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A solution consists of only water and alcohol such that the ratio of alcohol to water in the solution is 7:3. How much amount of water should be added to the solution (in milliliters) so that the resulting solution contains 60% alcohol?
1. Total quantity of the resulting solution is 350 milliliters.
2. The original solution contains 10.5 milliliters of alcohol for every 4.5 milliliters of water.
OA A.
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A solution consists of only water and alcohol such that the
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fskilnik@GMATH
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In the solution below, the unit considered is always milliliters.AAPL wrote:e-GMAT
A solution consists of only water and alcohol such that the ratio of alcohol to water in the solution is 7:3. How much amount of water should be added to the solution (in milliliters) so that the resulting solution contains 60% alcohol?
1. Total quantity of the resulting solution is 350 milliliters.
2. The original solution contains 10.5 milliliters of alcohol for every 4.5 milliliters of water.
Let´s use the k technique, one of the best tools of our method when dealing with ratios/proportions!
$$\left\{ \matrix{
\,{\rm{water}}\,\,\left( w \right)\,\,\, = \,\,3k \hfill \cr
\,{\rm{alcohol}}\,\,\left( a \right) = 7k \hfill \cr} \right.\,\,\,\,\,\,\left( {k > 0} \right)\,\,\,\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\,\,\left\{ \matrix{
\,{\rm{water}}\,\,\left( w \right)\,\,\, = \,\,3k + x \hfill \cr
\,{\rm{alcohol}}\,\,\left( a \right) = 7k \hfill \cr} \right.\,\,\,\,\,\,\,{\rm{such}}\,\,{\rm{that}}\,\,\,\,\,\,{{7k} \over {10k + x}} = {3 \over 5}\,\,\,\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{{\rm{cross - multiply}}} \,\,\,\,\,\,5k = 3x\,\,\,\,\,\left( * \right)$$
$$? = x$$
$$\left( 1 \right)\,\,\,10k + x = 350\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,6x + x = 350\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\,\,\,\,\,\,\,$$
$$\left( 2 \right)\,\,{a \over w} = {{10.5} \over {4.5}}\,\,\left( { = {{105} \over {45}} = {7 \over 3}} \right)\,\,\,{\rm{already}}\,\,{\rm{known}}\,\,{\rm{pre - statements}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.\,\,\,$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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Scott@TargetTestPrep
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We can let the amount of water to be added be w and the original amount of alcohol and water be 7x and 3x, respectively. We can create the equation:AAPL wrote:e-GMAT
A solution consists of only water and alcohol such that the ratio of alcohol to water in the solution is 7:3. How much amount of water should be added to the solution (in milliliters) so that the resulting solution contains 60% alcohol?
1. Total quantity of the resulting solution is 350 milliliters.
2. The original solution contains 10.5 milliliters of alcohol for every 4.5 milliliters of water.
(3x + w)/(7x + 3x + w) = 6/10
(3x + w)/(10x + w) = 3/5
5(3x + w) = 3(10x + w)
15x + 5w = 30x + 3w
2w = 15x
We need to determine the value of w. We see that w = 15x/2 or x = 2w/15. Therefore, if we know one of the two variables, then we know the other.
Statement One Only:
Total quantity of the resulting solution is 350 milliliters.
This means 10x + w = 350. Since x = 2w/15, we have:
10(2w/15) + w = 350
4w/3 + w = 350
4w + 3w = 1050
7w = 1050
w = 150
Statement one alone is sufficient.
Statement Two Only:
The original solution contains 10.5 milliliters of alcohol for every 4.5 milliliters of water.
This means original amount of alcohol and water are 10.5y and 4.5y, respectively. Therefore, we have:
7x = 10.5y and 3x = 4.5y
Either way, we have x = 1.5y. However, since y can be any positive number (for example, if y = 2, then x = 3 and if y = 4, then x = 6), we can't determine a unique value of x, and therefore, we can't determine the value of w. Statement two alone is not sufficient.
Answer: A
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