There are two sets A and B each of which consists of three numbers in GP whose product is 64 and R and r are the common ratios such that `R=r+2`. If `(p)/(q)=(3)/(2)`, where p and q are sum of numbers taken two at a time respectively in the two sets.
The value of p is

Options

A. 66
B. 72
C. 78
D. 84

Explanation

To solve the problem step-by-step, we will follow the given conditions and derive the necessary values.

### Step 1: Define the terms of the two sets
Let the three terms of set A be:
- \( a/r \)
- \( a \)
- \( ar \)

Let the three terms of set B be:
- \( b/r \)
- \( b \)
- \( br \)

### Step 2: Use the product condition
The product of the terms in set A is given as:
\[ \left(\frac{a}{r}\right) \cdot a \cdot (ar) = \frac{a^3}{r} = 64 \]
This implies:
\[ a^3 = 64r \]

Similarly, for set B:
\[ \left(\frac{b}{r}\right) \cdot b \cdot (br) = \frac{b^3}{r} = 64 \]
This implies:
\[ b^3 = 64r \]

### Step 3: Solve for \( a \) and \( b \)
From \( a^3 = 64r \) and \( b^3 = 64 \), we can find:
\[ b = 4 \quad (\text{since } b^3 = 64 \Rightarrow b = 4) \]
Substituting \( b = 4 \) into \( b^3 = 64r \):
\[ 64 = 64r \Rightarrow r = 1 \]

### Step 4: Find the common ratio \( R \)
We are given that \( R = r + 2 \):
\[ R = 1 + 2 = 3 \]

### Step 5: Calculate \( p \) and \( q \)
Now we need to find \( p \) and \( q \) where:
- \( p \) is the sum of products of terms taken two at a time from set A.
- \( q \) is the sum of products of terms taken two at a time from set B.

For set A:
\[ p = \left(\frac{a}{r} \cdot a\right) + \left(a \cdot ar\right) + \left(ar \cdot \frac{a}{r}\right) \]
Substituting \( a = 4 \) and \( r = 1 \):
\[ p = \left(\frac{4}{1} \cdot 4\right) + \left(4 \cdot 4\right) + \left(4 \cdot \frac{4}{1}\right) = 16 + 16 + 16 = 48 \]

For set B:
\[ q = \left(\frac{b}{r} \cdot b\right) + \left(b \cdot br\right) + \left(br \cdot \frac{b}{r}\right) \]
Substituting \( b = 4 \) and \( r = 1 \):
\[ q = \left(\frac{4}{1} \cdot 4\right) + \left(4 \cdot 4\right) + \left(4 \cdot \frac{4}{1}\right) = 16 + 16 + 16 = 48 \]

### Step 6: Use the ratio condition
We know that \( \frac{p}{q} = \frac{3}{2} \):
\[ \frac{48}{q} = \frac{3}{2} \]
Cross multiplying gives:
\[ 3q = 96 \Rightarrow q = 32 \]

### Step 7: Find the value of \( p \)
Now substituting back to find \( p \):
\[ p = 48 \]

### Final Answer
Thus, the value of \( p \) is:
\[ \boxed{48} \]