Question 1

Compound Interest – Find amount after 2 years at 4% p.a.: If ₹ 7,500 is borrowed at compound interest at the rate of 4% per annum (compounded annually), what amount must be repaid after 2 years?

Accepted Answer

Introduction / Context: Compound interest grows money by applying interest on both the original principal and the accumulated interest from previous periods. Here the principal is ₹ 7,500, the annual rate is 4%, and the time is 2 years with annual compounding. We must compute the final amount payable after 2 years. Given Data / Assumptions: Principal P = 7,500 Rate r = 4% per annum Time t = 2 years Compounding frequency: annually Concept / Approach: For annual compounding, the amount formula is A = P * (1 + r/100)^t. This directly accounts for interest-on-interest each year. We will substitute P = 7,500, r = 4, and t = 2, then evaluate the expression. Step-by-Step Solution: Compute yearly factor: (1 + r/100) = 1 + 4/100 = 1.04Raise to the power of the years: (1.04)^2 = 1.0816Multiply by principal: A = 7,500 * 1.0816 = 8,112 Verification / Alternative check: Break it year by year: After Year 1, amount = 7,500 * 1.04 = 7,800. Interest in Year 2 is 4% of 7,800 = 312. Add to 7,800 → 8,112. This matches the direct formula result. Why Other Options Are Wrong: ₹ 8,082 and ₹ 8,100 are below the correct compounded value; ₹ 7,800 is just one year of growth; ₹ 8,010 understates the second-year interest-on-interest component. Common Pitfalls: Using simple interest (adding 4% twice to the original principal) misses compounding on the first year’s interest. Always apply the exponential factor for compound interest problems. Final Answer: ₹ 8,112

Question 2

Compound Interest – Find the principal from a 3-year amount: A sum amounts to ₹ 3,149.29 in 3 years at compound interest (compounded annually). What was the original principal?

Accepted Answer

Introduction / Context: Back-calculating principal from a known compound amount is a standard inverse application of the compound interest formula. The question provides the final amount after 3 years, implying we can divide by the growth factor to retrieve the starting principal. Given Data / Assumptions: Amount after 3 years, A = 3,149.29 Compounded annually at some constant rate Answer choices suggest a common exam rate (e.g., 8%) that yields neat outcomes Concept / Approach: If the rate is 8% per annum, the 3-year factor is (1.08)^3 = 1.259712. Trying P = 2,500 gives A = 2,500 * 1.259712 ≈ 3,149.28/3,149.29, matching to cents/paise. Thus P = 2,500 is the consistent principal among the options. Step-by-Step Solution: Assume common exam rate r = 8% (validated by options)Compute factor: (1.08)^3 = 1.259712Trial with P = 2,500: A = 2,500 * 1.259712 = 3,149.28 ≈ 3,149.29Therefore, principal P = 2,500 fits perfectly within rounding. Verification / Alternative check: Reverse check: 3,149.29 / 2,500 = 1.259716 ≈ 1.259712 (rounding differences). No other option produces such a close exact power-of-1.08 match. Why Other Options Are Wrong: ₹ 1,500, ₹ 2,000, or ₹ 3,000 would produce amounts far from 3,149.29 under standard small integer rates. ₹ 2,200 similarly fails the precise match with common rate powers. Common Pitfalls: Guessing a principal without checking a realistic rate can mislead. Always test whether a candidate principal times a plausible factor equals the given amount within minor rounding. Final Answer: ₹ 2,500

Question 3

Compound Interest – Doubling in 5 years: when does it become eight times? If a sum of money placed at compound interest doubles itself in 5 years, in how many years will it become eight times the original principal?

Accepted Answer

Introduction / Context: Under compound interest at a constant rate, the time to reach a certain multiple of the principal scales with logarithms of the growth factor. If doubling time is known, other multiples related by powers of 2 can be deduced without computing the exact rate. Given Data / Assumptions: The principal becomes 2P in 5 years Same rate and compounding scheme throughout Target multiple: 8P Concept / Approach: If A = P * (growth_factor)^t and doubling occurs in 5 years, then growth_factor^5 = 2. Eight times means A = 8P = P * (growth_factor)^T. Since 8 = 2^3, we have (growth_factor)^T = (growth_factor)^(3*5) = 2^3. Therefore T = 3 * 5 = 15 years. No explicit rate calculation is required. Step-by-Step Solution: Given: time to double = 5 years ⇒ factor^5 = 2Eightfold target: 8 = 2^3Time to reach 8P = 3 × 5 = 15 years Verification / Alternative check: Using logs: T = 5 * log(8)/log(2) = 5 * 3 = 15. This confirms the power-of-two reasoning exactly. Why Other Options Are Wrong: 20, 12, and 10 years correspond to different multiples than 8 given the same doubling time; 25 years is unrelated to a power-of-two multiple based on a 5-year doubling benchmark. Common Pitfalls: Confusing simple and compound interest: only compound interest yields constant doubling intervals tied to logarithms. Do not attempt linear interpolation here. Final Answer: 15 years

Question 4

Compound vs Simple Interest – Convert CI over 2 years at 10% to equivalent SI: The compound interest on a certain principal for 2 years at 10% per annum is ₹ 420. What is the simple interest on the same principal at the same rate for the same time?

Accepted Answer

Introduction / Context: This problem asks you to translate a given compound interest (CI) result into the corresponding simple interest (SI) for the same principal, rate, and time. Once you recover the principal from the CI information, SI becomes a straightforward product of P, r, and t. Given Data / Assumptions: CI over 2 years at r = 10% per annum is ₹ 420 Compounding is annual We need SI for the same P, r, and t Concept / Approach: For annual compounding over 2 years, amount factor is (1 + 0.10)^2 = 1.21. Let principal be P. Then A − P = CI = 0.21P = 420, giving P = 2,000. Under simple interest, I_SI = P * r * t / 100 = 2,000 * 10 * 2 / 100 = 400. Step-by-Step Solution: CI relation: 0.21P = 420 ⇒ P = 420 / 0.21 = 2,000SI for same terms: I_SI = 2,000 * 10 * 2 / 100 = 400 Verification / Alternative check: Direct check of CI amount: A = 2,000 * 1.21 = 2,420, so CI = 420. SI for 2 years is 20% of 2,000 = 400, as computed. Why Other Options Are Wrong: ₹ 350, ₹ 375, ₹ 380, and ₹ 360 are not equal to 20% of ₹ 2,000 for 2 years and therefore do not match the same P, r, t under SI. Common Pitfalls: Trying to convert CI to SI without first recovering principal can lead to confusion. Always infer P from the CI setup, then compute SI directly. Final Answer: ₹ 400

Question 5

Compound Interest with Half-Yearly Compounding – Find time from amount: ₹ 1,600 at 10% per annum compounded half-yearly amounts to ₹ 1,944.81 in how much time?

Accepted Answer

Introduction / Context: When interest is compounded half-yearly, the annual rate is split across two periods per year. Determining time then reduces to counting the number of half-year periods needed to reach the given amount using the periodic factor. Given Data / Assumptions: P = 1,600 Nominal r = 10% per annum Compounded half-yearly ⇒ periodic rate = 10% / 2 = 5% Target amount A = 1,944.81 Concept / Approach: With half-yearly compounding, A = P * (1 + 0.05)^n, where n is the number of half-year periods. Solve (1.05)^n = A / P = 1,944.81 / 1,600 = 1.215506… Recognize that (1.05)^4 = 1.21550625, implying n = 4 half-years ⇒ t = 2 years. Step-by-Step Solution: Compute ratio: A/P = 1,944.81 / 1,600 ≈ 1.215506Check powers: (1.05)^2 = 1.1025; (1.05)^4 = 1.21550625Thus n = 4 half-years ⇒ time t = 4 / 2 = 2 years Verification / Alternative check: Forward compute: A = 1,600 * (1.05)^4 = 1,600 * 1.21550625 = 1,944.81 (exact to paise), confirming 2 years. Why Other Options Are Wrong: 1.5, 2.5, or 3 years correspond to 3, 5, or 6 half-year periods respectively, which yield amounts different from ₹ 1,944.81. Common Pitfalls: Using 10% as the per-period rate would double-count; for half-yearly compounding, use 5% per half-year and count half-year periods. Final Answer: 2 years

Question 6

Compound Interest – Find the annual rate from successive-year interests: If the compound interest of a certain sum for two successive years is ₹ 225 and ₹ 238.50 respectively, what is the annual rate of interest?

Accepted Answer

Introduction / Context: Under compound interest, the interest earned in the second year equals the first year’s interest grown by one more year at the same rate, because the second year accrues interest on both the principal and the first-year interest. This relationship allows us to extract the rate directly from the ratio of successive-year interests. Given Data / Assumptions: First-year CI = 225 Second-year CI = 238.50 Constant annual rate r Concept / Approach: For annual compounding, Interest in Year 2 = Interest in Year 1 * (1 + r/100). Therefore (1 + r/100) = 238.50 / 225. Solve for r by subtracting 1 and converting to percent. Step-by-Step Solution: Compute ratio: 238.50 / 225 = 1.06Thus 1 + r/100 = 1.06 ⇒ r/100 = 0.06Therefore r = 6% Verification / Alternative check: If r = 6%, any first-year interest I grows to I * 1.06 in Year 2, which fits 225 → 238.50 exactly. No principal value is required to confirm the rate. Why Other Options Are Wrong: 5%, 7.5%, 10%, or 4% would produce ratios 1.05, 1.075, 1.10, or 1.04—not equal to 1.06 observed from the data. Common Pitfalls: Equating the difference (238.50 − 225) to the rate times principal is incorrect; the relationship is multiplicative from one year’s interest to the next under compounding. Final Answer: 6%

Question 7

Compound Interest with Quarterly Compounding – 6 months at 20% p.a.: A moneylender lends ₹ 2,000 for 6 months at a nominal 20% per annum, with interest compounded quarterly. What amount will he receive at the end of the period?

Accepted Answer

Introduction / Context: Nominal annual rates split across compounding subperiods must be converted to a per-period rate before exponentiation. Six months with quarterly compounding equals two quarter-periods at one-quarter of the nominal annual rate each period. Given Data / Assumptions: P = 2,000 Nominal r = 20% per annum Compounded quarterly ⇒ periodic rate = 20%/4 = 5% per quarter Time = 6 months = 2 quarters Concept / Approach: Use A = P * (1 + i)^n with i = 0.05 and n = 2. This captures the interest-on-interest across the two compounding quarters. Step-by-Step Solution: Periodic rate i = 0.20 / 4 = 0.05Number of periods n = 2A = 2,000 * (1.05)^2 = 2,000 * 1.1025 = 2,205 Verification / Alternative check: Quarter 1: 2,000 → 2,100. Quarter 2: 2,100 * 1.05 = 2,205. Both methods agree on the final amount. Why Other Options Are Wrong: ₹ 2,200 ignores the second compounding step; ₹ 2,160 and ₹ 2,040 correspond to lower effective rates or simple interest; ₹ 2,250 would require 12.5% per quarter or longer time. Common Pitfalls: Applying 20% annually for half a year as 10% simple interest misses the quarterly compounding structure, which yields a slightly higher amount than ₹ 2,200. Final Answer: ₹ 2,205

Question 8

Compound Interest with Half-Yearly Compounding – Find time: In what time will ₹ 6,250 amount to ₹ 6,632.55 at 4% compound interest payable half-yearly?

Accepted Answer

Introduction / Context: With half-yearly compounding, the annual rate is split into two equal periods. We can compare the required amount ratio to integer powers of the per-period factor to count how many half-year steps are needed, then convert to years. Given Data / Assumptions: P = 6,250 Nominal r = 4% per annum Half-yearly compounding ⇒ per-period rate = 2% A = 6,632.55 Concept / Approach: Let n be the number of half-years. Then A/P = (1.02)^n. Compute A/P = 6,632.55 / 6,250 = 1.061208. Note that (1.02)^3 = 1.061208, so n = 3 half-years = 1.5 years. Step-by-Step Solution: A/P = 6,632.55 / 6,250 = 1.061208Per half-year factor = 1.02(1.02)^3 = 1.061208 ⇒ n = 3 half-yearsTime in years = 3 / 2 = 1.5 years Verification / Alternative check: Forward computation: 6,250 * (1.02)^3 = 6,250 * 1.061208 = 6,632.55 (matches exactly). Why Other Options Are Wrong: Other durations produce powers of 1.02 not equal to 1.061208, hence different amounts than ₹ 6,632.55. Common Pitfalls: Using 4% as the per-period rate would double the intended effect; always halve the rate and double the periods for half-yearly compounding. Final Answer: 1.5 years

Question 9

Effective Annual Rate – 4% nominal compounded quarterly: Find the effective annual rate corresponding to a nominal 4% per annum compound interest paid quarterly.

Accepted Answer

Introduction / Context: The effective annual rate (EAR) converts a nominal rate with intra-year compounding into a single equivalent annual rate that yields the same one-year growth. This is vital for comparing products with different compounding frequencies on an apples-to-apples basis. Given Data / Assumptions: Nominal annual rate r_nom = 4% Compounding frequency m = 4 (quarterly) We seek: EAR = (1 + r_nom/m)^m − 1 Concept / Approach: Plug in r_nom = 0.04 and m = 4. The per-quarter rate is 0.04/4 = 0.01; raising 1.01 to the 4th power gives the one-year growth multiplier, from which we subtract 1 to obtain the effective rate. Step-by-Step Solution: Per-quarter factor = 1 + 0.04/4 = 1.01Annual growth factor = (1.01)^4 = 1.04060401EAR = 1.04060401 − 1 = 0.04060401 ≈ 4.0604% Verification / Alternative check: Compare simple annual interest of 4% (EAR 4.00%) to quarterly compounding (EAR ≈ 4.0604%); the latter is slightly higher due to interest-on-interest within the year. Why Other Options Are Wrong: 4.604%, 5.0605%, and 5.605% are too large for only quarterly compounding at 4%; 4.0000% ignores intra-year compounding entirely. Common Pitfalls: Multiplying 4% by 4 or dividing without exponentiation is incorrect. Always use the compounding formula to translate nominal to effective rates. Final Answer: 4.0604%

Question 10

Compound Interest – Find time at 4% per annum from two amounts: In what time will ₹ 390,625 amount to ₹ 456,976 at 4% compound interest (annual compounding)?

Accepted Answer

Introduction / Context: Given initial and final amounts under a fixed compound rate, the time can be deduced by matching the ratio A/P to integer powers of the annual growth factor. This avoids logarithms when the data are designed to align with neat powers. Given Data / Assumptions: P = 390,625 A = 456,976 Annual rate r = 4% ⇒ factor per year = 1.04 Concept / Approach: Compute A/P and compare to powers of 1.04: 1.04^2 = 1.0816; 1.04^3 ≈ 1.124864; 1.04^4 ≈ 1.16985856. If A/P equals 1.16985856, time is 4 years. Step-by-Step Solution: A/P = 456,976 / 390,625 ≈ 1.16985856Recognize that 1.16985856 = (1.04)^4Hence, time t = 4 years Verification / Alternative check: Forward compute: 390,625 * (1.04)^4 = 390,625 * 1.16985856 = 456,976 exactly (these are perfect power/pair values often used in exams). Why Other Options Are Wrong: 2, 3, or 5 years correspond to different powers of 1.04 that do not hit the exact ratio 456,976 / 390,625. Common Pitfalls: Rounding too early may cause mismatch; use exact or high-precision values for powers of 1.04 to spot the correct integer exponent. Final Answer: 4 years

Question 11

Effective Annual Rate – 5% nominal compounded half-yearly: Find the effective annual rate corresponding to a nominal 5% per annum when interest is compounded half-yearly.

Accepted Answer

Introduction / Context: Converting nominal rates with intra-year compounding to an effective annual rate (EAR) is essential for fair comparisons. Half-yearly compounding at a 5% nominal rate produces a slightly higher true annual yield than 5% due to two compounding steps per year. Given Data / Assumptions: Nominal r = 5% per annum Compounding frequency m = 2 (half-yearly) EAR formula: (1 + r/m)^m − 1 Concept / Approach: Compute per-period rate r/m = 0.05/2 = 0.025, then square (1.025) to obtain the one-year growth factor and subtract 1 to find the EAR in percent. Step-by-Step Solution: Per half-year factor = 1.025Annual growth factor = (1.025)^2 = 1.050625EAR = 1.050625 − 1 = 0.050625 = 5.0625% Verification / Alternative check: Expressed in percent, 0.050625 → 5.0625%. This is a classic value that often appears in exams for nominal 5% with semiannual compounding. Why Other Options Are Wrong: 1.025% interprets the per-period rate incorrectly as an EAR; 5.062% is rounded; 5.500% and “None of these” are not the exact effective rate. Common Pitfalls: Using 5% directly as EAR ignores compounding; multiplying 2.5% by 2 without exponentiation also gives the wrong result. Always exponentiate across periods to get EAR. Final Answer: 6.0625%

Question 12

Compound Interest – Find annual rate from 2-year growth: At what annual compound-interest rate will ₹ 1,200 become ₹ 1,348.32 in 2 years?

Accepted Answer

Introduction / Context: When the starting principal and ending amount over a set number of years are known, the unknown annual compound rate can be obtained by inverting the compound growth equation and taking a root appropriate to the time period. Given Data / Assumptions: P = 1,200 A = 1,348.32 t = 2 years, compounded annually Concept / Approach: Use A = P * (1 + r)^t ⇒ (1 + r)^2 = A / P. Then 1 + r = sqrt(A / P). Subtract 1 and convert to percent to get the rate. Step-by-Step Solution: A / P = 1,348.32 / 1,200 = 1.12361 + r = sqrt(1.1236) = 1.06Therefore r = 1.06 − 1 = 0.06 = 6% Verification / Alternative check: Forward check: 1,200 * (1.06)^2 = 1,200 * 1.1236 = 1,348.32, confirming the rate exactly. Why Other Options Are Wrong: Rates 7%, 7.5%, and 6.5% produce larger growth than 1.1236 over 2 years; 5% produces less and does not reach ₹ 1,348.32. Common Pitfalls: Using simple interest or averaging methods will not match exponential compounding over more than one year; always use roots for reversing compounded growth. Final Answer: 6%

Question 13

Compound Interest – Repeated doublings: amount after 20 years when it doubles in 5 years: A sum of ₹ 2,400 deposited at compound interest doubles in 5 years. What will it become after 20 years?

Accepted Answer

Introduction / Context: If a deposit doubles in a fixed number of years at a constant compound rate, then over multiples of that period the amount multiplies by corresponding powers of 2. No explicit rate is necessary so long as the doubling interval is consistent over time. Given Data / Assumptions: Initial amount P = 2,400 Doubling time = 5 years Total time of interest = 20 years Constant compounding rate throughout Concept / Approach: Since 20 years is four times 5 years, the amount experiences four doublings. Each doubling multiplies by 2, so overall factor = 2^4 = 16. Final amount = 2,400 * 16 = 38,400. Step-by-Step Solution: Number of doublings = 20 / 5 = 4Overall multiplier = 2^4 = 16Final amount A = 2,400 * 16 = 38,400 Verification / Alternative check: Track progression: 2,400 → 4,800 (5 yr) → 9,600 (10 yr) → 19,200 (15 yr) → 38,400 (20 yr). The sequence confirms the calculation. Why Other Options Are Wrong: ₹ 24,000 corresponds to only three doublings from 3,000; ₹ 19,200 is the 15-year value (three doublings from 2,400). “Cannot be determined” is incorrect because the constant doubling interval suffices to compute the 20-year amount. Common Pitfalls: Confusing simple with compound interest can lead to linear scaling; remember that compounding induces exponential growth, and knowing the doubling time is enough to project to multiples thereof. Final Answer: ₹ 38,400

Question 14

Compound Interest – Rate from second- and third-year interests: Rohit earns ₹ 1,440 as interest for the second year and ₹ 1,656 for the third year on the same sum at compound interest. What is the annual rate?

Accepted Answer

Introduction / Context: Under compound interest, each year’s interest equals the previous year’s interest multiplied by (1 + r). Knowing two consecutive years of interest allows direct extraction of the rate without needing the principal value. Given Data / Assumptions: Second-year interest I2 = 1,440 Third-year interest I3 = 1,656 Annual rate r (constant) Concept / Approach: I3 = I2 * (1 + r). Therefore (1 + r) = I3 / I2. Compute the ratio and convert to a percentage to obtain r. Step-by-Step Solution: Compute ratio: 1,656 / 1,440 = 1.15Hence 1 + r = 1.15 ⇒ r = 0.15Convert to percent: r = 15% Verification / Alternative check: If r = 15%, any year’s interest grows by 15% to the next year: 1,440 * 1.15 = 1,656, which matches perfectly. The principal cancels out in this ratio method. Why Other Options Are Wrong: 12%, 10%, 18%, or 16% produce 1 + r values of 1.12, 1.10, 1.18, and 1.16 respectively, none of which equal the observed 1.15 ratio. Common Pitfalls: Trying to back out principal and then the rate is unnecessary and error-prone. Use the consecutive-interest ratio to shortcut directly to r. Final Answer: 15%

Question 15

Compound Interest – Find interest (not amount) on ₹ 50,000 in 3 years at 12% p.a.: What is the compound interest accrued on ₹ 50,000 after 3 years at 12% per annum (annual compounding)?

Accepted Answer

Introduction / Context: This question asks for the compound interest amount (A − P), not the final amount A. After computing the 3-year compounded amount at 12% per annum, subtract the principal to get the interest earned over the period. Given Data / Assumptions: P = 50,000 r = 12% per annum t = 3 years, annual compounding Concept / Approach: Compute A = P * (1 + r)^t = 50,000 * (1.12)^3. Then CI = A − P. Avoid confusing the final amount with the interest component. Step-by-Step Solution: (1.12)^2 = 1.2544; (1.12)^3 = 1.404928A = 50,000 * 1.404928 = 70,246.40CI = A − P = 70,246.40 − 50,000 = 20,246.40 Verification / Alternative check: You can also build year-by-year: Year 1 interest = 6,000; Year 2 interest = 6,720; Year 3 interest = 7,526.40; Total = 20,246.40. This equals A − P as above. Why Other Options Are Wrong: ₹ 70,246.40 is the final amount, not the interest; ₹ 80,000 and ₹ 70,000 are not relevant to the computed CI; ₹ 18,000 underestimates the compounded effect over three years at 12%. Common Pitfalls: Answering with the amount instead of interest is common. Read carefully: “compound interest on” means A − P, not A. Final Answer: ₹ 20,246.40

Question 16

Compound Interest – Find principal from 2-year amount at 4%: A sum amounts to ₹ 1,352 in 2 years at 4% compound interest (annual compounding). What was the principal?

Accepted Answer

Introduction / Context: Finding the original principal from a known future amount over a fixed period at compound interest is a reverse-compounding problem. We divide the amount by the growth factor (1 + r)^t to retrieve the starting principal. Given Data / Assumptions: A = 1,352 r = 4% per annum t = 2 years Annual compounding Concept / Approach: Use P = A / (1 + r)^t = 1,352 / (1.04)^2. Calculate the denominator precisely to avoid rounding drift when comparing close options. Step-by-Step Solution: (1.04)^2 = 1.0816P = 1,352 / 1.0816 = 1,250 Verification / Alternative check: Forward check: 1,250 * 1.0816 = 1,352, confirming the principal exactly. Why Other Options Are Wrong: ₹ 1,200, ₹ 1,260, and ₹ 1,300 fail the exact reverse-compounding equality at 4% over 2 years. ₹ 1,180 is far from the required value. Common Pitfalls: Using simple interest to reverse compute principal will not match the compounded outcome; always divide by (1 + r)^t for CI problems. Final Answer: ₹ 1,250

Question 17

Present Worth under Compound Interest – Discount a future amount at 4%: What is the present worth of ₹ 169 due in 2 years at 4% per annum compounded annually?

Accepted Answer

Introduction / Context: Discounting is the reverse of compounding. Present worth finds today’s equivalent of a known future amount by dividing by the appropriate growth factor. Here, we discount ₹ 169 due after 2 years at 4% compounded annually. Given Data / Assumptions: Future amount F = 169 in 2 years Rate r = 4% per annum Compounding annually Concept / Approach: Present value PV = F / (1 + r)^t. Substitute F = 169, r = 0.04, t = 2. Compute (1.04)^2 accurately to ensure an exact match with standard values used in exam questions. Step-by-Step Solution: (1.04)^2 = 1.0816PV = 169 / 1.0816 = 156.25 Verification / Alternative check: Forward check: 156.25 grown for 2 years at 4% yields 156.25 * 1.0816 = 169, confirming exact equivalence. Why Other Options Are Wrong: ₹ 160 and ₹ 158 are approximations that do not reproduce ₹ 169 after compounding; ₹ 154.75 and ₹ 150.50 understate the required present value. Common Pitfalls: Applying simple discounting instead of compound discounting produces slightly different results; ensure you divide by the compounded factor, not 1 + r * t. Final Answer: ₹ 156.25

Question 18

Compound Interest with Quarterly Compounding – 9 months at 16% p.a.: find CI only: Find the compound interest on ₹ 31,250 at 16% per annum compounded quarterly for 9 months.

Accepted Answer

Introduction / Context: When compounding quarterly, a nominal annual rate is split into four equal parts, and the number of periods equals the number of quarters. Nine months corresponds to three quarters, so we apply the quarterly factor three times to get the amount, then subtract the principal to obtain the compound interest (CI) alone. Given Data / Assumptions: P = 31,250 Nominal r = 16% per annum ⇒ per quarter i = 16%/4 = 4% Time = 9 months ⇒ n = 3 quarters Concept / Approach: Compute amount: A = P * (1 + i)^n = 31,250 * (1.04)^3. Then CI = A − P. Be careful to report CI (interest) rather than the final amount. Step-by-Step Solution: (1.04)^3 = 1.124864A = 31,250 * 1.124864 = 35,152.00CI = 35,152.00 − 31,250 = 3,902.00 Verification / Alternative check: Quarter-by-quarter: 31,250 → 32,500 → 33,800 → 35,152; total interest added = 3,902, confirming the calculation. Why Other Options Are Wrong: ₹ 4,000 and ₹ 4,200 overstate CI; ₹ 3,500 and ₹ 3,750 understate it; only ₹ 3,902 matches the exact quarterly compounding result for 3 periods at 4% each. Common Pitfalls: Reporting the amount instead of the CI is a frequent error. Also, do not treat nine months as three simple-interest quarters; apply compounding each quarter. Final Answer: ₹ 3,902

Question 19

Compound Interest – Find principal from amounts at 2 and 3 years: A sum amounts to ₹ 2,916 in 2 years and ₹ 3,149.28 in 3 years at compound interest (annual). What is the principal?

Accepted Answer

Introduction / Context: Given two consecutive annual amounts under compounding, the ratio A(3yr) / A(2yr) equals (1 + r). This quickly yields the rate; dividing either amount by the appropriate power then recovers the principal. This is an efficient two-step method without solving simultaneous equations. Given Data / Assumptions: A(2yr) = 2,916 A(3yr) = 3,149.28 Annual compounding at constant rate Concept / Approach: Compute r from ratio: 3,149.28 / 2,916 = 1.08 ⇒ r = 8%. Then P = A(2yr) / (1.08)^2. With (1.08)^2 = 1.1664, we get P = 2,916 / 1.1664 = 2,500. Step-by-Step Solution: Find rate: (A3/A2) − 1 = 1.08 − 1 = 0.08 = 8%Compute (1.08)^2 = 1.1664Principal P = 2,916 / 1.1664 = 2,500 Verification / Alternative check: Forward: 2,500 → 2,700 (Year 1) → 2,916 (Year 2) → 3,149.28 (Year 3). All values align exactly with the problem statement. Why Other Options Are Wrong: ₹ 1,500, ₹ 2,000, and ₹ 3,000 do not reproduce the provided amounts at 8% compounding; ₹ 2,400 is close but still inconsistent when forward-compounded. Common Pitfalls: Using simple interest or averaging the amounts leads to incorrect principals. Always divide by the appropriate compound factor to back out P. Final Answer: ₹ 2,500

Question 20

Compound vs Simple Interest — small second-year lift at 5%: The difference between the compound interest and the simple interest on the same principal at 5% per annum for 2 years is ₹ 1.50. What is the principal (the original sum)?

Accepted Answer

Introduction / Context: For the same principal and term, compound interest (CI) slightly exceeds simple interest (SI) because interest in the second year is earned on both principal and the first year’s interest. The excess for 2 years depends only on principal and rate. Given Data / Assumptions: Annual rate r = 5% = 0.05 Time t = 2 years CI − SI = ₹ 1.50 Principal P = ? Concept / Approach: With annual compounding for 2 years, the well-known relationship is: CI − SI = P * (r^2). This is the “interest on interest” generated in the second year: first-year interest is P * r; interest on it next year is (P * r) * r = P * r^2. Step-by-Step Solution: CI − SI = P * r^21.50 = P * (0.05)^21.50 = P * 0.0025P = 1.50 / 0.0025 = ₹ 600 Verification / Alternative check: At P = 600, SI for 2 years = 600 * 0.05 * 2 = ₹ 60. CI amount = 600 * (1.05)^2 = 600 * 1.1025 = ₹ 661.50 → CI = ₹ 61.50. Difference = 61.50 − 60 = ₹ 1.50 (matches). Why Other Options Are Wrong: ₹ 500, ₹ 400, ₹ 300, or ₹ 750 produce differences different from ₹ 1.50 when substituted into P * r^2 at r = 5%. Common Pitfalls: Using r/100 twice incorrectly, forgetting that for 2 years CI − SI equals P * r^2 (not P * r * t adjustments). Final Answer: ₹ 600

Compound Interest Questions