Problem Solving

Q. From a bag containing 12 identical blue balls, y identical yellow balls, and no other balls, one ball will be removed at random. If the probability is less than 2/5 that the ball removed will be blue, what is the least number of yellow balls that must be in the bag?

A. 17
B. 18
C. 19
D. 20
E. 21

Ans: C

Probability of blue = number of blue / total = 12 / 12+y.

If p(blue) is less than 2/5, then 12 / 12+y < 2/5.
Cross multiply by 5(12+y). You can do that, because y is obviously positive, so no need to flip the sign because of multiplying by negative.

12*5 < 2(12+y)
60 < 24+2y
36 < 2y
18<y
So y has to be greater than 18 - the only answer that matches that is 19.

Alternative approach: reverse Plug in the answer choices. Each of the five answer choices is a possible number of yellow balls. Since we want the least, I'd probably start with the smallest one and work my way upwards. For each answer choice, we need to check if blue / total is less than 2/5 - otherwise the answer choice isn;t the right one, and we eliminate and move on:

A If y=17, then there are 12 blue + 17 yellow in the bag. P(blue) blue is therefore 12/12+17 = 12/29, which is greater than 12/30 = 2/5. Eliminate this answer choice and go higher.

B If y=18, then there are 12 blue + 18 yellow in the bag. P(blue) blue is therefore 12/12+18 = 12/30, which is exactly equal to 2/5. We want less than 2/5, so we need another yellow to drop the number of blues to below 2/5 of the total - Eliminate this answer choice, choose the next higher one and move on.

Thanks for the explanation Geva, I followed the first approach but must have made a mistake with the inequality sign (substituted '<' for '=') and hence ended up with 18 !

Geva@MasterGMAT wrote:Alternative approach: reverse Plug in the answer choices. Each of the five answer choices is a possible number of yellow balls. Since we want the least, I'd probably start with the smallest one and work my way upwards. For each answer choice, we need to check if blue / total is less than 2/5 - otherwise the answer choice isn;t the right one, and we eliminate and move on:

A If y=17, then there are 12 blue + 17 yellow in the bag. P(blue) blue is therefore 12/12+17 = 12/29, which is greater than 12/30 = 2/5. Eliminate this answer choice and go higher.

B If y=18, then there are 12 blue + 18 yellow in the bag. P(blue) blue is therefore 12/12+18 = 12/30, which is exactly equal to 2/5. We want less than 2/5, so we need another yellow to drop the number of blues to below 2/5 of the total - Eliminate this answer choice, choose the next higher one and move on.

pratyoosh wrote:Thanks for the explanation Geva, I followed the first approach but must have made a mistake with the inequality sign (substituted '<' for '=') and hence ended up with 18 !

People keep looking for "harder GMAT questions" in the hope of finding a question so hard they don't know how to solve. This is understandable - If you learn how to solve a question you didn't know how to solve before, you may get it right on the real test and score points.
The problem is that there's another silent killer, far more prevalent, and therefore dangerous, than the question you didn't know how to solve, and those are questions such as this one - the ones you are perfectly capable of solving, but still get incorrectly because you rushed on it and missed out on an important detail.

The GMAT isn't HARD, people. There's nothing in the test that is over your head. If you had a full day to solve the test, you would get each and every question right. The problem is that under two minutes per question, test-takers tend to switch off their brains and work in a panic to get the question out of the way as soon as possible.

The solution is to keep a cool head, and focus on accuracy, not only speed. Read the question carefully before diving into the solution. Make a note of every potential problem/sticky point in the question (e.g. less than 2/5). Above all, THINK before you act. Decide what you need to do to solve the question, then do it, calmly, collectedly, addressing the sticky points you noted before. Think like a test writer, not like a test solver - why did whoever wrote the question use "less" here?

Getting off my soapbox now.

Agreed. I have noticed its the silly mistakes where I am getting caught out more often, and need to work on it. Thanks for your help once again.

Geva@MasterGMAT wrote:

pratyoosh wrote:Thanks for the explanation Geva, I followed the first approach but must have made a mistake with the inequality sign (substituted '<' for '=') and hence ended up with 18 !

People keep looking for "harder GMAT questions" in the hope of finding a question so hard they don't know how to solve. This is understandable - If you learn how to solve a question you didn't know how to solve before, you may get it right on the real test and score points.
The problem is that there's another silent killer, far more prevalent, and therefore dangerous, than the question you didn't know how to solve, and those are questions such as this one - the ones you are perfectly capable of solving, but still get incorrectly because you rushed on it and missed out on an important detail.

The GMAT isn't HARD, people. There's nothing in the test that is over your head. If you had a full day to solve the test, you would get each and every question right. The problem is that under two minutes per question, test-takers tend to switch off their brains and work in a panic to get the question out of the way as soon as possible.

The solution is to keep a cool head, and focus on accuracy, not only speed. Read the question carefully before diving into the solution. Make a note of every potential problem/sticky point in the question (e.g. less than 2/5). Above all, THINK before you act. Decide what you need to do to solve the question, then do it, calmly, collectedly, addressing the sticky points you noted before. Think like a test writer, not like a test solver - why did whoever wrote the question use "less" here?

Getting off my soapbox now.

as we know that probability of selecting blue ball is less than 2/5 = less than 40% (2 divided by 5. it means that the number of yellow balls must be greater than 60%
thus the number of yellow balls is greater than 12.
Lets take blue ball 12 and yellow 18 (answer choice B)= 12+18 = 30
18/30 = 60%
We have to choose the least value of yellow balls, choose answer choice C as 19 yellow balls is slightly greater than 60%.