One boy can solve `60%`of the problem in a book and another can solve `80%`. The probability that at least one of the two can solve a problem chosen at random from the book is .

Options

A. `2//25`
B. `23//25`
C. `4//25`
D. `9//10`.

Answer

To solve the problem, we need to find the probability that at least one of the two boys can solve a randomly chosen problem from the book. Let's denote the probabilities for each boy as follows:

- Let \( P(A) \) be the probability that Boy A can solve a problem.
- Let \( P(B) \) be the probability that Boy B can solve a problem.

Given:
- \( P(A) = 60\% = \frac{60}{100} = 0.6 \)
- \( P(B) = 80\% = \frac{80}{100} = 0.8 \)

### Step 1: Calculate the probability that each boy cannot solve the problem.
- The probability that Boy A cannot solve the problem is:
\[ P(A') = 1 - P(A) = 1 - 0.6 = 0.4 \]

- The probability that Boy B cannot solve the problem is:
\[ P(B') = 1 - P(B) = 1 - 0.8 = 0.2 \]

### Step 2: Calculate the probability that neither boy can solve the problem.
- The probability that neither Boy A nor Boy B can solve the problem is:
\[ P(A' \cap B') = P(A') \times P(B') = 0.4 \times 0.2 = 0.08 \]

### Step 3: Calculate the probability that at least one boy can solve the problem.
- The probability that at least one of them can solve the problem is the complement of the probability that neither can solve it:
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - 0.08 = 0.92 \]

### Step 4: Convert the probability to a fraction.
- To express \( 0.92 \) as a fraction:
\[ 0.92 = \frac{92}{100} = \frac{23}{25} \]

### Final Answer:
Thus, the probability that at least one of the two boys can solve a problem chosen at random from the book is:
\[ \frac{23}{25} \]