If a 30% alcohol solution is combined with a 50% alcohol solution to

Question

If a 30% alcohol solution is combined with a 50% alcohol solution to produce 10 liters of a 45% alcohol solution, how many liters of the 30% solution were used?

A. 2 liters
B. 2.5 liters
C. 2.7 liters
D. 3 liters
E. 3.2 liters

Archit3110 · Answer

use allegiation method
we get
30: 45: 50
15:5
3:1 
4x alcohol = 10 ltrs
x= 2.5 ltr
IMO B

moropa2018 · Answer

Archit3110  Can you please elaborate on this allegiation method?

VeritasPrepErika · Answer

The method that  Archit3110  used has a bunch of different names, but it's one of the best strategies out there for weighted averages! A really good way to use it is to visualize the concentrations of our original mixtures (30% and 50%) and the concentration of our final mixture (45%) on a number line:

<--30%---------------45%-----50%-->

Notice how the resulting concentration is much closer to 50% — this indicates that there is more of the 50% alcohol mixture than of the 30% alcohol mixture in the final 45% mixture.

At this point, we can definitively rule out E as an answer, as it would means we'd have more of the 30% alcohol mixture than the 55% alcohol mixture. But we can actually do more than that using ratios! 45% is 15% away from 30% vs. 5% away from 50%. In other words, the final concentration is 3 times as close to 50% than it is to 30% (as 15:5 = 3:1). This means that  there is 3 times as much of the 50% alcohol mixture than of the 30% alcohol mixture in the final mixture .

So the final alcohol mixture is 3 parts of the 50% mixture to 1 part of the 30% mixture, giving 4 parts total. There are 10 liters of the final 45% mixture in total, so each of the four parts is 10/4 = 2.5 liters. So the 1 part of the 30% mixture used is 2.5 liters, or answer choice B.

As shown by some of the other users here, this particular problem is pretty quick to solve using weighted averages equations, but this number line/ratio method is a great trick 1) if you aren't as comfortable building out the equations, 2) if you want to logically estimate/confirm your answer, or 3) when the numbers are a little uglier and harder to calculate as decimals.

Shef08 · Answer

Thank you Karishma for the easy solution, I am struggling to understand when the 10 litre mixture has 45% alcohol. So shouldn’t we be doing 45/100*10= 4.5 litres alcohol content and then it would be 4.5/4. I know I am missing out something but can’t understand what is it. Please can you help in this..

Posted from my mobile device

VeritasPrepErika · Answer

Shef08  I'm not Karishma (she's a colleague of mine at Veritas Prep), but I'm pretty sure this message is meant for me...

You're absolutely right that there are 4.5 liters of  pure alcohol  in the final mixture. However, our math shows that the 30% mixture makes up 1 of 4 parts of the  total volume  of the final mixture. This is why we take 1/4 of the 10 total liters to get our answer. The number line analysis that we already did  uses the concentrations  of the different mixtures to come up with the ratios of the total volumes of the mixtures, which is why we don't need to consider concentration in the final step — it's already been incorporated in the math.

Let's do a quick unit analysis on the scenario you're presenting to prove to ourselves that the math doesn't quite add up — if we took the 4.5 liters of pure alcohol, then took 1/4 of it, this would tell us that we need 1.125 liters of pure alcohol from the 30% mixture (again, this is not actually true). However, the 30% mixture  isn't  pure alcohol, so it doesn't make sense that we would only add 1.125 liters of it! This should indicate to us that we've gone off-track somewhere in our work.

ScottTargetTestPrep · Answer

Let x be the amount (in liters) of the 30% mixture and y be the amount (in liters) of the 50% mixture. We can create the equations:

0.3x + 0.5y = 0.45(x + y)

30x + 50y = 45x + 45y

5y = 15x

y = 3x

And

x + y = 10

Substituting, we have:

x + 3x = 10

4x = 10

x = 10/4 = 5/2 = 2.5

Alternate Solution:

We have x liters of a 30% mixture, and we add (10 - x) liters of a 50% mixture to it, yielding 10 liters of a 45% mixture. We can summarize this in an equation as:

0.30x + 0.50(10 - x) = 0.45(10)

30x + 50(10-x) = 45(10)

30x + 500 - 50x = 450

50 = 20x

2.5 = x

Answer: B

Bunuel · Answer

If a 30% alcohol solution is combined with a 50% alcohol solution to produce 10 liters of a 45% alcohol solution, how many liters of the 30% solution were used?

A. 2 liters
B. 2.5 liters
C. 2.7 liters
D. 3 liters
E. 3.2 liters

Assume there are  x  liters of the 30% alcohol solution. This implies there are  10-x  liters of the 50% alcohol solution.
 
Next, equate the total alcohol content in the 10-liter mixture:  0.3x + 0.5(10 - x) = 4.5 . Solving this equation yields  x = 2.5  liters.

Answer: B

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