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

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.

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..

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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.

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

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