A bag contains only quarters, dimes, and nickels. The probability of randomly selecting a quarter is 1/6. The probability of randomly selecting a nickel is 1/4. Which of the following could be the total number of coins in the bag?a)15b)24c)30d)32e)40Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Problem:
A bag contains only quarters, dimes, and nickels. The probability of randomly selecting a quarter is 1/6. The probability of randomly selecting a nickel is 1/4. Which of the following could be the total number of coins in the bag?

Solution:
To solve this problem, we need to consider the probabilities given and determine the possible number of coins in the bag. Let's analyze each answer option one by one.

a) 15 coins:
If there are 15 coins in the bag, the probability of randomly selecting a quarter would be 1/15, which is not equal to 1/6. Therefore, option (a) is not possible.

b) 24 coins:
If there are 24 coins in the bag, the probability of randomly selecting a quarter would be 1/24, which is not equal to 1/6. However, we can calculate the probability of randomly selecting a nickel. Let's assume there are q quarters, d dimes, and n nickels in the bag. According to the given probabilities, we have the following equations:

q/(q + d + n) = 1/6 (probability of selecting a quarter)
n/(q + d + n) = 1/4 (probability of selecting a nickel)

Simplifying the equations, we get:
q = (q + d + n)/6
n = (q + d + n)/4

Multiplying both equations by 6 and 4 respectively, we get:
6q = q + d + n
4n = q + d + n

Rearranging the equations, we have:
5q - d - n = 0
- q - d + 3n = 0

Solving these equations, we find that q = 12, d = 12, and n = 8. Therefore, it is possible to have 24 coins in the bag. Option (b) is correct.

c) 30 coins:
If there are 30 coins in the bag, the probability of randomly selecting a quarter would be 1/30, which is not equal to 1/6. Therefore, option (c) is not possible.

d) 32 coins:
If there are 32 coins in the bag, the probability of randomly selecting a quarter would be 1/32, which is not equal to 1/6. Therefore, option (d) is not possible.

e) 40 coins:
If there are 40 coins in the bag, the probability of randomly selecting a quarter would be 1/40, which is not equal to 1/6. Therefore, option (e) is not possible.

Conclusion:
Based on the analysis, the only possible number of coins in the bag is 24. Therefore, option (b) is the correct answer.

Answer

Since you cannot have a partial coin, the total number of coins in the bag must be divisible by both 6 and 4 (1/6 are quarters and 1/4 are nickels). The only answer choice that is divisible by both 6 and 4 is 24.

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