<p>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. Find the four parts.</p>
Step-by-Step Solution
Key Concept: Let the four parts in A.P. be (a-3d), (a-d), (a+d), (a+3d) so their sum equals 28. The ratio condition on products gives a quadratic in d² that determines the common difference.
<p><strong>Step 1:</strong> Let the four parts in A.P. be <strong>(a−3d), (a−d), (a+d), (a+3d)</strong> where a and d are parameters.</p><p><strong>Step 2:</strong> Sum condition: (a−3d) + (a−d) + (a+d) + (a+3d) = 28 → <strong>4a = 28 → a = 7</strong></p><p><strong>Step 3:</strong> The four parts are now: <strong>(7−3d), (7−d), (7+d), (7+3d)</strong></p><p><strong>Step 4:</strong> Product ratio condition: <br>$$\frac{(7-3d)(7+d)}{(7-d)(7+3d)} = \frac{8}{15}$$</p><p><strong>Step 5:</strong> Expand numerator: (7−3d)(7+d) = 49 + 7d − 21d − 3d² = 49 − 14d − 3d²<br>Expand denominator: (7−d)(7+3d) = 49 + 21d − 7d − 3d² = 49 + 14d − 3d²</p><p><strong>Step 6:</strong> Cross multiply: 15(49 − 14d − 3d²) = 8(49 + 14d − 3d²)<br>735 − 210d − 45d² = 392 + 112d − 24d²<br>343 − 322d − 21d² = 0<br>21d² + 322d − 343 = 0</p><p><strong>Step 7:</strong> Using quadratic formula or factoring: (3d − 2)(7d + 49) = 0 ... or simply: d² + (322/21)d − (343/21) = 0<br>Testing d = 1: 21 + 322 − 343 = 0 ✓ <br>So d = 1</p><p><strong>Step 8:</strong> The four parts are: (7−3), (7−1), (7+1), (7+3) = <strong>4, 6, 8, 10</strong></p><p><strong>Verification:</strong> Sum = 28 ✓; Ratio = (4×8)/(6×10) = 32/60 = 8/15 ✓</p>
Correct Answer: 4, 6, 8, 10