Data Sufficiency

A ball is randomly selected from a box containing white balls and black balls only. If the probability of randomly selecting a white ball is 4/5, how many white balls must be added to the box so that the probability of randomly drawing a white ball is 7/8?

(1) The ratio of white balls to black balls is 4:1
(2) There are 27 more white balls than black balls

Source: GMAT Prep Now
Difficulty level: 700

Answer: B

Brent@GMATPrepNow wrote:A ball is randomly selected from a box containing white balls and black balls only. If the probability of randomly selecting a white ball is 4/5, how many white balls must be added to the box so that the probability of randomly drawing a white ball is 7/8?

(1) The ratio of white balls to black balls is 4:1
(2) There are 27 more white balls than black balls

Here's an approach that uses two variables....

Target question: How many white balls must be added to the box so that the probability of randomly drawing a white ball is 7/8?

Given: The probability of randomly selecting a white ball is 4/5
Let W = number of white balls in the box
Let B = number of black balls in the box
So, the TOTAL number of balls = W+B
If the probability of selecting a white ball = 4/5, then we can write: W/(W+B) = 4/5
Cross multiply to get: 4(W+B) = 5W
Expand: 4W + 4B = 5W
Simplify: 4B = W

Statement 1: The ratio of white balls to black balls is 4:1
In other words: W/B = 4/1
Cross multiply to get 4B = W
This MATCHES the given information. In other words, statement 1 provides no new information.
As such, statement 1 is NOT SUFFICIENT

Statement 2: There are 27 more white balls than black balls
In other words, W = B + 27
Now take 4B = W and replace W with B+27, to get: 4B = B + 27
Solve to get: B = 9
So, there are presently 9 blacks in the box, which means there are 36 white balls in the box.
Now that we know exactly how many white and black balls are in the box, we can just keep add white balls to the box until P(selecting a white ball) = 7/8
So, we COULD answer the target question with certainty.
As such, statement 2 is SUFFICIENT

ASIDE: For "fun," let's determine how many white balls we need to add.
We presenty have 36 white balls and 9 black balls for a total of 45 balls.
Let's add x white balls to the box
So, 36+x = the new number of white balls in the box
And 45+x = TOTAL number of balls in the box
We want P(white ball) = 7/8
We can write: (36+x)/(45+x) = 7/8
Cross multiply to get: 8(36+x) = 7(45+x)
Expand: 288 + 8x = 315 + 7x
Solve: x = 27
So, we must add 27 white balls to the box so that P(white ball) = 7/8

Answer: B

ASIDE: Although it may take a little longer to do so, I prefer to use two variables when trying to solve these kinds of ratio/proportion questions. Otherwise, I end up forgetting what the variables stand for.

Cheers,
Brent

Brent@GMATPrepNow wrote:A ball is randomly selected from a box containing white balls and black balls only. If the probability of randomly selecting a white ball is 4/5, how many white balls must be added to the box so that the probability of randomly drawing a white ball is 7/8?

(1) The ratio of white balls to black balls is 4:1
(2) There are 27 more white balls than black balls

We are given that in a box containing only white and black balls, the probability of selecting a white ball is 4/5 and thus the probability of selecting a black ball is 1/5. We must determine how many white balls must be added to the box so the probability of drawing a white ball is 7/8.

Statement One Alone:

The ratio of white balls to black balls is 4:1.

This means that for some positive integer x, there are 4x white balls and x black balls in the box. Thus, there are a total of 5x balls in the box and the probability of selecting a white ball is (4x)/(5x) = 4/5. However, since we already know that the probability of selecting a white ball is 4/5, statement one does not provide any new information and thus is not sufficient to answer the question.

Statement Two Alone:

There are 27 more white balls than black balls.

We can let b = the number of black balls and w = the number of white balls, and thus:

w = b + 27

Furthermore we know:

w/(b+w) = 4/5

(b + 27)/(b + b + 27) = 4/5

(b + 27)/(2b + 27) = 4/5

5(b + 27) = 4(2b + 27)

5b + 135 = 8b + 108

27 = 3b

b = 9

Since b = 9 and w = 9 + 27 = 36, we can determine the number of white marbles that must be added to the box so the probability of selecting a white marble is 7/8.

Answer: B