Two kinds of Vodka are mixed in the ratio 1:2 and 2:1 and they are sold fetching the profit 10% and 20% respectively. If the vodkas are mixed in equal ratio and the individual profit percent on them are increased by 4/3 and 5/3 times respectively, then the mixture will fetch the profit of

A. 18%
B. 20%
C. 21%
D. 23%
E. Cannot be determined
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Mixture 1 and 2 having individual profit % of 10 and 20 respectively.
individual profits of vodka A for mixture 1 = 1/3 * 10 = 3.33
individual profits of vodka B for mixture 1 = 2/3 * 10 = 6.77

Individual profits for A and B are increased 4/3 and 5/3 times.

Means profits of mixtures 1 and 2 will increase too.

Increased profits of Vodka A for mixture 1 = 4/3 * 3.33 = 4.44
Increased profits of Vodka B for mixture 2 = 5/3 * 6.77 = 10.55
total increased profits for mixture 1 = 14.99

Similarly
total increased profits for mixture 2 = 24.42

using allegation formula

Va: Vb = 1:1 = [24.42 - (profit of mixture)] / [(profit of mixture)- 14.99]

hence profit of mixture = 19.705 approx = 20%

B

I did not do any calculation on this one.

There are two ingredients in V1 and V2 and we know from the stem that the first ingredient is expensive than the second.

Case I - V1 profit is 10% when mix is 1:2 => profit from second ingredient is more
Case II - V2 profit is 20% when mix is 2:1 => profit from first ingredient is more. Decreasing the first ingredient results in less profit - Case I. Hence first ingredient is more expensive.

If V1 and V2 are mixed in equal ratio then the profit will be closer to 20%. Even though we have equal quantity of each ingredient but first ingredient is more expensive.

Now the trick. When the profit on V1 is increased by 4/3 and V2 is increased by 5/3. This is equivalent to increasing the quantity of V2 as 1.33 < 1.67. It means the weighted mean will move even closer to 20%

Hence the final mean will be almost 20.

Hi Bunuel,

I am not quite familiar with the allegation formula amit2k9 has used to solve this problem. Could you please show a simpler way of solving this problem?

Thanks,
Ishan

Hi Bunuel, When we say Increased By -- Don't we add the values i.e. previous + current... in this case if the profit increased by 4/3 times the previous value shouldn't we add 40 + 30.. i.e. increased by 40% means 70%..

If the question had been increased to 4/3 times then we could have taken 40% i.e. from 30% it has changed to 40%.

Please help!!!!

since we do not know in what ratio the first 1:1 and second 1:1 mixtures are mixed we will not be able to determine the resultant profit.
Hence, the option is (E)

Hi Bunuel,
I got your method except how you got 30%. Will you explain, please, how you have solved and got 30%?
Thanks in advance.

Bunuel, I believe there is a wording problem with the question. It should be Increased TO and not BY.

Kindly confirm.
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I think the above solution is wrong. The method here is assuming that profit is the same for both Vodka A and Vodka B. If you want to do allegation you can do the following:


1. Allegation says that \(\frac{\text{cheaper quantity}}{\text{greater quantity}} = \frac{\text{profit of greater - average profit}}{\text{average profit - profit of cheaper}}\)
Let the vodkas be A and B respectively. Since we know 2:1 has a greater profit than 1:2, we know that A must be the greater quantity.
1:2 => profit 10% gives \(\frac{x - 10}{10-y} = \frac{2a}{1a}\) where \(a\) is an integer. You can break this down further into the two equations\(x-10 = 2a\) and \(10-y = 1a\).
2:1 => profit 20% gives \(\frac{x - 20}{20-y} = \frac{1a}{2a}\). You can also break this down further into the two equations \(x-20=a\) and \(20-y=2a\). Now you have 3 variables and 4 equations so you can solve for a,x, and y. You get that x = 30 and y = 0.

2. since vodkas A and B are increased by 4/3 and 5/3 respectively, you get that A' = 40 and B' = 0.

3. Using allegation once again, you have \(\frac{1}{1} = \frac{40-x}{x-0} \implies x = 40-x \implies x = 20.\).

Hence B.