<p>A carpenter was hired to build 192 window frames. The first day he made five frames and each day, thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?</p>
Step-by-Step Solution
Key Concept: The carpenter's daily production forms an arithmetic sequence (5, 7, 9, 11, ...). We need to find how many terms of this sequence sum to 192 using the arithmetic series formula S_n = n/2[2a + (n-1)d].
<p><strong>Step 1:</strong> Identify the arithmetic sequence. Day 1: 5 frames, Day 2: 7 frames, Day 3: 9 frames, etc. Here, first term a = 5 and common difference d = 2.</p><p><strong>Step 2:</strong> Set up the arithmetic series formula. The sum of frames made in n days is: S_n = n/2[2a + (n-1)d] = n/2[2(5) + (n-1)(2)] = n/2[10 + 2n - 2] = n/2[8 + 2n] = n(4 + n)</p><p><strong>Step 3:</strong> Set the sum equal to 192 (total frames needed). n(4 + n) = 192 ⟹ n² + 4n = 192 ⟹ n² + 4n - 192 = 0</p><p><strong>Step 4:</strong> Solve the quadratic equation using factorization or the quadratic formula. We need two numbers that multiply to -192 and add to 4. These are 16 and -12. Therefore: (n + 16)(n - 12) = 0</p><p><strong>Step 5:</strong> Find the valid solution. n = -16 (rejected, as days cannot be negative) or n = 12 (valid). We verify: S₁₂ = 12(4 + 12) = 12(16) = 192 ✓</p><p><strong>∴ Answer:</strong> c</p>
Correct Answer: c