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Modified B/C ratio for the required-receipts problem

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.

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).

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.)

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
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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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