A ball is randomly selected from a box containing white balls and

Question

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

* kudos for all correct solutions

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

* kudos for all correct solutions

BrentGMATPrepNow · Accepted Answer

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 t arget 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

Cheers,
Brent

vitaliyGMAT · Answer

Hi

We are given (w - # of white blls, b - # of black balls, x - # of white balls to be added):

\frac{w}{w+b} = \frac{4}{5}

w = 4b  or  w:b = 4:1

and

\frac{w + x}{w + b + x} = \frac{7}{8}

\frac{4b + x}{5b + x} = \frac{7}{8}

x = 3b

Now let's lok at our options:

(1) does not give us any additional information, we already know that w:b is 4:1. Insufficient.

(2) w = b + 27, combinig this with w=4b we get b=9 and x=27. Sufficient.

Answer B

chetan2u · Answer

Hi,

Let the ratio be  \frac{4x}{5x} ...
Add y white balls...
 \frac{4x+y}{5x+y}=\frac{7}{8} ..
 32x+8y=35x+7y.......3x=y ..
So y has to be multiple of 3.

Let's see the statements.
(1) The ratio of white balls to black balls is 4:1
 the probability 4/5 itself means the ratio to be 4:1 ..
Nothing new..
Insufficient

(2) There are 27 more white balls than black balls

So 4x-x=27.... x=9..
Substitute in 3x=y...So y = 3*9=27
B

Edited..

dorane · Answer

Ans. A tells me nothing new, so, just noodling with multiples of 4 & differences of 27 got me, eventually, to 36 & 9, whose sum is a multiple of 5, so there's the 4/5 probabilty . . . but Bunuel, et. al. will no doubt have a more elegant way. Also to get to the required 7/8, you will add . . . 27! This stuff is  deep!

BrentGMATPrepNow · Answer

It took my quite a while to find the issue, but I believe there's a problem with your solution (highlighted above in red)

In your expression 4x/5x, the 4x represents the number of white balls and 5x represents the total number of balls
So, if we add y white balls, 4x+y represents the NEW number of white balls and  5x+y  represents the NEW total number of balls
Your solution has just  5x  as the NEW total number of balls

My solution (above this post) shows that the given numbers yield integer values for the numbers of balls.

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

ScottTargetTestPrep · Answer

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

Ross2018 · Answer

Great post thank you !

shekyonline · Answer

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

* kudos for all correct solutions

(1) The ratio of white balls to black balls is 4:1

ratio 4:1
the no of balls can be as below:
w : b
4 1
 8 2 - in this case if we add 6 w balls ratio the probability of randomly drawing a white ball is 7/8.
12 3 - in this case if we add 9 w balls ratio the probability of randomly drawing a white ball is 7/8.
....... 
.......
 
 36 9

no definite answer.

NS

(2) There are 27 more white balls than black balls

And, .  If the probability of randomly selecting a white ball is 4/5. .
definite no of w, b balls are known.

w : b

36 9- in this case if we add  27  w balls ratio the probability of randomly drawing a white ball is 7/8.

suff.
 
 Ans. B

umabharatigudipalli · Answer

Hi,

I have a question here. We already know that w:b=4:1 - agreed. But in statement 2, we use w=4b for the solution. As w=4b was also given in statement 1, I landed up opting choice C. I hope you understood my question. Please help

Thanks,
Uma

Thanks, Uma

Nunuboy1994 · Answer

Tricky

You don't need statement 1 because it is already implied in the question stem

B

rulingbear · Answer

Statement 1 says nothing new, i.e 4b= w.

Statement 2 says  w-b  = 27, combined with  4b=w  (not exclusively from 1)  b = 9,  w = 36 and  b+w  = 45. Also makes sense, multiple of 5, just need to add  3b  (27) worth of  w  to change the probability from 4/5 to 7/8.

Hence B.

BrentGMATPrepNow · Answer

Hi Uma,

A key point here is that w:b = 4:1 is exactly the same as w = 4b
Here's why.

Take: w:b = 4:1
Rewrite as w/b = 4/1
Simplify: w/b = 4
Multiply both sides by b to get: w = 4b

Does that help?

Cheers,
Brent

Fdambro294 · Answer

If you understand the concepts of ratios and that the probability of pulling a white ball out of a box containing only white and black balls JUST depends on the relative amounts of such balls, you could answer this question rather quickly.

We are given that the probability of drawing a white ball out of a box that contains just white and black balls is = 4/5

From a theoretical probability perspective, this means that if we organized the unknown total of balls on the floor, for every 4 white balls we can gather there will be exactly 1 black ball that can be grouped with it.

To answer the question of how many white balls must be added to the box, we are looking for an actual total number of white balls: not a relative value.

In order to Determine how many actual white balls we need to add, we need to have an “actual” value regarding one part of the “ratio”

Either:

(1) the total actual count of balls

(2) the total count of white or black balls alone

Or

(3) some other relationship that can help us determine either of the above.

Statement 1: this information is already known and given to us in the question stem, as described above.

Statement 2: if we know that there are 27 more white balls than black balls, we can determine how many actual balls of each color are in the box. The statement is therefore sufficient and you can answer B without doing any calculations.

For each group of 5 balls that we laid out on the floor, there were 3 more white balls than black balls. (i.e., 4 white and 1 black)

If there are 27 more white balls IN TOTAL, then we must be able to create 9 groups of these balls (4 white and 1 black)

We can determine everything about the actual number of balls in the box and could answer how much would be needed to change the Ratio of white - to - black and alter the probability of drawing a white to (7/8)

B - s2 sufficient alone

Note: there are 36 white, 9 black, and 45 total balls in the box.

We would have to add +27 white balls to get the probability of picking a white ball to = (7/8)

Posted from my mobile device

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A ball is randomly selected from a box containing white balls and

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