Divide 28 into four parts in an A.P. so that the ratio of the product of first and third with the product of second and fourth is 8:15.

Explanation

To solve the problem of dividing 28 into four parts in an Arithmetic Progression (A.P.) such that the ratio of the product of the first and third parts to the product of the second and fourth parts is 8:15, we can follow these steps:

### Step-by-Step Solution:

1. **Let the Four Parts be Defined**:
Let the four parts in A.P. be represented as:
- First part: \( a - 3d \)
- Second part: \( a - d \)
- Third part: \( a + d \)
- Fourth part: \( a + 3d \)

2. **Set Up the Equation for the Sum**:
Since the sum of these four parts equals 28, we can write the equation:
\[ (a - 3d) + (a - d) + (a + d) + (a + 3d) = 28 \]
Simplifying this gives:
\[ 4a = 28 \]
Therefore, solving for \( a \):
\[ a = \frac{28}{4} = 7 \]

3. **Set Up the Ratio of Products**:
According to the problem, the ratio of the product of the first and third parts to the product of the second and fourth parts is given as:
\[ \frac{(a - 3d)(a + d)}{(a - d)(a + 3d)} = \frac{8}{15} \]

4. **Cross-Multiply to Eliminate the Fraction**:
Cross-multiplying gives:
\[ 15 \cdot (a - 3d)(a + d) = 8 \cdot (a - d)(a + 3d) \]

5. **Expand Both Sides**:
Expanding both sides:
- Left-hand side:
\[ 15 \cdot (a^2 + ad - 3ad - 3d^2) = 15(a^2 - 2ad - 3d^2) \]
- Right-hand side:
\[ 8 \cdot (a^2 + 3ad - ad - 3d^2) = 8(a^2 + 2ad - 3d^2) \]

6. **Set Up the Equation**:
This leads to:
\[ 15a^2 - 30ad - 45d^2 = 8a^2 + 16ad - 24d^2 \]
Rearranging gives:
\[ 7a^2 - 46ad - 21d^2 = 0 \]

7. **Substituting the Value of \( a \)**:
Substitute \( a = 7 \):
\[ 7(7^2) - 46(7)d - 21d^2 = 0 \]
Simplifying:
\[ 49 - 46d - 21d^2 = 0 \]

8. **Rearranging the Quadratic Equation**:
Rearranging gives:
\[ 21d^2 + 46d - 49 = 0 \]

9. **Using the Quadratic Formula**:
Using the quadratic formula \( d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 21, b = 46, c = -49 \):
\[ d = \frac{-46 \pm \sqrt{46^2 - 4 \cdot 21 \cdot (-49)}}{2 \cdot 21} \]
Calculate the discriminant:
\[ 46^2 + 4 \cdot 21 \cdot 49 = 2116 + 4116 = 6232 \]
Thus:
\[ d = \frac{-46 \pm \sqrt{6232}}{42} \]

10. **Finding the Value of \( d \)**:
Solving gives \( d = 1 \) (only the positive root is meaningful in this context).

11. **Finding the Four Parts**:
Substitute \( d = 1 \) back into the expressions for the parts:
- First part: \( 7 - 3(1) = 4 \)
- Second part: \( 7 - 1 = 6 \)
- Third part: \( 7 + 1 = 8 \)
- Fourth part: \( 7 + 3(1) = 10 \)

12. **Final Answer**:
The four parts are \( 4, 6, 8, 10 \).