One might think:

P(blue) = 12/(12+y)< 2/5
so (12+y)/12 < 5/2
12 + y < 30
y < 18

However, a major rule of inequalities is that we should never multiply by a denominator containing the unknown y, although, in this case as P can never be negative, it does not apply.

Nonetheless, proceed as follows and all will become evident:

P(blue) = 12/(12+y)< 2/5
so 12/(12+y)- 2/5 < 0

Now test given answers (by substitution for y):
A)17 -> 12/(12 + 17) - 2/5 = 0.138 > 0
B)18 -> 12/(12 + 18) - 2/5 = 0
C)19 -> 12/(12 + 19) - 2/5 = -... < 0
D)20 -> 12/(12 + 20) - 2/5 = -... < 0
E)21 -> 12/(12 + 21) - 2/5 = -... < 0

So you see, the first method was a solution for P(blue) = 0
instead of P(blue) < 0.

If the question had stated P(blue) <=0, then y = 18 would be correct.

So the answer is y = 19.

I hope this helps.

j_shreyans wrote: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 removed ball will be blue, what is the leat no. of yellow balls that must be in a bag?

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

Let T = the total number of marbles.
Since the probability of selecting a blue marble is less than 2/5, the 12 blue marbles in the bag must constitute less than 2/5 of the total:
12 < (2/5)T
30 < T.

Since the total number of marbles must be at least 31, the least number of yellow marbles that must be added to the 12 blue marbles is 19.

The correct answer is C.

I wonder if this would be correct (the method, I mean):

P(blue) = 12/(12+y)< 2/5
so (12+y)/12 > 5/2 "Because inverting the fractions, reverses the inequality sign"
12 + y > 30
y > 18

So y = 19 is the smallest given answer.

Could someone please comment on the "inversion" technique above?

Thanks.