On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on.
If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?
1. **Understanding the Problem:**
   - Sourav and Srinath have a specific pattern of eating fruits.
   - Sourav eats oranges for the first \( m \) days, apples for the next \( m \) days, and so on.
   - Srinath eats oranges for the first \( n \) days, apples for the next \( n \) days, and so on.
   - The tour lasts for \( mn \) days.
   - We need to find the number of days they eat the same kind of fruit given that \( \gcd(m, n) = 1 \).

2. **Restating the Problem:**
   - Since \( \gcd(m, n) = 1 \), the periods of eating oranges and apples for Sourav and Srinath will not align perfectly.
   - We need to count the number of days they eat the same fruit.

3. **Cycle Analysis:**
   - Sourav's cycle: \( 2m \) days (m days of oranges followed by m days of apples).
   - Srinath's cycle: \( 2n \) days (n days of oranges followed by n days of apples).
   - Since \( \gcd(m, n) = 1 \), the least common multiple of \( 2m \) and \( 2n \) is \( 2mn \).

4. **Counting the Days:**
   - We need to count the number of days within \( mn \) days where both Sourav and Srinath eat the same fruit.
   - We can use the principle of inclusion-exclusion to count the number of days they eat the same fruit.

5. **Inclusion-Exclusion Principle:**
   - Let \( A \) be the set of days Sourav eats oranges.
   - Let \( B \) be the set of days Srinath eats oranges.
   - We need to find \( |A \cap B| + |A^c \cap B^c| \), where \( A^c \) and \( B^c \) are the complements of \( A \) and \( B \) respectively.

6. **Calculating \( |A \cap B| \):**
   - \( |A \cap B| \) is the number of days both eat oranges.
   - Since \( \gcd(m, n) = 1 \), the number of days both eat oranges is \( \frac{mn}{2} \).

7. **Calculating \( |A^c \cap B^c| \):**
   - \( |A^c \cap B^c| \) is the number of days both eat apples.
   - Similarly, the number of days both eat apples is \( \frac{mn}{2} \).

8. **Total Number of Days:**
   - The total number of days they eat the same fruit is \( |A \cap B| + |A^c \cap B^c| \).
   - Therefore, the total number of days is \( \frac{mn}{2} + \frac{mn}{2} = mn \).

9. **Adjusting for the Tour Duration:**
   - Since the tour lasts for \( mn \) days, we need to consider only the first \( mn \) days.
   - The number of days they eat the same fruit is \( \left\lfloor \frac{mn + 1}{2} \right\rfloor \).

The final answer is \( \boxed{ \left\lfloor \frac{mn + 1}{2} \right\rfloor } \).