 5 percent butterfat and the cream is 75 percent butterfat, approximately how many liters of milk are in the mixture?

A. 20.00
B. 21.43
C. 30.00
D. 38.57
E. 42.75

PS20433

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

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Originally posted by BrentGMATPrepNow on 22 May 2020, 17:07.
Last edited by BrentGMATPrepNow on 23 May 2020, 04:58, edited 1 time in total.
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Re: Milk and cream are combined to make a 60-liter mixture that is 50 perc  [#permalink]

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22 May 2020, 20:59
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
---------------------------
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.
---------------------------
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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Milk and cream are combined to make a 60-liter mixture that is 50 perc  [#permalink]

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23 May 2020, 03:55
Quote:
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 milk 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

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?

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Re: Milk and cream are combined to make a 60-liter mixture that is 50 perc  [#permalink]

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23 May 2020, 05:02
Top Contributor
WBogey wrote:
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?

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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Re: Milk and cream are combined to make a 60-liter mixture that is 50 perc  [#permalink]

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23 May 2020, 07:10
BrentGMATPrepNow wrote:
WBogey wrote:
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?

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?

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!
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Re: Milk and cream are combined to make a 60-liter mixture that is 50 perc  [#permalink]

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23 May 2020, 14:17
1
1
B?
Milk =x
Cream = 60-x

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

Solving: x = 150/7 = 21.xx
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Re: Milk and cream are combined to make a 60-liter mixture that is 50 perc  [#permalink]

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30 May 2020, 21:16

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.

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Re: Milk and cream are combined to make a 60-liter mixture that is 50 perc   [#permalink] 30 May 2020, 21:16