A box contains some green marbles and exactly four red marbles. The probability of selecting a red marble is x%. If the number of green marbles is doubled, the probability of selecting one of the four red marbles from the box is (x − 15)%. How many green marbles are in the box before the number of green marbles is doubled?
At first:
Number of red marbles: R
Number of green marbles: 4
Probability of drawing a red marble: R / (R + 4) = x
After doubling the number of green marbles:
Number of red marbles: R
Number of greeen marbles: 8
Probability of drawing a red marble: R / (R + 8) = x - 0.15
--- solving this equation for x: R / (R + 8) + 0.15 = x
Setting the two equations equal to each other:
R / (R + 4) = R / (R + 8) + 0.15
Multiplying all terms by (R + 4) · (R + 8)
R · (R + 8) = R · (R + 4) + (0.15) · (R + 8) · (R + 4)
Multipling out:
R2 + 8R = R2 + 4R + 0.15(R2 + 12R + 32)
R2 + 8R = R2 + 4R + 0.15R2 + 1.8R + 4.8
Simplifying:
0 = 0.15R2 - 2.2R + 4.8
Multiplying by 100:
0 = 15R2 - 220R + 480
Dividing by 5:
0 = 3R2 - 44R + 96
Factoring:
0 = (3R + 8)(R - 12)
Answers: either -8/3 (impossible) or 12