The probability of choosing a caramel from a certain bag of candy is `(1)/(5)`, and the probability of choosing a butterscotch is `(5)/(8)`. If the bag contains 40 pieces of candy, and the only types of candy in the bag are caramel, butterscotch, and fudge, how many pieces of fudge are in the bag?

Options

A. `5`
B. `7`
C. `8`
D. `25`

Answer

To find the number of pieces of fudge in the bag of candy, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( C \) = number of caramels
- \( B \) = number of butterscotches
- \( F \) = number of fudges
- Total pieces of candy = 40

### Step 2: Write the Probabilities
We know the probabilities:
- Probability of choosing a caramel \( P(C) = \frac{1}{5} \)
- Probability of choosing a butterscotch \( P(B) = \frac{5}{8} \)

The probability of choosing fudge \( P(F) \) can be expressed as:
\[ P(F) = \frac{F}{40} \]

### Step 3: Set Up the Probability Equation
According to the rule of probabilities, the sum of the probabilities of all events must equal 1:
\[ P(C) + P(B) + P(F) = 1 \]
Substituting the known probabilities:
\[ \frac{1}{5} + \frac{5}{8} + \frac{F}{40} = 1 \]

### Step 4: Find a Common Denominator
The least common multiple (LCM) of 5, 8, and 40 is 40. We will convert each term to have a denominator of 40:
- Convert \( \frac{1}{5} \):
\[ \frac{1}{5} = \frac{8}{40} \]
- Convert \( \frac{5}{8} \):
\[ \frac{5}{8} = \frac{25}{40} \]

### Step 5: Substitute and Simplify
Substituting these values back into the equation:
\[ \frac{8}{40} + \frac{25}{40} + \frac{F}{40} = 1 \]
Combine the fractions:
\[ \frac{8 + 25 + F}{40} = 1 \]
This simplifies to:
\[ \frac{33 + F}{40} = 1 \]

### Step 6: Solve for Fudge
Now, multiply both sides by 40 to eliminate the denominator:
\[ 33 + F = 40 \]
Subtract 33 from both sides:
\[ F = 40 - 33 = 7 \]

### Conclusion
The number of pieces of fudge in the bag is \( \boxed{7} \).

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