If `Q = (p + q + r )/(2) & (Q-p ) : (Q-q) : (Q -r)::17:5:8.` then `p :q:r = ?`

Options

A. `16:19:21`
B. `7:8:5`
C. `14:19:21`
D. `13:25:22`

Answer

To solve the problem, we need to find the ratio \( p : q : r \) given the relationships involving \( Q \).

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We are given that:
\[ Q = \frac{p + q + r}{2} \]
and the ratios:
\[ (Q - p) : (Q - q) : (Q - r) = 17 : 5 : 8 \]

2. **Expressing the Ratios in Terms of a Common Variable:**
Let \( k \) be a common factor. We can express the differences as:
\[ Q - p = 17k, \quad Q - q = 5k, \quad Q - r = 8k \]

3. **Finding Expressions for \( p, q, \) and \( r \):**
From the above equations, we can express \( p, q, \) and \( r \) in terms of \( Q \) and \( k \):
\[ p = Q - 17k \]
\[ q = Q - 5k \]
\[ r = Q - 8k \]

4. **Adding the Expressions for \( p, q, \) and \( r \):**
Now, we can add these three equations:
\[ p + q + r = (Q - 17k) + (Q - 5k) + (Q - 8k) \]
Simplifying this gives:
\[ p + q + r = 3Q - 30k \]

5. **Using the Expression for \( Q \):**
We know from the first equation that:
\[ p + q + r = 2Q \]
Setting the two expressions for \( p + q + r \) equal to each other:
\[ 3Q - 30k = 2Q \]

6. **Solving for \( Q \):**
Rearranging gives:
\[ 3Q - 2Q = 30k \implies Q = 30k \]

7. **Substituting Back to Find \( p, q, \) and \( r \):**
Now we substitute \( Q \) back into the expressions for \( p, q, \) and \( r \):
\[ p = 30k - 17k = 13k \]
\[ q = 30k - 5k = 25k \]
\[ r = 30k - 8k = 22k \]

8. **Finding the Ratio \( p : q : r \):**
Thus, we have:
\[ p : q : r = 13k : 25k : 22k \]
Cancelling \( k \) gives:
\[ p : q : r = 13 : 25 : 22 \]

### Final Answer:
The ratio \( p : q : r \) is \( 13 : 25 : 22 \).