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A spirit and water solution is sold in a market. The cost pe

by gmatter2012 » Mon Jul 09, 2012 7:48 am
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
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by Anurag@Gurome » Mon Jul 09, 2012 7:51 am
gmatter2012 wrote: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?
Fraction of spirit by volume in a solution of 1 liter of spirit and 1 liter of water = 1/(1 + 1) = 1/2
Fraction of spirit by volume in a solution of 1 liter of spirit and 2 liters of water = 1/(1 + 2) = 1/3

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.
Last edited by Anurag@Gurome on Mon Jul 09, 2012 8:29 am, edited 1 time in total.
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by Bill@VeritasPrep » Mon Jul 09, 2012 7:51 am
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.
Last edited by Bill@VeritasPrep on Mon Jul 09, 2012 8:37 am, edited 1 time in total.
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by gmatter2012 » Mon Jul 09, 2012 8:24 am
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.
Last edited by gmatter2012 on Mon Mar 17, 2014 10:03 pm, edited 1 time in total.
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by Anurag@Gurome » Mon Jul 09, 2012 8:30 am
Hello gmatter2012,

The solution is correct.
I missed the phrase "per liter".
Edited my reply. See if that helps.
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by Bill@VeritasPrep » Mon Jul 09, 2012 8:35 am
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!
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by gmatter2012 » Mon Jul 09, 2012 8:53 am
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
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by Jeff@TargetTestPrep » Mon Dec 18, 2017 10:49 am
gmatter2012 wrote: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
When a solution has 1 liter of spirit and 1 liter of water, the solution has 2 liters of liquid with a spirit concentration of 1/2. Since the cost is 50 cents for 2 liters, the cost per liter is 25 cents. Now we can create an equation to find the proportional constant k:

(1/2) x k = 25

k = 50

Now we need to determine the cost of a solution that has 1 liter of spirit and 2 liters of water. We see that the solution has 3 liters of liquid with a spirit concentration of 1/3. Let's first find the cost per liter, c:

(1/3) x k = c

(1/3) x 50 = c

50/3 = c

Since the cost per liter is 50/3 cents and there are 3 liters of solution, the cost of the solution is:

50/3 x 3 = 50 cents

Answer: C

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