Word Problem - A solution consists of only water and alcohol such that

(in 350 ml --> 245 ml is alcohol + 105ml is water(given ratio 7:3)
let w be the quantity of water added to be 40% water(or 60% alcohol) in final solution
thus 105 +w =40%(350+w)
thus w can be calculated
suff

(2) ony gives ratio 10.5/4.5= 7/3 which is given in question
No idea of volume...
insuff

Ans A

Solution

Step 1 & Step 2: Understanding the Question statement and Drawing Inferences

Given info:

• Ratio of alcohol to water in original solution = 7:3
• Let the quantity of original solution be x ml.

⇒ Quantity of alcohol = \(\frac{7\mathrm x}{10}\)
⇒ Quantity of water = \(\frac{3\mathrm x}{10}\)

To find:

• We have to find the quantity of water to be added to the solution, such that the resulting solution contains 60% of alcohol.
• Let the quantity of water that is to be added to the solution be y millilitres.
• So total quantity of final solution will be x+y millilitres
• Now since only water is being added we can infer that the quantity of alcohol in the solution will not change.
• So quantity of alcohol in the final solution will be
• Percentage of alcohol in the final solution should be 60%

⇒ \(\frac{\frac{7\mathrm x}{10}}{(\mathrm x+\mathrm y)}\ast100=60\)
⇒ \(\frac{\frac{7\mathrm x}{10}}{(\mathrm x+\mathrm y)}=\frac35\)
Multiplying both sides of the equation by 5, we get
⇒ \(\frac{\left(\frac{7\mathrm x}{10}\right)\ast5}{(\mathrm x+\mathrm y)}=3\)
⇒ \(\frac{\left(\frac{7\mathrm x}2\right)}{(\mathrm x+\mathrm y)}=3\)
⇒\(3.5x = 3x+3y\)
⇒\(0.5x = 3y\) ...(Equation 1)

• Since we do not know the value of x, we will not be able to determine the value of y.
• So we need a relation in x and y, or value of x or y, to be able to determine the amount of water to be added to the solution.

Step 3: Analyze statement 1 independently

• Total quantity of the resulting solution is 350ml.
• Total quantity of the resulting solution = x+y ml (Already established above)

⇒ \(x+y=350ml\) ...(Equation 2)

• Since we have another relation in x and y, we will be able to determine the value of y, and thus the amount of water to be added to the solution.
• Hence statement 1 is sufficient to answer the question.

Step 4: Analyze statement 2 independently

• The original solution contains 10.5ml of alcohol for every 4.5ml of water.
• So, \(\frac{\mathrm{Alcohol}}{\mathrm{Water}}=\frac{10.5}{4.5}=\frac73\)
• The ratio of alcohol to water from this statement comes out to be 7:3, which is the same as given to us in the question stem.
• So we do not get any additional information from which we can determine the values of x or y.
• Hence statement 2 is not sufficient to answer the question.

Step 5: Analyze the two statements together

• Since from statement 1, we are able to arrive at a unique answer, combining and analysing statements together is not required.
• Hence the correct answer is Option A

Thanks,
Saquib
Quant Expert
e-GMAT

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Did it in 2:30 minutes (i guess way over time)
a/w = 7/3 (total parts 10)
we need a/w = 6/4

1) total ML is 350 --> easily we will know 350/10 = 35, divided each into 6/4 ratios. So exact amount of a/w can be achieved.

2) ratio of a/w= 10.5/ 4.5 --> great, but this would apply to each part of the mixture. How do we know the total milliliters? Hence insuff.

Ans A

In the solution below, the unit considered is always milliliters.

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.

A - sufficient
B - mistakanely considered it as sufficient - missed that its the same information as metnioned in the question - no new information

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:

(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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