Find the value of p and q respectively which satisfies the given equation.
`p(1)/(3) + q(1)/(3) = 7 (2)/(3)`

Options

A. 4,8
B. 2,3
C. 3,4
D. 3,5

Answer

To solve the equation \( p\frac{1}{3} + q\frac{1}{3} = 7\frac{2}{3} \), we will follow these steps:

### Step 1: Convert the mixed fraction on the right-hand side to an improper fraction.
The mixed fraction \( 7\frac{2}{3} \) can be converted to an improper fraction using the formula:
\[ \text{Improper Fraction} = \left( a \times c + b \right) / c \]
where \( a \) is the whole number, \( b \) is the numerator of the fraction, and \( c \) is the denominator.

For \( 7\frac{2}{3} \):
- \( a = 7 \)
- \( b = 2 \)
- \( c = 3 \)

Using the formula:
\[ 7\frac{2}{3} = \frac{7 \times 3 + 2}{3} = \frac{21 + 2}{3} = \frac{23}{3} \]

### Step 2: Rewrite the equation with the improper fraction.
Now we can rewrite the original equation:
\[ p\frac{1}{3} + q\frac{1}{3} = \frac{23}{3} \]

### Step 3: Combine the left-hand side.
The left-hand side can be combined:
\[ \frac{p + q}{3} = \frac{23}{3} \]

### Step 4: Eliminate the denominator by multiplying both sides by 3.
Multiply both sides by 3 to eliminate the fraction:
\[ p + q = 23 \]

### Step 5: Find suitable values for \( p \) and \( q \).
Now we need to find pairs of integers \( (p, q) \) that satisfy the equation \( p + q = 23 \). We can check the options provided (if any) or find suitable pairs manually.

### Step 6: Check possible pairs.
Let's consider pairs:
- If \( p = 3 \), then \( q = 20 \) (3 + 20 = 23)
- If \( p = 4 \), then \( q = 19 \) (4 + 19 = 23)
- If \( p = 10 \), then \( q = 13 \) (10 + 13 = 23)
- If \( p = 11 \), then \( q = 12 \) (11 + 12 = 23)
- If \( p = 12 \), then \( q = 11 \) (12 + 11 = 23)
- If \( p = 13 \), then \( q = 10 \) (13 + 10 = 23)
- If \( p = 19 \), then \( q = 4 \) (19 + 4 = 23)
- If \( p = 20 \), then \( q = 3 \) (20 + 3 = 23)

### Conclusion:
From the calculations, the pairs \( (3, 20), (4, 19), (10, 13), (11, 12), (12, 11), (13, 10), (19, 4), (20, 3) \) all satisfy the equation \( p + q = 23 \).