A box contains 9 blue stones, 6 white stones and some black stones. A stone is selected at random, and the probability that it is black is 1/4. What is the total number of stones in the box?

Introduction / Context:
This is a basic probability and algebra question. We know the numbers of blue and white stones, but the number of black stones is unknown. The probability that a randomly selected stone is black is given, and we use this to find the total number of stones in the box.

Given Data / Assumptions:

• Number of blue stones = 9.
• Number of white stones = 6.
• Number of black stones is unknown, let it be k.
• Probability of drawing a black stone at random = 1/4.
• All stones are equally likely to be selected.

Concept / Approach:
Probability is defined as favourable outcomes divided by total outcomes. Here, black stones are favourable outcomes. We express the probability of drawing a black stone in terms of k and the total number of stones, and then equate it to 1/4 to find k. From that, we compute the total number of stones.

Step-by-Step Solution:
Let the number of black stones be k.Total stones in the box = 9 + 6 + k = 15 + k.Probability of selecting a black stone = k / (15 + k).This probability is given as 1/4, so set up the equation k / (15 + k) = 1 / 4.Cross multiply: 4k = 15 + k.Rearrange: 4k - k = 15, so 3k = 15.Solve: k = 15 / 3 = 5.Total stones = 15 + k = 15 + 5 = 20.

Verification / Alternative check:
Check the result by recomputing the probability. If there are 5 black stones and 20 stones in total, then the probability of drawing a black stone is 5 / 20 = 1 / 4. This matches the given probability, confirming that the total number of stones is correct.

Why Other Options Are Wrong:
If the total were 24, the number of black stones would need to be 6 to give probability 1/4, but then 9 + 6 + 6 = 21, not 24. For a total of 18, black stones would have to be 4.5 to give probability 1/4, which is impossible since the number of stones must be an integer. For a total of 15, there would be no black stones, giving probability 0 instead of 1/4. Therefore only 20 is consistent.

Common Pitfalls:
A common mistake is to confuse the given probability with the fraction of non black stones, or to assume that 1/4 of 9 or 6 gives the number of black stones. Another error is to forget to include both blue and white stones when forming the total number in the denominator. Always carefully build the probability expression as favourable outcomes divided by total outcomes, then solve the resulting algebraic equation.

Final Answer:
The total number of stones in the box is 20.