A bag contains 5 apples and 7 oranges and another basket contains 4 apples and 8 oranges. One fruit is picked out from each basket. Find the probability that the fruits are both apples or both oranges.

Options

A. `(24)/(144)`
B. `(56)/(144)`
C. `(68)/(144)`
D. `(76)/(144)`

Explanation

To solve the problem, we need to find the probability that the fruits picked from both baskets are either both apples or both oranges.

### Step-by-step Solution:

1. **Identify the total number of fruits in each basket:**
- **First Basket:** 5 apples + 7 oranges = 12 fruits
- **Second Basket:** 4 apples + 8 oranges = 12 fruits

2. **Calculate the probability of picking apples from both baskets:**
- Probability of picking an apple from the first basket:
\[ P(\text{Apple from 1st Basket}) = \frac{5 \text{ apples}}{12 \text{ total fruits}} = \frac{5}{12} \]
- Probability of picking an apple from the second basket:
\[ P(\text{Apple from 2nd Basket}) = \frac{4 \text{ apples}}{12 \text{ total fruits}} = \frac{4}{12} = \frac{1}{3} \]
- Therefore, the probability of picking apples from both baskets is:
\[ P(\text{Both Apples}) = P(\text{Apple from 1st}) \times P(\text{Apple from 2nd}) = \frac{5}{12} \times \frac{1}{3} = \frac{5}{36} \]

3. **Calculate the probability of picking oranges from both baskets:**
- Probability of picking an orange from the first basket:
\[ P(\text{Orange from 1st Basket}) = \frac{7 \text{ oranges}}{12 \text{ total fruits}} = \frac{7}{12} \]
- Probability of picking an orange from the second basket:
\[ P(\text{Orange from 2nd Basket}) = \frac{8 \text{ oranges}}{12 \text{ total fruits}} = \frac{8}{12} = \frac{2}{3} \]
- Therefore, the probability of picking oranges from both baskets is:
\[ P(\text{Both Oranges}) = P(\text{Orange from 1st}) \times P(\text{Orange from 2nd}) = \frac{7}{12} \times \frac{2}{3} = \frac{14}{36} = \frac{7}{18} \]

4. **Calculate the total probability of picking either both apples or both oranges:**
- Total probability:
\[ P(\text{Both Apples or Both Oranges}) = P(\text{Both Apples}) + P(\text{Both Oranges}) = \frac{5}{36} + \frac{7}{18} \]
- To add these fractions, convert \(\frac{7}{18}\) to have a common denominator of 36:
\[ \frac{7}{18} = \frac{14}{36} \]
- Now add:
\[ P(\text{Both Apples or Both Oranges}) = \frac{5}{36} + \frac{14}{36} = \frac{19}{36} \]

### Final Answer:
The probability that the fruits are both apples or both oranges is \(\frac{19}{36}\).