<p>150 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped the second day, four more workers dropped the third day and so on. Because of this, it took 8 extra days to finish the work. Find the number of days in which the work is completed.</p>
Step-by-Step Solution
Key Concept: The total work done equals the sum of (number of workers × days worked), which can be expressed as an arithmetic series. Set up equations for both scenarios: original plan vs actual execution with dropping workers.
<p><strong>Step 1:</strong> Let the original number of days = n</p><p>Total work W = 150n worker-days</p><p><strong>Step 2:</strong> With workers dropping, day-wise workers: Day 1: 150, Day 2: 146, Day 3: 142, ... (decreasing by 4 each day)</p><p>Work is completed in (n + 8) days with this pattern.</p><p><strong>Step 3:</strong> Total work in (n + 8) days = 150 + 146 + 142 + ... (for n + 8 terms)</p><p>This is an A.P. with a = 150, d = -4, number of terms = n + 8</p><p>Sum = [(n + 8)/2][2(150) + (n + 8 - 1)(-4)]</p><p>= [(n + 8)/2][300 - 4(n + 7)]</p><p>= [(n + 8)/2][300 - 4n - 28]</p><p>= [(n + 8)/2][272 - 4n]</p><p><strong>Step 4:</strong> Equate both total works:</p><p>150n = [(n + 8)/2][272 - 4n]</p><p>300n = (n + 8)(272 - 4n)</p><p>300n = 272n - 4n² + 2176 - 32n</p><p>300n = 240n - 4n² + 2176</p><p>4n² + 60n - 2176 = 0</p><p>n² + 15n - 544 = 0</p><p><strong>Step 5:</strong> Using quadratic formula: n = [-15 ± √(225 + 2176)]/2 = [-15 ± √2401]/2 = [-15 ± 49]/2</p><p>n = 17 or n = -32 (rejected)</p><p>∴ Original days = 17, Work completed in = 17 + 8 = <strong>25 days</strong></p>
Correct Answer: 25 days