Question 1

Compound vs Simple Interest — Find the difference (rounded to 2 decimals) between compound interest and simple interest on ₹ 16,200 for 3 years at 25% per annum.

Accepted Answer

Introduction / Context: The excess of compound interest over simple interest across multiple years can be computed directly from rate and time without first finding the principal from amounts. Given Data / Assumptions: P = ₹ 16,200 r = 25% = 0.25 per annum t = 3 years Concept / Approach: For 3 years, CI − SI = P * (3r^2 + r^3) = P * r^2 * (3 + r). This comes from expanding (1 + r)^3 − 1 = 3r + 3r^2 + r^3 and subtracting 3r (the SI part). Step-by-Step Solution: r^2 = 0.25^2 = 0.0625; (3 + r) = 3.25Multiplier = 0.0625 * 3.25 = 0.203125Difference = 16,200 * 0.203125 = 3,290.625 ⇒ ₹ 3,290.63 (rounded to 2 decimals) Verification / Alternative check: Compute SI and CI separately and subtract; the above closed form should match exactly, saving time. Why Other Options Are Wrong: They correspond to using a wrong formula such as P*r^2 (valid only for 2 years) or arithmetic slips when multiplying. Common Pitfalls: Using 2-year difference formula P*r^2 for a 3-year case, or forgetting to round to 2 decimals. Final Answer: ₹ 3,290.63

Question 2

Compound Interest — Divide ₹ 2,602 between M and N so that M’s amount after 7 years equals N’s amount after 9 years at 4% per annum compounded annually. Find individual shares.

Accepted Answer

Introduction / Context: We split a fixed principal between two people so that their future amounts (at different times) are equal under the same compound-interest rate. Given Data / Assumptions: Total = ₹ 2,602. Rate = 4% p.a. compounded annually. M grows for 7 years; N grows for 9 years. Concept / Approach: Let M's share be x ⇒ N's share = 2,602 − x. The equality of future amounts is x*(1.04)^7 = (2,602 − x)*(1.04)^9. Canceling the common factor (1.04)^7 gives x = (2,602 − x)*(1.04)^2. Step-by-Step Solution: (1.04)^2 = 1.0816x = 1.0816*(2,602 − x) ⇒ x + 1.0816x = 2,602*1.08162.0816x = 2,814.3232 ⇒ x = ₹ 1,352N's share = 2,602 − 1,352 = ₹ 1,250 Verification / Alternative check: Check equality: 1,352*(1.04)^7 equals 1,250*(1.04)^9 (since multiplying M by (1.04)^2 matches N's longer term). Why Other Options Are Wrong: Only ₹ 1,352 and ₹ 1,250 exactly satisfy the compounding relation; other pairs do not. Common Pitfalls: Forgetting to cancel common powers or using simple interest instead of compound interest. Final Answer: ₹ 1,352, ₹ 1,250

Question 3

Compound Interest — SBI lends ₹ 1,331 lakh to TATA and receives ₹ 1,728 lakh after 3 years with annual compounding. Find the annual rate of interest.

Accepted Answer

Introduction / Context: When both the principal and amount are perfect cubes, recognizing the cube ratio directly yields the growth factor per year. Given Data / Assumptions: P = 1,331 lakh; A (3 years) = 1,728 lakh. Annual compounding for 3 years. Concept / Approach: (1 + r)^3 = A / P. Note 1,331 = 11^3 and 1,728 = 12^3. Therefore (1 + r)^3 = (12/11)^3 ⇒ 1 + r = 12/11. Step-by-Step Solution: 1 + r = 12/11 ⇒ r = (12/11) − 1 = 1/11 ≈ 0.0909r ≈ 9.09% per annum Verification / Alternative check: Apply 9.09% for 3 years: P*(1 + 1/11)^3 = 11^3*(12/11)^3 = 12^3, matching A. Why Other Options Are Wrong: 11% and 12% overshoot; 8.33% undershoots; 10% gives 1.1^3 = 1.331, not 1.299... ratio required. Common Pitfalls: Using simple interest formula or comparing linear differences instead of cube roots. Final Answer: 9.09%

Question 4

Compound Interest — A sum invested for 2 years at a nominal 20% p.a. yields ₹ 964 more when interest is payable half-yearly than when payable annually. Find the principal.

Accepted Answer

Introduction / Context: Nominal 20% p.a. compounded semiannually means 10% every half-year across 4 periods in 2 years. Annual compounding at 20% uses 2 periods of 20% each. The difference in amounts is stated. Given Data / Assumptions: Rate (nominal): 20% p.a. Time: 2 years. Amount difference (semiannual − annual) = ₹ 964. Concept / Approach: Let P be the principal. Annual factor: (1 + 0.20)^2 = 1.44. Semiannual factor: (1 + 0.10)^4 = 1.4641. Difference in amounts: P * (1.4641 − 1.44) = P * 0.0241 = 964. Step-by-Step Solution: P = 964 / 0.0241 = 964 * 10000 / 241 = ₹ 40,000 Verification / Alternative check: 40,000 * 1.4641 = 58,564; 40,000 * 1.44 = 57,600; difference = 964 ✔ Why Other Options Are Wrong: They do not satisfy the exact 0.0241 multiplier relation. Common Pitfalls: Confusing nominal with effective annual rate; here we compare compounding frequencies at the same nominal rate. Final Answer: ₹ 40,000

Question 5

Compound Interest — On ₹ 4,000 the compound interest earned in 9 months (interest compounded quarterly) is ₹ 630.50. Find the annual nominal rate (compounded quarterly).

Accepted Answer

Introduction / Context: Quarterly compounding over 9 months means 3 compounding periods. We can infer the per-quarter growth factor from the stated interest and back out the nominal annual rate. Given Data / Assumptions: P = ₹ 4,000. n = 3 quarters (9 months). Interest (CI over 3 quarters) = ₹ 630.50 ⇒ Amount factor over 3 quarters = 1 + 630.50/4,000 = 1.157625. Concept / Approach: Let quarterly rate be r_q. Then (1 + r_q)^3 = 1.157625. Recognize 1.157625 = 1.05^3 ⇒ r_q = 0.05 per quarter. Step-by-Step Solution: r_q = 5% per quarter ⇒ nominal annual rate (compounded quarterly) = 4 * 5% = 20% p.a. Verification / Alternative check: Compute amount: 4,000 * (1.05)^3 = 4,000 * 1.157625 = 4,630.50 ⇒ CI = 630.50 ✔ Why Other Options Are Wrong: They correspond to different quarterly rates that do not cube to 1.157625. Common Pitfalls: Using 9 months as 0.75 years with annual compounding instead of 3 quarterly periods. Final Answer: 20%

Question 6

Successive Percentage Changes — Shantanu’s income is ₹ 4,000. It decreases by 10% in the first year and by 5% in the second year, but increases by 15% in the third year. What is his income at the end of the third year?

Accepted Answer

Introduction / Context: Successive percentage changes multiply, they do not add. A decrease followed by an increase is evaluated by chaining the corresponding factors. Given Data / Assumptions: Initial income = ₹ 4,000. Year 1: −10% ⇒ factor 0.90. Year 2: −5% ⇒ factor 0.95. Year 3: +15% ⇒ factor 1.15. Concept / Approach: Final income = 4,000 * 0.90 * 0.95 * 1.15. Step-by-Step Solution: After Year 1: 4,000 * 0.90 = 3,600After Year 2: 3,600 * 0.95 = 3,420After Year 3: 3,420 * 1.15 = 3,933 Verification / Alternative check: The combined multiplier is 0.90 * 0.95 * 1.15 = 0.98325; 4,000 * 0.98325 = 3,933. Why Other Options Are Wrong: They arise from adding percentages (−10 −5 +15 = 0) which is incorrect; or from rounding at intermediate steps. Common Pitfalls: Adding percentage changes instead of multiplying the factors; ignoring order and compounding effect. Final Answer: ₹ 3,933

Question 7

Simple vs Compound Interest — The simple interest on a certain sum for 2 years at 4% p.a. is ₹ 80. What is the compound interest on the same sum for 2 years at 4% p.a.?

Accepted Answer

Introduction / Context: From the stated simple interest, we can infer the principal, then compute the corresponding compound interest over the same time at the same rate. Given Data / Assumptions: SI = ₹ 80 for 2 years at 4% p.a. Need CI for 2 years at 4% p.a. on the same principal. Concept / Approach: SI = P * r * t ⇒ P = SI / (r * t) = 80 / (0.04 * 2) = 1,000. CI (2 years) = P * [(1 + r)^2 − 1]. Step-by-Step Solution: P = 1,000(1 + 0.04)^2 − 1 = 1.0816 − 1 = 0.0816CI = 1,000 * 0.0816 = ₹ 81.60 Verification / Alternative check: Compute yearwise: Year-1 interest 40; Year-2 interest on amount 1,040 is 41.60; total 81.60. Why Other Options Are Wrong: ₹ 80 equals SI, not CI; others come from arithmetic slips. Common Pitfalls: Assuming SI and CI are equal over 2 years; they only match for t = 1 year. Final Answer: ₹ 81.60

Question 8

Variable Compound Rates — A sum is invested at 5%, 10%, and 20% successively for three years (compounded annually) and amounts to ₹ 16,632. Find the original principal.

Accepted Answer

Introduction / Context: When annual rates vary year to year, the amount is the product of each year's growth factor times the principal. Given Data / Assumptions: A = ₹ 16,632. Year-wise rates: 5%, then 10%, then 20%. Concept / Approach: Amount factor = 1.05 * 1.10 * 1.20 = 1.386. Thus A = P * 1.386. Step-by-Step Solution: P = 16,632 / 1.386 = ₹ 12,000 Verification / Alternative check: 12,000 * 1.386 = 16,632 exactly. Why Other Options Are Wrong: They do not match the precise division by 1.386. Common Pitfalls: Adding rates instead of multiplying annual growth factors. Final Answer: ₹ 12,000

Question 9

Loan Amortization (Compound Interest) — ₹ 5,100 is to be repaid in two equal yearly installments at 4% p.a. compounded annually. Find each installment.

Accepted Answer

Introduction / Context: For equal annual installments under compound interest, the present value of all installments at the loan rate must equal the loan amount. Given Data / Assumptions: Principal (loan) = ₹ 5,100. Rate i = 4% p.a. (compounded yearly). Installments: two, paid at ends of years 1 and 2. Concept / Approach: Let each installment be A. Present value: PV = A/(1 + i) + A/(1 + i)^2. Set PV = 5,100 and solve for A. Equivalently, PV = A * [1 − (1 + i)^(−2)] / i. Step-by-Step Solution: 5,100 = A * [1 − (1.04)^(−2)] / 0.04[1 − (1.04)^(−2)] = 1 − 1/1.0816 = 0.075444…A = 5,100 * 0.04 / 0.075444… ≈ 5,100 * 0.53023… ≈ ₹ 2,704 Verification / Alternative check: PV check: 2,704/1.04 + 2,704/1.0816 ≈ 2,600 + 2,500 = 5,100 (rounded). Why Other Options Are Wrong: They do not equate PV to ₹ 5,100 at 4% with two payments. Common Pitfalls: Summing installments without discounting or using simple interest instead of compounding for PV. Final Answer: ₹ 2,704

Question 10

Compound Interest — An amount invested at a compound rate yields totals of ₹ 1,200 after year 1 and ₹ 1,587 after year 3. Find the annual rate of interest.

Accepted Answer

Introduction / Context: Knowing the amount after 1 year and after 3 years under the same compound rate allows us to cancel the principal and isolate the growth factor. Given Data / Assumptions: A1 = ₹ 1,200 = P*(1 + r) A3 = ₹ 1,587 = P*(1 + r)^3 Concept / Approach: Divide A3 by A1: (1 + r)^2 = 1,587 / 1,200 = 1.3225. Hence 1 + r = √1.3225. Step-by-Step Solution: √1.3225 = 1.15 ⇒ r = 0.15 = 15% p.a. Verification / Alternative check: Check: A1 = P*1.15; A3 = P*1.15^3 = P*1.520875; ratio to A1 is 1.15^2 = 1.3225 matching 1,587/1,200. Why Other Options Are Wrong: They do not square to 1.3225 when added to 1. Common Pitfalls: Attempting to subtract amounts to find r rather than using the ratio method. Final Answer: 15%

Question 11

Loan Amortization (Compound Interest) — ₹ 11,000 is to be repaid in two equal annual installments at 20% p.a. compounded annually. Find the value of each installment.

Accepted Answer

Introduction / Context: Equal annual installments under compound interest are determined by equating the present value of all installments to the loan amount at the loan rate. Given Data / Assumptions: Loan = ₹ 11,000. Annual rate i = 20%. Number of installments n = 2 (end of years 1 and 2). Concept / Approach: PV factor for an n-year annuity-immediate: PV = A * [1 − (1 + i)^(−n)] / i. Set PV = 11,000 and solve for A. Step-by-Step Solution: A = 11,000 * i / [1 − (1 + i)^(−2)] with i = 0.20(1 + i)^(−2) = 1 / 1.44 = 0.694444… ⇒ denominator = 1 − 0.694444… = 0.305555…A = 11,000 * 0.20 / 0.305555… ≈ 11,000 * 0.654545… ≈ ₹ 7,200 Verification / Alternative check: PV check: 7,200/1.20 + 7,200/1.44 = 6,000 + 5,000 = 11,000 ✔ Why Other Options Are Wrong: They fail the PV equality at 20% with two payments. Common Pitfalls: Using simple interest discounting or forgetting to divide each installment by the appropriate compound factor. Final Answer: ₹ 7,200

Question 12

Compound Interest — ₹ 8,448 is to be divided between A (age 18) and B (age 19) so that if their present shares are invested at 6.25% per annum compounded annually, they will receive equal amounts when each turns 21 years old. What is A’s present sha…

Accepted Answer

Introduction / Context: This is a present-worth allocation under compound interest. Two people receive money today but will let it grow for different numbers of years before comparing outcomes at a common target age. Because the annual rate is the same for both, we can relate their present shares by requiring equal future amounts on their respective “turning-21” dates. Given Data / Assumptions: Total to divide now = ₹ 8,448. A is 18, so money grows for 3 years (to age 21). B is 19, so money grows for 2 years (to age 21). Rate r = 6.25% = 0.0625 per annum, compounded annually. Growth factor g = 1 + r = 1.0625. Concept / Approach: Let present shares be A0 for A and B0 for B. Equal future amounts at age 21 mean A0 * g^3 = B0 * g^2. Thus A0 / B0 = g^(2−3) = g^(−1) = 1 / g. Therefore A0 : B0 = 1 : g = 16 : 17, using g = 17/16. Step-by-Step Solution: Ratio A0 : B0 = 16 : 17Total parts = 16 + 17 = 33Value per part = 8,448 / 33 = 256A0 = 16 × 256 = ₹ 4,096B0 = 17 × 256 = ₹ 4,352 Verification / Alternative check: Future amounts at 21: A's future = 4,096 * (1.0625)^3; B's future = 4,352 * (1.0625)^2. Since (1.0625)^3 / (1.0625)^2 = 1.0625, and A0/B0 = 1/1.0625, both futures match exactly. Why Other Options Are Wrong: ₹ 4,352 is B's present share, not A's. ₹ 4,225 and ₹ 4,180 break the 16:17 ratio. ₹ 4,000 also fails to satisfy the equality of future amounts at the given rate. Common Pitfalls: Using simple interest instead of compound growth; forgetting that A grows for 3 years while B grows for only 2; or dividing the total equally without adjusting for time. Final Answer: ₹ 4,096

Question 13

Simple vs Compound Interest — The simple interest (SI) on a certain sum for 3 years at 8% per annum is half the compound interest (CI) on ₹ 8,000 for 2 years at 10% per annum. Find the sum on which SI is calculated.

Accepted Answer

Introduction / Context: We are asked to compare a given simple-interest amount to half of a known compound-interest amount. The trick is to compute the CI on ₹ 8,000 over 2 years at 10% first, then halve it, and finally back-solve the principal that generates that simple interest at 8% for 3 years. Given Data / Assumptions: CI base: ₹ 8,000, rate = 10% per annum, time = 2 years, compounding annually. SI case: unknown principal P at 8% per annum for 3 years. SI (3 years, 8%) = (1/2) × CI(₹ 8,000, 2 years, 10%). Concept / Approach: For 2 years, compound-interest factor is (1 + 0.10)^2 = 1.21. CI = P_ci * (1.21 − 1) = 8,000 * 0.21 = ₹ 1,680. Half of that is ₹ 840, which equals the SI on the unknown principal for 3 years at 8%. Step-by-Step Solution: Half of CI = 1,680 / 2 = ₹ 840SI formula: SI = P * r * t840 = P * 0.08 * 3P = 840 / 0.24 = ₹ 3,500 Verification / Alternative check: Check SI: 3,500 * 0.08 * 3 = 840, exactly half of 1,680. Hence consistent. Why Other Options Are Wrong: ₹ 3,800, ₹ 3,600, and ₹ 3,200 result from algebra or arithmetic slips; ₹ 4,000 overestimates the principal and would give SI = 960, not 840. Common Pitfalls: Confusing CI with SI formulas; using 10% for 3 years instead of 2; or forgetting to halve the computed CI amount. Final Answer: ₹ 3,500

Question 14

Population Change — A village population increases by 5% in the first year and then decreases by 5% in the second year. If the population after the second year is 47,880, what was the population at the beginning of the first year?

Accepted Answer

Introduction / Context: Successive percentage changes are multiplicative. An increase followed by a decrease of the same rate does not return to the initial value because the base changes after the first year. We exploit multiplicative factors to back-compute the original population from the final value. Given Data / Assumptions: Year 1: +5% ⇒ multiply by 1.05 Year 2: −5% ⇒ multiply by 0.95 Final population after two years = 47,880 Concept / Approach: Let initial population be P0. After two years: P2 = P0 * 1.05 * 0.95 = P0 * 0.9975. Therefore P0 = P2 / 0.9975. Step-by-Step Solution: P0 = 47,880 / 0.99751 / 0.9975 ≈ 1.002506…P0 ≈ 47,880 * 1.002506… ≈ 48,000 Verification / Alternative check: Forward check: 48,000 * 1.05 = 50,400; then 50,400 * 0.95 = 47,880 ✔ Why Other Options Are Wrong: 45,500 and 43,500 are too low; 53,000 is too high; 47,760 is a rounding distraction not matching the exact reversal. Common Pitfalls: Adding percentages (+5 − 5 = 0) and assuming no change, which is incorrect because the 5% decrease applies to the larger base after the increase. Final Answer: 48,000

Question 15

Compound Interest with Fractional Year — Find the amount receivable on ₹ 1,750 in 2.5 years at 8% per annum when interest is compounded annually (assume the standard rule for fractional years).

Accepted Answer

Introduction / Context: When compounding is specified annually but the total time includes a fractional part (e.g., 0.5 year), the standard treatment is: compound for the full integer years, then add simple interest for the remaining fractional part on the amount after the integer years, computed at the same annual rate pro-rated for the fraction. Given Data / Assumptions: Principal P = ₹ 1,750. Annual rate r = 8% = 0.08. Total time = 2.5 years: 2 full years + 0.5 year fractional part. Concept / Approach: Step 1: Compound for 2 years. Step 2: Apply simple interest for 0.5 year on the amount obtained after 2 years at the same rate, proportionally for 0.5 year. Step-by-Step Solution: Amount after 2 years: A2 = 1,750 * (1.08)^2 = 1,750 * 1.1664 = ₹ 2,041.20Interest for remaining 0.5 year: I_half = A2 * r * 0.5 = 2,041.20 * 0.08 * 0.5 = 2,041.20 * 0.04 = ₹ 81.648Total amount: A = A2 + I_half = 2,041.20 + 81.648 = ₹ 2,122.848 ≈ ₹ 2,122.85 Verification / Alternative check: If we incorrectly compounded for 2.5 years as (1.08)^(2.5), we would implicitly assume sub-annual compounding, which is not stated. The standard textbook rule used here matches the provided options. Why Other Options Are Wrong: ₹ 2,125 and ₹ 2,118 round or mis-apply the half-year step. ₹ 2,100 and ₹ 2,200 are rough estimates not consistent with precise calculation. Common Pitfalls: Treating 0.5 year as another compounding period when only annual compounding is specified; or applying the 8% to the original principal for the last half-year rather than to the amount after 2 years. Final Answer: ₹ 2,122.85

Question 16

Threshold Year under Compound Interest — At 10% per annum compounded annually, in which first year will a sum become more than double its original amount?

Accepted Answer

Introduction / Context: The question asks for the earliest integer year n such that the compound-interest growth factor (1.10)^n exceeds 2. This is a threshold or crossing-point problem under geometric growth, common in time-value-of-money topics. Given Data / Assumptions: Growth per year g = 1 + 10% = 1.10. We seek smallest integer n with (1.10)^n > 2. Concept / Approach: Either compute successive powers or use logarithms. Powers are quick here: 1.1^7 ≈ 1.9487, still below 2; 1.1^8 ≈ 2.1436, above 2. Hence the first year exceeding double is year 8. Step-by-Step Solution: 1.1^5 ≈ 1.610511.1^6 ≈ 1.771561.1^7 ≈ 1.948721.1^8 ≈ 2.14359 > 2 ⇒ threshold crossed at n = 8 Verification / Alternative check: Log method: n > ln(2) / ln(1.1) ≈ 0.6931 / 0.09531 ≈ 7.27 ⇒ smallest integer n = 8. Why Other Options Are Wrong: 6th and 7th years are below the threshold; “Data inadequate” is incorrect because g is fully specified; 9th year is not the first crossing. Common Pitfalls: Confusing “at least double” with “more than double,” which changes the inequality if a power equaled 2 exactly (it does not here). Final Answer: 8th year

Question 17

Sharon Stone deposits 2000 dollars at the end of each year into an account that earns 10% interest compounded annually for 25 years. Approximately how much interest will she have earned on her deposits by the end of 25 years?

Accepted Answer

Introduction / Context: This question is a compound interest annuity problem. Instead of investing a single lump sum, Sharon Stone deposits 2000 dollars at the end of every year, and the money grows at 10% interest compounded annually. We are asked to find the total interest earned after 25 years. Such problems are important in personal finance and engineering economics as they model recurring investments like retirement contributions. Given Data / Assumptions: - Regular deposit each year = 2000 dollars. - Deposits are made at the end of each year (ordinary annuity). - Annual interest rate r = 10% = 0.10 per year. - Number of years n = 25. - Interest is compounded annually. - We need the total interest earned, not just the final balance. Concept / Approach: For an ordinary annuity, the future value after n periods at interest rate r is given by the formula: FV = P * ((1 + r)^n - 1) / rwhere P is the periodic payment. Once we compute FV, the total deposits made are P * n. The total interest earned is then FV minus the sum of all deposits. This separates principal from interest growth due to compounding. Step-by-Step Solution: Step 1: Identify P = 2000, r = 0.10, n = 25.Step 2: Use the future value formula for an ordinary annuity: FV = 2000 * ((1 + 0.10)^25 - 1) / 0.10.Step 3: Compute (1 + 0.10)^25 which is approximately 10.8347.Step 4: Then ((1 + 0.10)^25 - 1) = 10.8347 - 1 = 9.8347.Step 5: Divide by r: 9.8347 / 0.10 = 98.347.Step 6: Multiply by P: FV = 2000 * 98.347 which is approximately 196694.12 dollars.Step 7: Total deposits over 25 years = 2000 * 25 = 50000 dollars.Step 8: Total interest earned = FV - total deposits = 196694.12 - 50000 = 146694.12 dollars. Verification / Alternative Check: We can roughly verify by observing that without compounding, simple interest on 50000 dollars at 10% for 25 years would be 50000 * 0.10 * 25 = 125000 dollars. Because the deposits are staggered over time rather than invested all at once, the interest should be somewhat near but a bit higher than this simple estimate due to early deposits compounding longer. The calculated interest of approximately 146694.12 dollars is reasonable and aligns with the annuity formula result. Why Other Options Are Wrong: The amounts 13452, 18232 and 15627 dollars are far too small for interest earned on repeated contributions of 2000 dollars over 25 years at 10%. They are closer to one or two years of simple interest on a few deposits rather than a long term annuity. Only 146694.12 dollars matches the correct computation of future value minus total deposits. Common Pitfalls: Many learners mistakenly apply the simple interest formula, or they treat the problem as a single lump sum investment instead of a series of equal yearly deposits. Others forget that deposits are at the end of each year, which is the standard ordinary annuity case. It is also common to confuse the final accumulated amount with the interest portion. Always compute the future value using the annuity formula and then subtract the principal contributions to isolate the interest earned. Final Answer: The total interest earned by Sharon Stone after 25 years is approximately 146694.12 dollars.

Question 18

In a compound interest calculation, find the compound interest on Rs 8000 at the rate of 15% per annum for a total time period of 2 years and 4 months, when the interest is compounded annually.

Accepted Answer

Introduction: This question tests your understanding of compound interest with a non integral time period, where interest is compounded annually but the total time includes years and additional months. You must know how to treat the full years using the compound interest formula and the remaining fractional year using simple interest on the accumulated amount. Given Data / Assumptions: Principal (P) = Rs 8000. Rate of interest (r) = 15% per annum. Total time = 2 years and 4 months. Interest is compounded annually. For the extra 4 months, interest is taken as simple interest on the amount at the end of 2 years. Concept / Approach: For full years, use the compound interest formula A = P * (1 + r/100)^n. For the fractional part of the year, treat the interest as simple interest on the amount already accrued. Finally, subtract the original principal from the final amount to get the compound interest earned over the entire period. Step-by-Step Solution: Amount after 2 years: A2 = 8000 * (1 + 15/100)^2A2 = 8000 * (1.15)^2 = 8000 * 1.3225 = Rs 10580Interest for remaining 4 months (which is 4/12 year) at 15% on Rs 10580:I4m = 10580 * 15/100 * (4/12)I4m = 10580 * 0.15 * 1/3 = 10580 * 0.05 = Rs 529Total amount at the end of 2 years 4 months = 10580 + 529 = Rs 11109Total compound interest = 11109 - 8000 = Rs 3109 Verification / Alternative Check: You can separately compute the interest for 2 years: I2 = 10580 - 8000 = Rs 2580. Then add the extra 4 months interest of Rs 529 to get 2580 + 529 = Rs 3109, which confirms the result without redoing the full calculation. Why Other Options Are Wrong: Rs 2109: This ignores some part of the compounding or the extra 4 months.Rs 4109: This overestimates interest, possibly by compounding for 3 full years.Rs 6109: This is far too large and not consistent with a 15% rate for just over 2 years.Rs 2709: This underestimates the extra fractional year interest. Common Pitfalls: Students often either treat the whole 2 years 4 months as 2.33 years in the compound interest formula or forget that the extra months are to be treated using simple interest on the amount after full years. Another mistake is using the principal instead of the amount after 2 years to calculate interest for the last 4 months. Final Answer: The compound interest on Rs 8000 at 15% per annum for 2 years and 4 months, compounded annually, is Rs 3109.

Question 19

A sum of money is lent at compound interest for 2 years at the rate of 20% per annum. If the interest were payable half yearly instead of annually, the amount of interest earned over 2 years would be Rs 482 more. What is the principal (original sum …

Accepted Answer

Introduction: This question examines your understanding of how compounding frequency affects the amount of compound interest. The nominal rate remains 20% per annum, but the interest may be compounded annually or semi annually, creating different accumulated amounts and hence a difference in interest earned. Given Data / Assumptions: Annual rate of interest r = 20% per annum. Time period = 2 years. Case 1: Interest compounded annually. Case 2: Interest compounded half yearly (every 6 months). Difference in interest between Case 2 and Case 1 over 2 years = Rs 482. We must find the principal P. Concept / Approach: For compound interest, the amount depends on both the nominal rate and the compounding frequency. With half yearly compounding, the rate per period is half the annual rate and the number of periods doubles. We compute the interest expressions in both cases, form an equation for the given difference and solve for the principal amount P. Step-by-Step Solution: Case 1 (annual compounding): A1 = P * (1 + 20/100)^2 = P * (1.2)^2 = P * 1.44Interest1 = A1 - P = 0.44PCase 2 (half yearly compounding): rate per half year = 20/2 = 10%Number of half year periods in 2 years = 4A2 = P * (1 + 10/100)^4 = P * (1.1)^4(1.1)^2 = 1.21 and (1.1)^4 = 1.21^2 = 1.4641Interest2 = A2 - P = 0.4641PDifference in interest: Interest2 - Interest1 = 0.4641P - 0.44P = 0.0241PGiven 0.0241P = 482, so P = 482 / 0.0241 = Rs 20000 Verification / Alternative Check: Using P = 20000, Interest1 = 0.44 * 20000 = Rs 8800. Interest2 = 0.4641 * 20000 = Rs 9282. Difference = 9282 - 8800 = Rs 482, which matches the given data, confirming that the principal is correct. Why Other Options Are Wrong: Rs 10000: Gives a difference of about Rs 241, only half of what is required.Rs 40000 and Rs 50000: Produce differences that are too large compared to Rs 482.Rs 15000: Also fails to match the required difference when substituted in the formula. Common Pitfalls: Students sometimes forget to double the number of compounding periods for half yearly interest or incorrectly use 20% directly as the half yearly rate. Another frequent mistake is equating the amounts instead of the difference in interest as specified in the question. Final Answer: The principal (original sum of money) is Rs 20000.

Question 20

For a principal of Rs 30000 invested at a compound interest rate of 7% per annum, the compound interest earned is Rs 4347. For how many complete years was the money invested?

Accepted Answer

Introduction: This problem checks whether you can use the basic compound interest formula to relate principal, rate, time and interest. We know the principal, the rate and the total compound interest earned, and we must determine the number of years the money was invested. Given Data / Assumptions: Principal (P) = Rs 30000. Rate of interest (r) = 7% per annum. Compound interest (CI) = Rs 4347. Compounding is assumed to be annual. We must find the time period n in years. Concept / Approach: The compound interest formula for amount is A = P * (1 + r/100)^n. The compound interest earned is CI = A - P. By substituting the known values and trying integer values of n, we can find which year count gives the required interest of Rs 4347. Step-by-Step Solution: Let n be the number of years.A = P * (1 + 7/100)^n = 30000 * (1.07)^nCI = A - P = 30000 * (1.07)^n - 30000We need CI = 4347, so 30000 * (1.07)^n - 30000 = 4347Try n = 2: A2 = 30000 * (1.07)^2 = 30000 * 1.1449 = Rs 34347CI for 2 years = 34347 - 30000 = Rs 4347, which matches the given interest.Hence, n = 2 years. Verification / Alternative Check: You can also test n = 1 and n = 3 to ensure they do not work. For 1 year, CI = 30000 * 0.07 = Rs 2100 (too low). For 3 years, CI would be even higher than Rs 4347. Therefore, only n = 2 years satisfies the condition. Why Other Options Are Wrong: 2.5 years and 3 years: These would give an amount larger than Rs 34347 and hence a greater CI.4 years: Gives an even larger compound interest, so it cannot be correct.1 year: Produces only Rs 2100 as interest, which is much less than Rs 4347. Common Pitfalls: Many learners forget that time in the standard formula is usually taken as an integer number of compounding periods unless a fractional period is clearly specified. Some also mistakenly equate CI directly to P * r/100 * n, which is the simple interest formula, not the compound interest formula. Final Answer: The money was invested for 2 years to earn compound interest of Rs 4347 at 7% per annum on Rs 30000.

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