--- title: "Modified B/C ratio for the required-receipts problem" subject: "Engineering Economics" chapter: 3 key: "semester_6/engineering_economics:modified-bcr-for-5a" source: https://www.bctengineer.com/q/semester_6/engineering_economics/modified-bcr-for-5a site: BCT Engineering --- # Modified B/C ratio for the required-receipts problem Engineering Economics · Chapter 3: Basic Methodologies of Engineering Economic Analysis - Asked in: 2065 Kartik, 8 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/engineering_economics/modified-bcr-for-5a - Whole chapter: https://www.bctengineer.com/md/semester_6/engineering_economics/chapter-3 ## Question Find the modified B/C ratio for the investment proposal below (problem 5(a): initial investment Rs. 10,00,000, salvage value Rs. 1,00,000, O & M expenses Rs. 50,000 per year, benefits of A+70,000, A+80,000, A+90,000, A+100,000, A+110,000 in years 1 to 5, using the value of A found there). ## Short answer Modified B/C = 1.00 at A ≈ Rs 223,923 (MARR 12%). A chosen so PW benefits equals PW costs, making ratio exactly 1 (break-even). Net benefits and net investment both equal Rs 943,257. ## Answer Assumption: the MARR is not printed with the question; take **MARR = 12%**. First find A (the value that makes the proposal just acceptable), then compute the modified B/C ratio. #### Cash-flow diagram (Rs.) ```text Benefits A+70k A+80k A+90k A+100k A+110k | | | | | 0 ------1------2------3------4-------5---> year | O&M -50,000 every year (1-5) ^ salvage +100,000 v 1,000,000 (investment) ``` #### Step 1: find A The extra amount over A is 70,000, 80,000, ..., 110,000. This is a uniform series of 70,000 plus a gradient of G = 10,000 (0, 10k, 20k, 30k, 40k). Factors at 12%, N = 5: - (P/A, 12%, 5) = 3.6048 - (P/G, 12%, 5) = 6.3970 - (P/F, 12%, 5) = 0.5674 ```text PW(benefits) = A(P/A) + 70,000(P/A) + 10,000(P/G) = 3.6048 A + 252,334 + 63,970 = 3.6048 A + 316,304 PW(costs) = 1,000,000 + 50,000(P/A) - 100,000(P/F) = 1,000,000 + 180,239 - 56,743 = 1,123,496 ``` Set PW(benefits) = PW(costs): 3.6048 A = 1,123,496 - 316,304 = 807,192 **A = Rs. 223,923 per year** (required annual receipts). #### Step 2: modified B/C ratio Modified B/C = [PW(benefits) - PW(O&M)] / [Investment - PW(salvage)] - PW(benefits) = 3.6048 × 223,923 + 316,304 = 1,123,496 - PW(O&M) = 50,000 × 3.6048 = 180,239 - Net benefit = 1,123,496 - 180,239 = 943,257 - Net investment = 1,000,000 - 100,000 × 0.5674 = 943,257 Modified B/C = 943,257 / 943,257 = **1.00** **Answer: Modified B/C = 1.00 (at A ≈ Rs. 223,923).** Because A was chosen so that PW of benefits equals PW of costs, the ratio is exactly 1, the break-even point. Any receipts larger than this A give a ratio above 1 and make the proposal acceptable. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the first step before computing the modified B/C? A. Find A so that PW of benefits equals PW of costs B. Ignore A C. Compute IRR D. Compute payback Answer: A. The answer first finds A (required receipts), then computes the modified ratio. (Key point: Step order) 2. Which is the modified B/C formula used? A. [PW(B) + PW(O&M)] / I B. PW(B) / [I + PW(O&M) - PW(S)] C. [PW(B) - PW(O&M)] / [I - PW(S)] D. PW(B) / PW(O&M) Answer: C. O&M is netted from benefits; salvage is netted from investment. (Key point: Modified B/C formula) 3. What is the modified B/C ratio at the A found? A. 0.76 B. 1.00 C. 1.48 D. 1.19 Answer: B. Net benefit and net investment are both 943,257, so the ratio is exactly 1. (Key point: Final value) 4. Why does the ratio come out exactly 1? A. O&M is zero B. MARR is 1% C. Salvage is zero D. A was chosen so PW benefits equal PW costs (break-even) Answer: D. Setting PW(benefits) = PW(costs) to find A is the break-even point, giving ratio 1. 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