A jar contains only black marbles and white marbles. If two thirds of the marbles are black, how many white marbles are

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Gmat_mission: Legendary Member; Posts: 1622; Joined: Thu Mar 01, 2018 7:22 amFollowed by: 2 members

by Gmat_mission » Thu Oct 01, 2020 7:05 am

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A jar contains only black marbles and white marbles. If two thirds of the marbles are black, how many white marbles are in the jar?

(1) If two marbles were to be drawn, simultaneously and at random, from the jar, there is a 5/12 probability that both would be black.

(2) If one white marble were removed from the jar, there would be a 1/4 probability that the next randomly-drawn marble would be white.

Answer: D

Source: Veritas Prep

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Source: Beat The GMAT — Data Sufficiency | Thu Oct 01, 2020 7:05 am

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

Re: A jar contains only black marbles and white marbles. If two thirds of the marbles are black, how many white marbles

by deloitte247 » Thu Oct 08, 2020 8:12 am

Let total marbles = x
Target question: how many white marbles are in the jar?
Therefore, black marbles = 2x/3 and white marbles = 1 - (2x/3) = x/3

Statement 1: If two marbles were to be drawn, simultaneously and at random, from the jar, there is a 5/12 probability that both would be black.
Probability of ball drawn both to be black = 5/12
$$\left(\frac{\frac{2x}{3}}{x}\right)\cdot\left(\frac{\left(\frac{2x}{3}-1\right)}{x-1}\right)=\frac{5}{12}$$
$$\left(\frac{2x}{3}\cdot\frac{1}{x}\right)\cdot\left(\frac{2x-3}{3}\right)\cdot\left(\frac{1}{x-1}\right)=\frac{5}{12}$$
$$\left(\frac{2}{3}\right)\cdot\left(\frac{2x-3}{3x-3}\right)=\frac{5}{12}$$
$$\frac{4x-6}{9x-9}=\frac{5}{12}$$
$$12\left(4x-6\right)=5\left(9x-9\right)$$
$$48x-72=45x-45$$
$$48x-45x=-45+72$$
$$3x=27$$
$$x=\frac{27}{3}=9$$
From the question stem,
$$White=\frac{x}{3}=\frac{9}{3}=3$$
White = 3 marbles. Therefore, statement 1 is sufficient.

Statement 2: If one white marble were removed from the jar, there would be a 1/4 probability that the next randomly-drawn marble would be white.
If (white marbles - 1) probability of next ball being white = 1/4
$$\frac{\left(\frac{x}{3}-1\right)}{x-1}=\frac{1}{4}$$
$$\frac{\left(x-3\right)}{3}\cdot\frac{1}{x-1}=\frac{1}{4}$$
$$\frac{x-3}{3x-3}=\frac{1}{4}$$
$$4\left(x-3\right)=3x-3$$
$$4x-12=3x-3$$
$$4x-3x=-3+12$$
$$x=9$$
From the question stem,
$$White=\frac{x}{3}=\frac{9}{3}=3$$
White = 3 marbles. Therefore, statement 2 is sufficient.

Conclusively, since each statement alone is SUFFICIENT, answer = option D.

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