A student studying the weather for d days observed that (i) it rained on 7 days. morning or afternoon, (ii) when it rained in the afternoon, it was clear in the morning, (iii) there were five clear afternoons and (iv) there were six clear morning. Then d equals

Options

A. 3
B. 7
C. 11
D. 9

Answer

To solve the problem step by step, we will analyze the given conditions and derive the total number of days (d).

### Step 1: Understand the given conditions
1. It rained on 7 days (either morning or afternoon).
2. When it rained in the afternoon, it was clear in the morning.
3. There were 5 clear afternoons.
4. There were 6 clear mornings.

### Step 2: Set up the equations
- Let’s denote:
- The number of days it rained in the morning as \( R_m \).
- The number of days it rained in the afternoon as \( R_a \).

From the information given:
- \( R_m + R_a = 7 \) (since it rained on 7 days)

### Step 3: Analyze clear days
- The total number of afternoons is equal to the total number of days (d).
- Since there are 5 clear afternoons, the number of rainy afternoons \( R_a = d - 5 \).
- Similarly, since there are 6 clear mornings, the number of rainy mornings \( R_m = d - 6 \).

### Step 4: Set up the equations based on clear days
From the above, we can write:
1. \( R_a = d - 5 \)
2. \( R_m = d - 6 \)

### Step 5: Substitute into the rainy days equation
Now we can substitute \( R_m \) and \( R_a \) into the equation \( R_m + R_a = 7 \):
\[ (d - 6) + (d - 5) = 7 \]
\[ 2d - 11 = 7 \]

### Step 6: Solve for d
Now, we solve for \( d \):
\[ 2d = 7 + 11 \]
\[ 2d = 18 \]
\[ d = \frac{18}{2} = 9 \]

### Conclusion
Thus, the total number of days \( d \) equals 9.