Set B Solutions

B1. Answer: (c) Rs. 186 Method: Direct 3-year difference shortcut. Diff(3 yrs) = PR²(300+R)/10⁶ = (6000 × 100 × 310)/1,000,000 = 186,000,000/1,000,000 = Rs. 186. Verification by direct computation: SI = 3×6000×10/100 = 1800. CI = 6000×[(1.1)³−1] = 6000×0.331 = 1986. Diff = 1986−1800 = 186. ✓ Both methods agree, confirming the shortcut formula.

B2. Answer: (c) Rs. 3,000 Method: Reverse-apply the 3-year difference formula to solve for P. 93 = P × 100 × 310/10⁶ = P × 31000/1,000,000 = P × 0.031. P = 93/0.031 = Rs. 3,000. Option (a) Rs. 2,700 is what a student gets by mistakenly using the 2-year difference formula instead of the 3-year one — always check the number of years stated in the question before selecting a difference formula.

B3. Answer: (c) Rs. 1,261 Method: Convert to half-yearly terms before applying CI. 10% p.a. half-yearly for 1.5 years → rate per period = 5%, periods = 3. A = 8000 × (1.05)³ = 8000 × 1.157625 = Rs. 9,261. CI = 9261 − 8000 = Rs. 1,261.

B4. Answer: (c) Rs. 808 Method: Convert to quarterly terms before applying CI. 8% p.a. quarterly for 6 months → rate per period = 2%, periods = 2. A = 20000 × (1.02)² = 20000 × 1.0404 = Rs. 20,808. CI = 20808 − 20000 = Rs. 808.

B5. Answer: (c) Rs. 10,000 Method: Reverse-apply the 2-year difference formula. 25 = P × 25/10000 = P × 0.0025. P = 25/0.0025 = Rs. 10,000.

B6. Answer: (d) 12.5% Method: SI doubling-time shortcut. For SI, a sum doubles (SI = P) when R = 100/T = 100/8 = 12.5%. This is a direct application of Trick 5 and needs no formula expansion.

B7. Answer: (c) 9 years Method: CI multiplying-time shortcut (Trick 6). Since 8 = 2³, becoming “8 times” requires three repetitions of the same “2 times in 3 years” growth step, giving 3 × 3 = 9 years. A student who tries to solve this algebraically using logarithms will take far longer than the shortcut allows for.

B8. Answer: (c) Rs. 1,800 Method: SI installment formula. Using P(1+nR/100) = x[n + n(n−1)R/200] with n=3, R=20, x=800: bracket = 3 + (3×2×20)/200 = 3 + 0.6 = 3.6. P × (1 + 60/100) = 800 × 3.6 → P × 1.6 = 2880 → P = 2880/1.6 = Rs. 1,800.

B9. Answer: (c) Rs. 2,100 Method: CI installment (present-value) formula, since the question specifies compounding. P = x/(1+r) + x/(1+r)² = 1210/1.1 + 1210/1.21 = 1100 + 1000 = Rs. 2,100. Using the SI installment formula here by mistake would instead give Rs. 2,117.5 — always check the interest type stated in the question stem before choosing the formula.

B10. Answer: (c) 24,200 Method: Population growth is CI in disguise. Population = 20000 × (1.1)² = 20000 × 1.21 = 24,200.

B11. Answer: (c) Rs. 40,500 Method: Depreciation uses (1 − R/100)^T. Value = 50000 × (0.9)² = 50000 × 0.81 = Rs. 40,500. Option (b) Rs. 40,000 is the trap for students who mistakenly compute simple 10%-per-year depreciation (i.e., subtract 5,000 twice) instead of applying it on the reducing balance.

B12. Answer: (c) 10% Method: Extract CI from the amount, then solve for rate using the 2-year CI relation. CI = 6050 − 5000 = 1050. (1+r)² − 1 = 1050/5000 = 0.21 → (1+r)² = 1.21 → 1+r = 1.1 → r = 10%.

B13. Answer: (c) Rs. 16,000 Method: Reverse-apply the 3-year difference formula. 122 = P × 25 × 305/10⁶ = P × 7625/1,000,000 = P × 0.007625. P = 122/0.007625 = Rs. 16,000.

B14. Answer: (c) 10%, Rs. 8,000 Method: Ratio shortcut (Trick 11) — divide the later-year amount by the earlier one to isolate (1 + R/100) in a single step. 1 + R/100 = 10648/9680 = 1.1 → R = 10%. Then Sum = 9680/(1.1)² = 9680/1.21 = Rs. 8,000. This avoids setting up two simultaneous equations from scratch.

B15. Answer: (c) 21% Method: Effective annual rate formula. R_eff = [(1 + 20/200)² − 1] × 100 = [(1.1)² − 1] × 100 = (1.21 − 1) × 100 = 21%. Note this matches the net-effect shortcut from the Percentage chapter: two successive 10% half-yearly rises give a net annual rise of 10 + 10 + (10×10)/100 = 21%.

B16. Answer: (b) 5% Method: Reverse-apply the 2-year difference shortcut for rate. 18 = 7200 × R²/10000 → R² = 180000/7200 = 25 → R = 5%.

B17. Answer: (c) Rs. 465 Method: Direct 3-year difference shortcut. Diff = PR²(300+R)/10⁶ = (15000 × 100 × 310)/1,000,000 = 465,000,000/1,000,000 = Rs. 465. Check: SI = 3×15000×10/100 = 4,500; CI = 15000×[(1.1)³−1] = 15000×0.331 = 4,965; difference = 465. ✓

B18. Answer: (c) Rs. 410 Method: Convert to half-yearly terms first. 10% p.a. half-yearly for 1 year → rate per period = 5%, periods = 2. A = 4000 × (1.05)² = 4000 × 1.1025 = Rs. 4,410. CI = 4410 − 4000 = Rs. 410.

B19. Answer: (c) Rs. 1,050 Method: Convert to quarterly terms first. 40% p.a. quarterly for 6 months → rate per period = 10%, periods = 2. A = 5000 × (1.1)² = 5000 × 1.21 = Rs. 6,050. CI = 6050 − 5000 = Rs. 1,050.

B20. Answer: (c) Rs. 3,850 Method: SI installment formula. Using P(1+nR/100) = x[n + n(n−1)R/200] with n=2, R=10, x=2200: bracket = 2 + (2×1×10)/200 = 2 + 0.1 = 2.1. P × 1.2 = 2200 × 2.1 = 4,620 → P = 4620/1.2 = Rs. 3,850.

B21. Answer: (c) Rs. 6,620 Method: CI installment (present-value) formula for 3 installments. P = 2662/1.1 + 2662/1.21 + 2662/1.331 = 2,420 + 2,200 + 2,000 = Rs. 6,620.

B22. Answer: (c) 10% Method: Population growth as CI. 12100/10000 = (1+R/100)² → 1.21 = (1+R/100)² → 1+R/100 = 1.1 → R = 10%.

B23. Answer: (c) Rs. 1,00,000 Method: Depreciation formula, solved backwards. 64000 = P × (0.8)² = P × 0.64. P = 64000/0.64 = Rs. 1,00,000.

B24. Answer: (c) 20% Method: Effective annual rate formula, solved for the nominal rate. (1 + R/200)² − 1 = 0.21 → (1 + R/200)² = 1.21 → 1 + R/200 = 1.1 → R/200 = 0.1 → R = 20%.

B25. Answer: (c) 12 years Method: CI multiplying-time shortcut (Trick 6). Since 27 = 3³, becoming “27 times” requires three repetitions of the “3 times in 4 years” growth step: 3 × 4 = 12 years.

B26. Answer: (c) 8% Method: Generalised SI multiplying-time shortcut (Trick 12). A sum becoming 5 times itself means SI = 4P, so R = 100(k−1)/T = 100 × 4/50 = 400/50 = 8%.

B27. Answer: (c) 10% Method: Ratio method for SI (Trick 4/generalised Trick 12). Amount = 8/5 P means SI = (8/5 − 1)P = (3/5)P. R = [(3/5)P × 100]/(P × 6) = 60/6 = 10%.

B28. Answer: (c) Rs. 5,625 Method: Changing-rate CI (chain the multiplying factors). A = 15000 × 1.1 × 1.25 = 15000 × 1.375 = Rs. 20,625. CI = 20625 − 15000 = Rs. 5,625.

B29. Answer: (c) Rs. 3,246 Method: Fractional-year CI (whole years compounded, then SI for the fraction). Whole 2 years: A₂ = 12000 × (1.1)² = 12000 × 1.21 = Rs. 14,520. Remaining 6 months = ½ year, SI on 14,520 at 10% for ½ year = 14520 × 10 × 0.5/100 = Rs. 726. Final amount = 14520 + 726 = Rs. 15,246. CI = 15246 − 12000 = Rs. 3,246.

B30. Answer: (c) 10% Method: Ratio shortcut (Trick 11) — divide the later-year amount by the earlier one. 1 + R/100 = 5819/5290 = 1.1 → R = 10%.