A truck travelled from town A to town B over several days. During the first day, it covered 1/p of the total distance, where p is a natural number. During the second day, it travelled 1/q of the remaining distance, where q is a natural number. During the third day, it travelled 1/p of the distance remaining after the second day, and during the fourth day, 1/q of the distance remaining after third day. By the end of the fourth day the truck had travelled 3/4 of the distance between A and B.
The value of p + q is

Options

A. A) 4
B. B) 5
C. C) 6
D. D) 7

Answer

To solve the problem step by step, we need to analyze the distances covered by the truck each day and set up equations based on the information provided.

Let the total distance between town A and town B be \( D \).

### Step 1: Distance covered on the first day
On the first day, the truck covered \( \frac{1}{p} \) of the total distance \( D \).
\[ \text{Distance covered on Day 1} = \frac{D}{p} \]
Remaining distance after Day 1:
\[ \text{Remaining distance} = D - \frac{D}{p} = D \left(1 - \frac{1}{p}\right) = D \left(\frac{p-1}{p}\right) \]

### Step 2: Distance covered on the second day
On the second day, the truck covered \( \frac{1}{q} \) of the remaining distance.
\[ \text{Distance covered on Day 2} = \frac{1}{q} \times D \left(\frac{p-1}{p}\right) = \frac{D(p-1)}{pq} \]
Remaining distance after Day 2:
\[ \text{Remaining distance} = D \left(\frac{p-1}{p}\right) - \frac{D(p-1)}{pq} = D \left(\frac{p-1}{p}\right) \left(1 - \frac{1}{q}\right) = D \left(\frac{p-1}{p}\right) \left(\frac{q-1}{q}\right) \]

### Step 3: Distance covered on the third day
On the third day, the truck covered \( \frac{1}{p} \) of the remaining distance after Day 2.
\[ \text{Distance covered on Day 3} = \frac{1}{p} \times D \left(\frac{p-1}{p}\right) \left(\frac{q-1}{q}\right) = \frac{D(p-1)(q-1)}{p^2q} \]
Remaining distance after Day 3:
\[ \text{Remaining distance} = D \left(\frac{p-1}{p}\right) \left(\frac{q-1}{q}\right) - \frac{D(p-1)(q-1)}{p^2q} = D \left(\frac{p-1}{p}\right) \left(\frac{q-1}{q}\right) \left(1 - \frac{1}{p}\right) = D \left(\frac{p-1}{p}\right) \left(\frac{q-1}{q}\right) \left(\frac{p-1}{p}\right) \]

### Step 4: Distance covered on the fourth day
On the fourth day, the truck covered \( \frac{1}{q} \) of the remaining distance after Day 3.
\[ \text{Distance covered on Day 4} = \frac{1}{q} \times D \left(\frac{p-1}{p}\right) \left(\frac{q-1}{q}\right) \left(\frac{p-1}{p}\right) = \frac{D(p-1)^2(q-1)}{p^2q^2} \]
Remaining distance after Day 4:
\[ \text{Remaining distance} = D \left(\frac{p-1}{p}\right) \left(\frac{q-1}{q}\right) \left(\frac{p-1}{p}\right) - \frac{D(p-1)^2(q-1)}{p^2q^2} \]

### Step 5: Total distance covered after four days
By the end of the fourth day, the truck had traveled \( \frac{3}{4}D \).
\[ \text{Total distance covered} = \frac{D}{p} + \frac{D(p-1)}{pq} + \frac{D(p-1)(q-1)}{p^2q} + \frac{D(p-1)^2(q-1)}{p^2q^2} = \frac{3}{4}D \]

### Step 6: Setting up the equation
Factoring out \( D \) from the left side, we get:
\[ \frac{1}{p} + \frac{(p-1)}{pq} + \frac{(p-1)(q-1)}{p^2q} + \frac{(p-1)^2(q-1)}{p^2q^2} = \frac{3}{4} \]

### Step 7: Finding \( p + q \)
Through simplification and testing natural numbers for \( p \) and \( q \), we find that \( p + q = 7 \).

### Final Answer
Thus, the value of \( p + q \) is:
\[ \boxed{7} \]