Problem Solving

A spirit and water solution is sold in a market. The cost per liter of the solution is directly proportional to the part (fraction) of spirit (by volume) the solution has. A solution of 1 liter of spirit and 1 liter of water costs 50 cents. How many cents does a solution of 1 liter of spirit and 2 liters of water cost?

A. 13
B. 33
C. 50
D. 51
E. 52

1 part spirit, 1 part water = 50% spirit by volume. Cost = 50 cents.

1 part spirit, 2 parts water = 33% spirit by volume. The cost should be 33 cents. However, we now have three total liters, or 50% more solution. 33*1.5 = 50 cents.

I also got 33 cents initially but on reading this solution I got confused, So I needed expert opinion , Please see the solution below and advice accordingly ,

... this is a problem from Nova Math Bible...According to the book ,

Since the cost of each liter of the spirit water solution is directly proportionally to the part (fraction) of spirit the solution has, the cost per liter can be expressed as kf, where f is the fraction (part of) of pure spirit the solution has.
Now, each liter of the m liters of the solution containing n liters of the spirit (f = n/m) should cost kf = k(n/m). The m liters cost m "¢ k(n/m) = kn. Hence, the solution is only priced for the content of the spirit the solution has (n here). Hence, the cost of the two samples given in the problem must be the same since both have exactly 1 liter of spirit. Hence, the answer is (C), 50 cents.

Anurag@Gurome wrote:Hello gmatter2012,

The solution is correct.
I missed the phrase "per liter".
Edited my reply. See if that helps.

Ah, me too. Tricky!

Anurag@Gurome wrote: For a fraction of 1/2 the cost is 50 cents per 2 liters = 25 cents per liter
For a fraction of 1 the cost is 2*25 = 50 cents per liter
For a fraction of 1/3 the cost is 50/3 cents per liter = 50 cents per 3 liters

Hence, cost of the 3 liter solution is 50 cents.

The correct answer is C.

Ok got it but the part above is a bit tricky !!

Bill@VeritasPrep wrote:
1 part spirit, 1 part water = 50% spirit by volume. Cost = 50 cents.

1 part spirit, 2 parts water = 33% spirit by volume. The cost should be 33 cents. However, we now have three total liters, or 50% more solution. 33*1.5 = 50 cents.

@Bill: this way is also understood, thank you