Four parts of 24 are in A.P. such that the ratio of product of extremes to products of means is `7:15`, then four parts are

Options

A. `(3)/(2), (9)/(2),(15)/(2),(21)/(2)`
B. `(11)/(2),(13)/(2),3,9`
C. `(5)/(2),(15)/(2),(9)/(2),(21)/(2)`
D. `(21)/(2),(9)/(2),(15)/(2),(7)/(2)`

Answer

To solve the problem, we need to find four parts of 24 that are in Arithmetic Progression (A.P.) such that the ratio of the product of extremes to the product of means is 7:15.

### Step-by-step Solution:

1. **Define the terms in A.P.**:
Let the four parts in A.P. be represented as:
\[ a - 3d, \quad a - d, \quad a + d, \quad a + 3d \]
Here, \( a \) is the middle value and \( d \) is the common difference.

2. **Set up the equation for the sum**:
The sum of these four parts is given to be 24:
\[ (a - 3d) + (a - d) + (a + d) + (a + 3d) = 24 \]
Simplifying this gives:
\[ 4a = 24 \implies a = 6 \]

3. **Substitute \( a \) back into the expressions**:
Now substituting \( a = 6 \) into the expressions for the four parts:
\[ 6 - 3d, \quad 6 - d, \quad 6 + d, \quad 6 + 3d \]

4. **Calculate the product of extremes and means**:
- The product of extremes:
\[ (6 - 3d)(6 + 3d) = 36 - 9d^2 \]
- The product of means:
\[ (6 - d)(6 + d) = 36 - d^2 \]

5. **Set up the ratio**:
According to the problem, the ratio of the product of extremes to the product of means is given as:
\[ \frac{36 - 9d^2}{36 - d^2} = \frac{7}{15} \]

6. **Cross-multiply to solve for \( d^2 \)**:
Cross-multiplying gives:
\[ 15(36 - 9d^2) = 7(36 - d^2) \]
Expanding both sides:
\[ 540 - 135d^2 = 252 - 7d^2 \]

7. **Rearranging the equation**:
Bringing all terms involving \( d^2 \) to one side and constants to the other:
\[ 540 - 252 = 135d^2 - 7d^2 \]
Simplifying gives:
\[ 288 = 128d^2 \]

8. **Solve for \( d^2 \)**:
\[ d^2 = \frac{288}{128} = \frac{9}{4} \implies d = \frac{3}{2} \text{ or } d = -\frac{3}{2} \]

9. **Finding the four parts**:
- If \( d = \frac{3}{2} \):
\[ 6 - 3d = 6 - 4.5 = 1.5, \quad 6 - d = 6 - 1.5 = 4.5, \quad 6 + d = 6 + 1.5 = 7.5, \quad 6 + 3d = 6 + 4.5 = 10.5 \]
So, the four parts are \( 1.5, 4.5, 7.5, 10.5 \).

- If \( d = -\frac{3}{2} \):
\[ 6 - 3d = 6 + 4.5 = 10.5, \quad 6 - d = 6 + 1.5 = 7.5, \quad 6 + d = 6 - 1.5 = 4.5, \quad 6 + 3d = 6 + 4.5 = 1.5 \]
The four parts are the same: \( 10.5, 7.5, 4.5, 1.5 \).

### Final Answer:
The four parts are \( 1.5, 4.5, 7.5, 10.5 \).