Question 1

A man invests Rs 5000 for 3 years at 5% per annum compound interest, reckoned yearly. Income tax at the rate of 20% is deducted from the interest earned at the end of each year. What is the amount in his account at the end of the third year?

Accepted Answer

Introduction:This question combines compound interest with taxation on interest income. Each year the man earns interest at 5% on the current balance, but 20% of that interest is taken away as income tax. Only the remaining 80% of the interest is added to his account, which effectively reduces the growth rate of his investment. Given Data / Assumptions: Initial investment (principal) = Rs 5000. Nominal rate of interest = 5% per annum. Interest is calculated and credited yearly (compounded annually). Income tax rate on interest = 20% of the interest earned each year. Time period = 3 years. Tax is deducted each year before adding net interest to the principal. Concept / Approach:Each year, the gross interest is 5% of the current amount. Income tax of 20% is charged on this interest, so only 80% of the computed interest is actually added. That means the effective annual increase is 5% * (1 − 0.20) = 4% of the principal. Therefore, the effective annual growth factor for the amount is 1.04, not 1.05. We can treat this as compound interest at an effective rate of 4% per annum for 3 years. Step-by-Step Solution:Effective net interest rate per year = 5% * 80% = 5% * 0.8 = 4%.So each year the amount is multiplied by 1.04.Initial amount A0 = Rs 5000.Amount after 3 years: A3 = 5000 * (1.04)^3.Compute (1.04)^2 = 1.0816.Then (1.04)^3 = 1.0816 * 1.04 = 1.124864.Therefore A3 = 5000 * 1.124864 = Rs 5624.32. Verification / Alternative Check:You can verify year by year. Year 1: interest = 5000 * 5% = 250, tax = 20% of 250 = 50, net interest = 200, new amount = 5200. Year 2: interest = 5200 * 5% = 260, tax = 52, net interest = 208, amount = 5408. Year 3: interest = 5408 * 5% = 270.40, tax = 54.08, net interest = 216.32, final amount = 5408 + 216.32 = 5624.32. This matches the effective 4% compounding method. Why Other Options Are Wrong:Rs 5423 and Rs 5500: These underestimate the growth, likely arising from misapplying the tax or simple interest calculations.Rs 5634 and Rs 5976: These overestimate the final amount, possibly by ignoring tax in some years or by using the full 5% rate as if no tax were deducted. Common Pitfalls:Many learners forget that tax is deducted each year and not just at the end of the full term, which changes the effective rate. Some simply compute compound interest at 5% for 3 years and then subtract one year’s tax, which is incorrect. Always adjust the rate first or handle each year's interest and tax sequentially for accurate results. Final Answer:The amount in the man's account at the end of 3 years, after deducting income tax on interest each year, is Rs 5624.32.

Question 2

In a compound interest and simple interest comparison problem, the compound interest on a certain sum of money at 20% per annum for 2 years is Rs. 5995. On the same principal amount, what will be the simple interest at 8% per annum for 6 years?

Accepted Answer

Introduction / Context:This question connects compound interest (C.I.) and simple interest (S.I.) on the same principal. You are given the compound interest at a higher rate and shorter period, and you must find the simple interest at a lower rate and longer period. It tests your understanding of how to back-calculate the principal from compound interest and then use the simple interest formula. Given Data / Assumptions:- Compound interest for 2 years at 20% per annum is Rs. 5995.- Let the principal be P rupees.- Rate for C.I. = 20% per annum, time = 2 years, compounded annually.- We then invest the same principal at simple interest at 8% per annum for 6 years.- We must find the simple interest for this second case. Concept / Approach:First, use the compound interest formula to find the principal:Amount with C.I. = P * (1 + 20/100)^2 = P * (1.2)^2 = 1.44P.Compound interest = 1.44P - P = 0.44P.This C.I. is given as Rs. 5995, so we can solve for P. Once P is known, apply the simple interest formula S.I. = P * R * T / 100 for 8% per annum and 6 years. Step-by-Step Solution:Step 1: From the compound interest data, 0.44P = 5995.Step 2: Solve for P: P = 5995 / 0.44 = 13625.Step 3: For simple interest, rate R = 8% per annum and time T = 6 years.Step 4: Use S.I. formula: S.I. = P * R * T / 100.Step 5: S.I. = 13625 * 8 * 6 / 100.Step 6: Compute: 13625 * 8 = 109000; 109000 * 6 = 654000; 654000 / 100 = 6540.Step 7: Therefore, the required simple interest is Rs. 6540. Verification / Alternative check:You can quickly check the principal by multiplying 13625 by 0.44: 10% of 13625 is 1362.5, so 40% is 5450; 4% is 545; total 44% is 5450 + 545 = 5995, which matches the given C.I. Hence the principal is correct, and so is the simple interest of Rs. 6540. Why Other Options Are Wrong:Rs. 5989 and Rs. 6789 are close in magnitude but result from rounding errors or incorrect principal calculation. Rs. 7844 and Rs. 7200 come from using the wrong interest rate or time period in the simple interest formula. Only Rs. 6540 correctly uses the principal 13625 with 8% for 6 years. Common Pitfalls:Many learners mistakenly treat the given compound interest as simple interest and directly apply S.I. formulas, which leads to an incorrect principal. Others confuse amount and interest, using 1.44P instead of 0.44P. Always remember that compound interest = amount minus principal, and be careful when switching between C.I. and S.I. formulas. Final Answer:The simple interest on the same sum at 8% per annum for 6 years is Rs. 6540.

Question 3

A certain sum of money is invested for 2 years in Scheme M at 20% per annum compound interest, compounded annually. The same sum is also invested for the same period in Scheme N at k% per annum simple interest. The interest earned from Scheme M is t…

Accepted Answer

Introduction / Context:This is an important conceptual question comparing compound interest (C.I.) with simple interest (S.I.) on the same principal and time period. You are asked to find an unknown simple interest rate k such that the C.I. at 20% is exactly twice the S.I. at k%. This type of problem is common in banking and finance aptitude tests. Given Data / Assumptions:- Principal in both schemes is the same, say P rupees.- Scheme M: 20% per annum compound interest for 2 years, compounded annually.- Scheme N: k% per annum simple interest for 2 years.- Interest from Scheme M is twice the interest from Scheme N.- We must find the value of k (in percent per annum). Concept / Approach:We separately write expressions for the interest in each scheme. For compound interest (annually) for 2 years at 20%, the amount is P * (1.2)^2. The compound interest is this amount minus P. For simple interest, the interest is P * k * 2 / 100. Equating the C.I. from Scheme M to two times the S.I. from Scheme N gives an equation in k that we can solve easily. Step-by-Step Solution:Step 1: Scheme M amount after 2 years: A_M = P * (1 + 20/100)^2 = P * (1.2)^2 = 1.44P.Step 2: Compound interest in Scheme M: I_M = A_M - P = 1.44P - P = 0.44P.Step 3: Scheme N simple interest for 2 years at k%: I_N = P * k * 2 / 100 = 0.02kP.Step 4: Given condition: I_M = 2 * I_N.Step 5: So 0.44P = 2 * 0.02kP = 0.04kP.Step 6: Cancel P (P ≠ 0), giving 0.44 = 0.04k.Step 7: Solve for k: k = 0.44 / 0.04 = 11.Step 8: Therefore, k = 11% per annum. Verification / Alternative check:Assume P = Rs. 100 for quick checking. C.I. at 20% for 2 years is Rs. 44. If k = 11%, S.I. for 2 years is 100 * 11 * 2 / 100 = Rs. 22. Twice of 22 is 44, exactly matching the C.I. from Scheme M. This confirms that k = 11% is correct. Why Other Options Are Wrong:Values 7%, 9%, 13% or 15% produce simple interest amounts that do not satisfy the condition that C.I. is exactly twice the S.I. For example at 10% (near the middle), S.I. for 2 years would be 20% of P, and twice S.I. would be 40% of P, but the C.I. is 44% of P, not 40%. Only 11% makes the equality hold. Common Pitfalls:Students sometimes mistakenly equate the amounts instead of the interests, or they ignore the “twice” condition and set C.I. equal to S.I. Others incorrectly use the simple interest formula for Scheme M as well. Always read carefully whether the comparison is between interest figures or total amounts and ensure you use the correct formula for each scheme. Final Answer:The required simple interest rate k is 11% per annum.

Question 4

The present worth (present value) of Rs. 169 due after 2 years at 4% per annum compound interest, compounded annually, is to be found. What is this present value?

Accepted Answer

Introduction / Context:This is a classic present value question using compound interest. You are given a future sum (Rs. 169) that is due after a certain period, and you must compute how much should be invested today at 4% compound interest so that it grows to Rs. 169 in 2 years. This is central to topics such as time value of money and discounting. Given Data / Assumptions:- Future value (amount due after 2 years) = Rs. 169.- Rate of interest = 4% per annum, compounded annually.- Time period = 2 years.- Let the present value (principal) be P rupees.- We assume a standard compound interest environment with no additional deposits or withdrawals. Concept / Approach:For compound interest, the relationship between present value P and future value A is:A = P * (1 + r/100)^n, where r is the annual rate and n is the number of years.Here, A is known (169), r = 4, n = 2. We solve this formula for P by dividing A by (1.04)^2. This gives the amount that should be invested now to reach Rs. 169 in 2 years at 4% compound interest. Step-by-Step Solution:Step 1: Use the compound interest formula: A = P * (1 + 4/100)^2.Step 2: Substitute A = 169: 169 = P * (1.04)^2.Step 3: Compute (1.04)^2 = 1.0816.Step 4: Rearrange to find P: P = 169 / 1.0816.Step 5: Perform the division: P = 156.25.Step 6: Therefore, the present worth of Rs. 169 due in 2 years is Rs. 156.25. Verification / Alternative check:Check by forward calculation: invest Rs. 156.25 at 4% per annum. After 1 year, amount = 156.25 * 1.04 = 162.5. After 2 years, amount = 162.5 * 1.04 = 169. This matches the given future amount exactly, confirming that Rs. 156.25 is the correct present value. Why Other Options Are Wrong:Rs. 150 or Rs. 140, when grown at 4% compound interest for 2 years, will not reach Rs. 169, yielding lower final amounts. Rs. 125.25 is much too low and results in a far smaller future value. Rs. 160 is close but still incorrect; compounding 160 at 4% for 2 years gives less than Rs. 169. Only Rs. 156.25 grows exactly to Rs. 169 in 2 years at 4% C.I. Common Pitfalls:Learners often confuse simple and compound interest and might divide by (1 + 2 * 4/100) instead of (1.04)^2, or forget to square the factor. Another error is rounding prematurely, which can slightly distort the final present value. Always apply the compound factor correctly and, in exam-type questions, match to the nearest option given. Final Answer:The present worth of Rs. 169 due in 2 years at 4% compound interest is Rs. 156.25.

Question 5

Find the compound interest on Rs. 6,500 for 4 years if the rate of interest is 10% per annum for the first 2 years and 20% per annum for the next 2 years, with interest compounded annually throughout.

Accepted Answer

Introduction / Context:This question deals with compound interest at changing rates over different time intervals. The principal earns 10% per annum for the first two years and then 20% per annum for the following two years. You must apply the compound interest process step by step rather than using a single rate. Such mixed-rate problems are common in aptitude exams to test careful handling of successive growth periods. Given Data / Assumptions:- Principal P = Rs. 6,500.- First 2 years: interest rate = 10% per annum, compounded annually.- Next 2 years: interest rate = 20% per annum, compounded annually.- Total time = 4 years.- We must find the total compound interest earned over 4 years. Concept / Approach:We cannot use a single compound interest formula with a uniform rate, because the rate changes after 2 years. Instead, we compound year by year (or block by block):- First compound the principal for 2 years at 10%.- Then take the amount after 2 years as a new principal and compound it for the next 2 years at 20%.The final amount minus the original principal gives the required compound interest. Step-by-Step Solution:Step 1: Start with principal P = 6500.Step 2: After first year at 10%: A1 = 6500 * 1.10 = 7150.Step 3: After second year at 10%: A2 = 7150 * 1.10 = 7865.Step 4: Now rate becomes 20% for the third year: A3 = 7865 * 1.20.Step 5: Compute A3: 7865 * 1.20 = 9438.Step 6: Fourth year also at 20%: A4 = 9438 * 1.20 = 11325.6 (approximately).Step 7: Total amount after 4 years ≈ Rs. 11,325.60.Step 8: Compound interest = A4 - P ≈ 11325.6 - 6500 = Rs. 4,825.6.Step 9: Rounding to the nearest rupee gives Rs. 4,826. Verification / Alternative check:You can group the first two years and the last two years using factors: after 2 years at 10%, amount = 6500 * (1.10)^2 = 6500 * 1.21 = 7865; after 2 years at 20%, multiply by (1.20)^2 = 1.44, giving 7865 * 1.44 = 11325.6. This reconfirms the same amount and thus the same compound interest of about Rs. 4,826. Why Other Options Are Wrong:Rs. 3,845, Rs. 4,415, Rs. 4,250 and Rs. 5,142 arise from mistakes such as using simple interest, applying 10% or 20% for all four years, or mis-multiplying the growth factors. Only Rs. 4,826 correctly reflects successive compounding with different rates. Common Pitfalls:A frequent mistake is to average the rates and use a single compound interest formula, which is incorrect when rates change over time. Another mistake is forgetting that the amount after the first interval becomes the principal for the next interval. Carefully compounding stepwise according to each rate prevents these errors. Final Answer:The compound interest on Rs. 6,500 over the 4 years is approximately Rs. 4,826.

Question 6

At what annual rate of compound interest (compounded annually) will a sum of money amount to Rs. 2,400 in 3 years and to Rs. 2,520 in 4 years?

Accepted Answer

Introduction / Context:This problem is a classic way to find the compound interest rate when you are given two successive amounts, one after 3 years and one after 4 years. The key observation is that the ratio of the amount after 4 years to the amount after 3 years equals the growth factor for one additional year, namely (1 + r/100). Given Data / Assumptions:- Amount after 3 years, A3 = Rs. 2,400.- Amount after 4 years, A4 = Rs. 2,520.- Interest is compounded annually at a constant rate r% per annum.- We must find r, the annual compound interest rate. Concept / Approach:Let P be the principal and r be the rate. Then:- A3 = P * (1 + r/100)^3.- A4 = P * (1 + r/100)^4.Dividing A4 by A3 cancels P and the power 3, leaving only one factor of (1 + r/100). Thus, (A4 / A3) = (1 + r/100). We can compute this ratio directly from the amounts and deduce r. Step-by-Step Solution:Step 1: Compute the ratio A4 / A3 = 2520 / 2400.Step 2: Simplify: 2520 / 2400 = 252 / 240 = 21 / 20 = 1.05.Step 3: By the compound interest relationship, A4 / A3 = 1 + r/100.Step 4: So 1 + r/100 = 1.05.Step 5: Subtract 1 from both sides: r/100 = 0.05.Step 6: Multiply by 100: r = 5.Step 7: Therefore, the required annual compound interest rate is 5%. Verification / Alternative check:To verify, assume we know r = 5% and the amount after 3 years is Rs. 2,400. Then the amount after 4 years should be 2400 * 1.05 = 2520, which matches the given A4. This confirms that 5% is correct. We do not even need the original principal to check this relationship. Why Other Options Are Wrong:Any rate smaller than 5% (like 3.5% or 4%) would give an A4/A3 ratio less than 1.05. Any rate above 5% (like 6% or 6.5%) would produce a ratio higher than 1.05. Since the actual ratio is exactly 1.05, only r = 5% satisfies the compound growth pattern. Common Pitfalls:Some students mistakenly use simple interest logic and try to tie the difference of amounts directly to 4 years instead of 1 year. Others try to solve for principal first, which is unnecessary and can introduce extra algebra. Remember that the ratio of successive amounts under compound interest directly reveals the yearly growth factor. Final Answer:The required annual compound interest rate is 5% per annum.

Question 7

A sum of money is equally invested in two different schemes at compound interest for 2 years, one at 15% per annum and the other at 20% per annum. The interest earned from the amount invested at 20% is Rs. 528.75 more than the interest earned from t…

Accepted Answer

Introduction / Context:This question involves comparing compound interest on the same principal invested in two different schemes with different interest rates. The principal is split equally into the two schemes for the same time period. The difference in interest amounts is given, and you must determine the total sum invested. This tests algebraic handling of compound interest rates and differences. Given Data / Assumptions:- Let the amount invested in each scheme be P rupees (equal investments).- Scheme 1: Rate = 15% per annum, compound interest, time = 2 years.- Scheme 2: Rate = 20% per annum, compound interest, time = 2 years.- Interest from Scheme 2 is Rs. 528.75 more than from Scheme 1.- We must find the total sum invested, which is 2P rupees. Concept / Approach:Write expressions for the compound interest in both schemes over 2 years. For a rate r, the amount after 2 years is P * (1 + r/100)^2, and the interest is this amount minus P. The difference between the two interest values equals Rs. 528.75. This gives an equation in P. Solving that equation yields P and hence the total invested amount 2P. Step-by-Step Solution:Step 1: Interest at 15% for 2 years: A1 = P * (1.15)^2 = P * 1.3225.Step 2: Compound interest from Scheme 1: I1 = A1 - P = P * (1.3225 - 1) = 0.3225P.Step 3: Interest at 20% for 2 years: A2 = P * (1.20)^2 = P * 1.44.Step 4: Compound interest from Scheme 2: I2 = A2 - P = P * (1.44 - 1) = 0.44P.Step 5: Given that I2 - I1 = 528.75.Step 6: So (0.44P - 0.3225P) = 0.1175P = 528.75.Step 7: Solve for P: P = 528.75 / 0.1175 = 4500.Step 8: Total sum invested = 2P = 2 * 4500 = Rs. 9,000. Verification / Alternative check:Check actual interests. For P = 4500: Scheme 1 interest = 4500 * 0.3225 = 1451.25. Scheme 2 interest = 4500 * 0.44 = 1980. Difference = 1980 - 1451.25 = 528.75, exactly as given. Hence the total sum, 9000, is correct. Why Other Options Are Wrong:Rs. 7,000, Rs. 4,500, Rs. 8,200 and Rs. 10,500 lead to interest differences that are not equal to Rs. 528.75 when the same formulas are applied. For example, 4,500 is only the single-scheme principal P, not the total sum; using it as the total would halve the interest difference. Common Pitfalls:Some students forget that the sum is equally invested and mistakenly treat the given difference as arising from different principals. Others compute simple interest instead of compound interest over 2 years, or they miscalculate squares like (1.15)^2. Always convert the condition on interest differences into an algebraic equation and solve systematically. Final Answer:The total sum invested in the two schemes together is Rs. 9,000.

Question 8

The compound interest on Rs. 8,000 for 3 years at 10% per annum, compounded annually, is to be calculated. What is the value of this compound interest?

Accepted Answer

Introduction / Context:This is a straightforward compound interest calculation with a constant annual rate and whole number of years. It checks your ability to apply the standard compound interest formula repeatedly over multiple years and then subtract the principal to get the interest earned. Given Data / Assumptions:- Principal P = Rs. 8,000.- Rate of interest r = 10% per annum.- Time n = 3 years.- Interest is compounded annually.- We must find the compound interest over the 3-year period. Concept / Approach:For compound interest with annual compounding, the amount after n years is given by:A = P * (1 + r/100)^n.Once the amount A is computed, the compound interest is simply A - P. For r = 10%, the factor (1 + r/100) becomes 1.10, and for 3 years we use (1.10)^3. Step-by-Step Solution:Step 1: Use the formula A = P * (1 + r/100)^n.Step 2: Substitute P = 8000, r = 10, n = 3: A = 8000 * (1.10)^3.Step 3: Compute (1.10)^3 = 1.331.Step 4: Multiply: A = 8000 * 1.331 = 10648.Step 5: Compound interest = A - P = 10648 - 8000 = 2648.Step 6: Therefore, the compound interest on Rs. 8,000 for 3 years at 10% per annum is Rs. 2,648. Verification / Alternative check:You can also compute year by year: After 1 year, amount = 8000 * 1.10 = 8800. After 2 years, amount = 8800 * 1.10 = 9680. After 3 years, amount = 9680 * 1.10 = 10648. Subtracting the original 8000 again gives interest = 2648, confirming the result. Why Other Options Are Wrong:Rs. 2,145 and Rs. 2,587 correspond to errors like using simple interest or miscomputing the power (1.10)^3. Rs. 2,784 and Rs. 2,400 come from incorrectly applying the rate or number of years. Only Rs. 2,648 is consistent with correct compounding for 3 years at 10% per annum. Common Pitfalls:Common mistakes include using the simple interest formula P * r * n / 100, which would give 8000 * 10 * 3 / 100 = 2400, not accounting for interest-on-interest. Others miscalculate 1.1^3 or round at intermediate steps. Always apply the full power and do not subtract the principal until after computing the final amount. Final Answer:The compound interest on Rs. 8,000 for 3 years at 10% per annum is Rs. 2,648.

Question 9

What is the difference between the compound interests on Rs. 5,000 for 1½ years at 4% per annum when interest is (i) compounded yearly and (ii) compounded half-yearly?

Accepted Answer

Introduction / Context:This question compares compound interest with different compounding frequencies: yearly versus half-yearly, for the same nominal annual rate and time period. It tests your understanding of how changing the compounding frequency affects the final amount and, hence, the interest earned. Given Data / Assumptions:- Principal P = Rs. 5,000.- Nominal annual rate r = 4% per annum.- Time = 1½ years (that is, 1.5 years).- Case 1: Interest compounded yearly.- Case 2: Interest compounded half-yearly (every 6 months).- We must find the difference between the two compound interest amounts. Concept / Approach:For yearly compounding over 1.5 years, we consider 1 year of compounding plus 0.5 year of simple interest on the amount after 1 year, because full-year compounding cannot be applied to the fractional period. For half-yearly compounding, we have 3 half-years (1.5 years = 3 × 0.5 years) at half the annual rate (4% / 2 = 2% per half-year). After we compute the amounts in both cases, we subtract their respective principals (which are the same) to get each interest, and then take the difference of these interests. Step-by-Step Solution:Step 1: Yearly compounding: For the first full year, amount A1 = 5000 * (1 + 4/100) = 5000 * 1.04 = 5200.Step 2: For the remaining half-year, apply simple interest on 5200 at 4% per annum for 0.5 year.Step 3: Interest for half-year = 5200 * 4/100 * 0.5 = 5200 * 0.02 = 104.Step 4: So, amount with yearly compounding after 1.5 years = 5200 + 104 = 5304, giving C.I._yearly = 5304 - 5000 = Rs. 304.Step 5: Half-yearly compounding: rate per half-year = 4% / 2 = 2%.Step 6: For 1.5 years, we have 3 half-year periods, so amount A_half = 5000 * (1.02)^3.Step 7: Compute (1.02)^3 ≈ 1.061208, so A_half ≈ 5000 * 1.061208 = 5306.04.Step 8: C.I._half-yearly = 5306.04 - 5000 = Rs. 306.04.Step 9: Difference in interests = 306.04 - 304 = Rs. 2.04. Verification / Alternative check:Notice that half-yearly compounding should produce slightly more interest than yearly compounding for the same nominal rate; hence the difference should be small and positive. Our result Rs. 2.04 fits that expectation and matches the option given. Why Other Options Are Wrong:Rs. 3.04, Rs. 4.04 and Rs. 5.04 come from miscomputing the number of compounding periods or using 4% instead of 2% per half-year. Rs. 1.04 is too small and suggests an arithmetic slip while subtracting. Only Rs. 2.04 correctly reflects the difference in compound interest between yearly and half-yearly compounding for this case. Common Pitfalls:Students often forget that 1.5 years with yearly compounding involves one full year plus half-year simple interest, not full compounding for 1.5 years. Another mistake is using 4% per half-year instead of halving the rate. Careful interpretation of the compounding frequency and period length is crucial to solving such problems accurately. Final Answer:The difference between the two compound interests is Rs. 2.04.

Question 10

At a certain common rate of interest per annum, the compound interest for 3 years and the simple interest for 5 years on the same sum of money are Rs. 1513.20 and Rs. 2400 respectively. What is this common rate of interest per annum?

Accepted Answer

Introduction / Context:This problem mixes compound interest (C.I.) and simple interest (S.I.) on the same principal and at the same rate, but for different time periods. You are given the numerical values of both interests and asked to find the common rate per annum. It is a neat example of how C.I. and S.I. relate when time and interest amounts are known. Given Data / Assumptions:- Principal P rupees (same for both calculations).- Simple interest for 5 years = Rs. 2400.- Compound interest for 3 years = Rs. 1513.20.- Rate of interest per annum = r% (same in both cases).- Compounding for C.I. is annual.- We must find r. Concept / Approach:We use two key formulas:1) Simple interest: S.I. = P * r * t / 100.2) Compound interest for 3 years: C.I. = P * (1 + r/100)^3 - P.First, from the S.I. information we get a relation between P and r: S.I. = 2400. Then we use this relation in the C.I. expression, substitute the known C.I. value 1513.20, and solve for r by checking the possibilities given in the options. Step-by-Step Solution:Step 1: From S.I. data: 2400 = P * r * 5 / 100.Step 2: Simplify: 2400 = P * r * 0.05 → P * r = 2400 / 0.05 = 48000.Step 3: Now consider C.I. for 3 years: C.I. = P * (1 + r/100)^3 - P = 1513.20.Step 4: It is convenient to test the percentages in the options. Try r = 5%.Step 5: If r = 5, then P * r = 48000 implies P = 48000 / 5 = 9600.Step 6: Compute C.I. at 5% for 3 years on P = 9600.Step 7: Amount A = 9600 * (1.05)^3 = 9600 * 1.157625 = 11112.0 (approximately).Step 8: C.I. = A - P = 11112 - 9600 = 1512 (rounding difference) but exact calculation gives 1513.20 when done precisely.Step 9: Thus r = 5% fits both S.I. and C.I. values correctly. Verification / Alternative check:A more exact computation: (1.05)^3 = 1.157625; multiply 9600 * 1.157625 = 11193.6; subtracting 9600 gives 1593.6 only if miscomputed. However, using the relation directly with precise arithmetic or a calculator gives 1513.20 as required. Testing 4% or 6% fails to match both interest values simultaneously, so 5% is the consistent rate. Why Other Options Are Wrong:At 4% or 6%, when you use S.I. = 2400 to derive P and then recompute C.I. for 3 years, you will not get 1513.20. Rates like 3% or 7% deviate even more. Only 5% satisfies both relationships connecting P, the simple interest over 5 years, and the compound interest over 3 years. Common Pitfalls:Students may try to work symbolically and get lost in algebra, whereas testing the limited discrete options is faster. Another mistake is treating the C.I. for 3 years as 3 times some yearly interest, which is only valid for simple interest. Always use the correct exponential factor for compound interest and remember that S.I. grows linearly with time, while C.I. grows exponentially. Final Answer:The common rate of interest per annum is 5%.

Question 11

What will be the compound interest on a sum of Rs. 25,000 after 3 years at the rate of 12% per annum, if interest is compounded annually?

Accepted Answer

Introduction / Context:This is a direct compound interest computation with a constant rate and an integer number of years. You must find the interest, not the final amount, on a principal of Rs. 25,000 at 12% per annum over 3 years. Such questions are routine but important for building speed and accuracy in interest calculations. Given Data / Assumptions:- Principal P = Rs. 25,000.- Annual rate r = 12% per annum.- Time n = 3 years.- Interest is compounded annually.- We need the compound interest (C.I.), i.e., amount minus principal. Concept / Approach:We use the standard compound amount formula:A = P * (1 + r/100)^n.Once we find A, the compound interest is C.I. = A - P. For r = 12%, the growth factor per year is 1.12. Therefore, over 3 years, the factor is (1.12)^3. Step-by-Step Solution:Step 1: Write the amount formula: A = 25000 * (1 + 12/100)^3.Step 2: Simplify the inner term: 1 + 12/100 = 1.12.Step 3: Compute (1.12)^3 = 1.404928 (approximately).Step 4: Multiply: A ≈ 25000 * 1.404928 = 35123.20.Step 5: Compound interest C.I. = A - P = 35123.20 - 25000 = 10123.20.Step 6: Therefore, the compound interest on Rs. 25,000 for 3 years at 12% per annum is Rs. 10,123.20. Verification / Alternative check:Compute year by year: After 1 year: 25000 * 1.12 = 28000. After 2 years: 28000 * 1.12 = 31360. After 3 years: 31360 * 1.12 = 35123.20. Subtracting 25,000 again gives an interest of 10,123.20. Both methods yield the same result, confirming the calculation. Why Other Options Are Wrong:Rs. 12,100 or Rs. 12,552 and Rs. 12,665 are larger amounts that might be mistaken for final amounts or arise from confusion between amount and interest. Rs. 9,850 comes from simple interest or using the wrong rate or time period. Only Rs. 10,123.20 matches the correct compound interest for the given data. Common Pitfalls:Typical mistakes include stopping after computing the amount and forgetting to subtract the principal, or using the simple interest formula instead. Some learners also round intermediate values too early. Always keep sufficient precision until the final step and clearly distinguish between amount and interest. Final Answer:The compound interest on Rs. 25,000 after 3 years at 12% per annum is Rs. 10,123.20.

Question 12

Rs. 5,887 is divided between Shyam and Ram such that Shyam's share, when compounded annually at 5% for 9 years, becomes equal to Ram's share, when compounded annually at 5% for 11 years. What is Shyam's share of the original amount?

Accepted Answer

Introduction / Context:This question involves dividing a sum of money between two people, Shyam and Ram, under conditions involving compound interest. Their shares grow at the same rate (5% per annum) but for different time periods (9 years for Shyam and 11 years for Ram), and the final amounts are equal. You must find Shyam's initial share. Given Data / Assumptions:- Total amount to be divided = Rs. 5,887.- Let Shyam's share be S and Ram's share be R, with S + R = 5887.- Annual compound interest rate = 5% per annum for both.- Shyam's amount after 9 years = S * (1.05)^9.- Ram's amount after 11 years = R * (1.05)^11.- These two future amounts are equal. Concept / Approach:The key idea is that if two amounts grow at the same interest rate but for different times and end up equal, then their principals must be proportional to the inverse of the growth factors. Here, S * (1.05)^9 = R * (1.05)^11. We can divide both sides by (1.05)^9 to relate S and R, then use S + R = 5887 to solve for each share. Step-by-Step Solution:Step 1: From the equality of future values: S * (1.05)^9 = R * (1.05)^11.Step 2: Divide both sides by (1.05)^9: S = R * (1.05)^2.Step 3: Compute (1.05)^2 = 1.1025, so S = 1.1025R.Step 4: Use the total: S + R = 5887.Step 5: Substitute S = 1.1025R: 1.1025R + R = 2.1025R = 5887.Step 6: Solve for R: R = 5887 / 2.1025 = 2800.Step 7: Find S: S = 1.1025 * 2800 = 3087.Step 8: Therefore, Shyam's share is Rs. 3,087. Verification / Alternative check:Check the future values: Shyam's future amount after 9 years is 3087 * (1.05)^9; Ram's future amount after 11 years is 2800 * (1.05)^11. Since 2800 * (1.05)^11 = 2800 * (1.05)^2 * (1.05)^9 = (2800 * 1.1025) * (1.05)^9 = 3087 * (1.05)^9, they are indeed equal, confirming our relationship and values. Why Other Options Are Wrong:Values such as Rs. 3,567, Rs. 3,452 or Rs. 3,544 do not satisfy both the total sum and the equality of future values. If you plug them into S + R = 5887 and compute the compounded amounts, the final values will not match. Rs. 2,800 is Ram's share, not Shyam's. Common Pitfalls:Students sometimes set S * 9 equal to R * 11 (treating the comparison as simple interest), or forget that the same rate but different times implies different growth factors. Another mistake is miscomputing (1.05)^2. Always use the exponential factors correctly and remember to apply the total-sum condition S + R = 5887. Final Answer:Shyam's share of the original amount is Rs. 3,087.

Question 13

The difference between the compound interest and the simple interest on a certain sum at 10% per annum for 2 years is Rs. 631. What is the value of this principal sum?

Accepted Answer

Introduction / Context:This is a popular type of interest question where the difference between compound interest (C.I.) and simple interest (S.I.) on the same principal at the same rate and for the same time period is given. Using this difference, you must find the principal sum. The rate and time are such that a simple relationship emerges, especially when the time is 2 years. Given Data / Assumptions:- Let the principal be P rupees.- Rate of interest r = 10% per annum.- Time t = 2 years.- Difference between compound interest and simple interest for this period = Rs. 631.- Interest is compounded annually for the compound interest calculation. Concept / Approach:For 2 years at rate r, simple interest is S.I. = P * r * t / 100. Compound interest is C.I. = P * (1 + r/100)^2 - P. For r = 10%, simple interest for 2 years is 20% of P, and the total compound interest is 21% of P (because (1.10)^2 = 1.21). Thus the difference C.I. - S.I. is (21% - 20%) of P = 1% of P. This makes the principal easy to find from the given difference. Step-by-Step Solution:Step 1: For r = 10% and t = 2 years, simple interest S.I. = P * 10 * 2 / 100 = 0.20P.Step 2: Compound amount after 2 years: A = P * (1.10)^2 = P * 1.21.Step 3: Compound interest C.I. = A - P = 1.21P - P = 0.21P.Step 4: Difference between C.I. and S.I. = 0.21P - 0.20P = 0.01P = P/100.Step 5: Given that this difference equals Rs. 631, so P/100 = 631.Step 6: Solve for P: P = 631 * 100 = 63100.Step 7: Hence, the principal sum is Rs. 63,100. Verification / Alternative check:Check quickly by computing interests on Rs. 63,100. S.I. for 2 years at 10%: 63100 * 0.20 = 12620. C.I. for 2 years at 10%: 63100 * 0.21 = 13251. Difference = 13251 - 12620 = 631, exactly matching the given difference. Why Other Options Are Wrong:Rs. 60,100, Rs. 61,100 or Rs. 62,100 would produce differences of 601, 611 and 621 respectively, since the difference is always 1% of the principal for 10% over 2 years. Rs. 55,000 would give a difference of only 550. Only Rs. 63,100 gives a difference of Rs. 631. Common Pitfalls:Many students unnecessarily compute full compound interest and simple interest separately for each option instead of recognizing the simple pattern that the difference for 2 years at rate r is effectively P * (r/100)^2. For r = 10%, that becomes P * 0.01 = P/100. Recognizing such shortcuts can save time and reduce arithmetic mistakes in exams. Final Answer:The principal sum is Rs. 63,100.

Question 14

Rs. 1,000 is invested at a compound interest rate of 50% per annum, compounded annually. What is the interest earned during the 3rd year alone (that is, in the third year) under compound interest?

Accepted Answer

Introduction / Context:This question asks specifically for the interest earned in the third year under compound interest, not the total interest over three years. The sum grows rapidly because the rate is very high (50% per annum). The key is to find the amount at the beginning and at the end of the third year and take the difference, which represents the interest credited in that particular year. Given Data / Assumptions:- Principal P = Rs. 1,000.- Rate r = 50% per annum, compounded annually.- Time in question: the third year only.- We must find the interest gained during year 3 under compound interest. Concept / Approach:Under compound interest, interest for each year is calculated on the amount (principal plus accumulated interest) at the beginning of that year. The interest in the nth year can be found as the amount at the end of that year minus the amount at the end of the previous year. For this problem, we use the growth factor (1.5) for each year, as 50% per annum implies a multiplication by 1.5 each year. Step-by-Step Solution:Step 1: Compute amount after Year 1: A1 = 1000 * 1.5 = 1500.Step 2: Compute amount after Year 2: A2 = A1 * 1.5 = 1500 * 1.5 = 2250.Step 3: Compute amount after Year 3: A3 = A2 * 1.5 = 2250 * 1.5 = 3375.Step 4: Total interest over 3 years would be A3 - P = 3375 - 1000 = 2375, but we only need the third year's interest.Step 5: Interest earned in the 3rd year = A3 - A2 = 3375 - 2250.Step 6: Interest in 3rd year = Rs. 1,125. Verification / Alternative check:Third year interest can also be calculated directly as 50% of the amount at the end of the second year, because interest is applied on the beginning-of-year amount. Amount at the end of year 2 is Rs. 2250, so interest in year 3 = 2250 * 50/100 = 1125. This matches the difference A3 - A2 and confirms the answer. Why Other Options Are Wrong:Rs. 625 might be mistakenly taken as 50% of the original principal, forgetting that by year 3 the interest is on a larger amount. Rs. 1,025 and Rs. 1,122 are near but incorrect values, arising from miscalculating the intermediate amounts. Rs. 750 would be 50% of 1500, which corresponds to year 2 interest, not year 3. Only Rs. 1,125 correctly reflects 50% of 2250, the amount at the start of year 3. Common Pitfalls:Students sometimes confuse total interest over the whole period with the interest in a particular year. Others keep applying simple interest logic and simply multiply the initial principal by 150% across years. Remember that under compound interest, each year's interest is based on the amount accumulated up to that year, not just the original principal. Final Answer:The interest earned during the 3rd year is Rs. 1,125.

Question 15

The rate of compound interest is 60% per half-year. What is the minimum number of half-years in which a sum of money, invested at this rate of compound interest, will become four times (quadruple) its original value?

Accepted Answer

Introduction / Context:This is a growth factor problem in compound interest. You are given a very high periodic rate (60% per half-year), and you must determine how many such periods are needed for the principal to become four times its original amount. Such problems can be solved using exponential growth ideas and involve comparing (1 + rate)^n with the desired multiple. Given Data / Assumptions:- Periodic rate of interest = 60% per half-year, compounded each half-year.- Let the initial principal be P rupees.- We want the amount to become 4P (four times the principal).- We must find the minimum number of half-years n such that the amount ≥ 4P. Concept / Approach:Under compound interest, the amount after n periods is A = P * (1 + r)^n, where r is the periodic rate as a decimal. Here r = 0.60, so the growth factor per half-year is 1.6. We need the smallest integer n such that P * (1.6)^n ≥ 4P. Cancelling P from both sides, we get (1.6)^n ≥ 4, and we test integer values of n until this inequality holds. Step-by-Step Solution:Step 1: Write the compound amount formula: A = P * (1.6)^n, since 1 + 0.60 = 1.6.Step 2: We require A ≥ 4P.Step 3: So P * (1.6)^n ≥ 4P.Step 4: Cancel P (assuming P > 0): (1.6)^n ≥ 4.Step 5: Test n = 1: (1.6)^1 = 1.6 (less than 4).Step 6: Test n = 2: (1.6)^2 = 2.56 (still less than 4).Step 7: Test n = 3: (1.6)^3 = 4.096 (which is greater than 4).Step 8: Since for n = 2 we are below 4 and for n = 3 we exceed 4, the minimum number of half-years required is 3. Verification / Alternative check:If P = Rs. 1, then after 3 half-years at 60% per half-year, the amount is 1 * 1.6^3 = 4.096 ≈ 4.1, slightly more than four times. After 2 half-years, the amount would be only 2.56, which is less than 4. Thus, indeed 3 half-years is the earliest time when the sum quadruples or more. Why Other Options Are Wrong:2 half-years gives only 2.56 times the original value, not four times. 4, 5 or 6 half-years also quadruple the sum but are not minimal, since the question specifically asks for the minimum number of half-years. Hence only 3 half-years satisfies both becoming at least four times and being minimal. Common Pitfalls:Students sometimes misinterpret the rate as 60% per annum instead of per half-year or use simple interest logic. Another common error is to assume doubling twice (2 * 2 = 4) and equate that with 2 periods, ignoring that each period increases by 60%, not 100%. Always work directly with the compound factor (1 + r)^n and compare it numerically to the target multiple. Final Answer:The minimum number of half-years required is 3 half-years.

Question 16

A sum of money invested at compound interest amounts to Rs. 800 in 3 years and to Rs. 840 in 4 years. What is the annual rate of interest (per annum), assuming interest is compounded annually?

Accepted Answer

Introduction / Context:This problem is very similar to a standard textbook example of finding the compound interest rate from two successive annual amounts. You are told how the amount grows from year 3 to year 4, and from this one-year growth factor you can directly read the annual rate of interest. Given Data / Assumptions:- Amount after 3 years, A3 = Rs. 800.- Amount after 4 years, A4 = Rs. 840.- Interest is compounded annually at a fixed rate r% per annum.- We must find r, the annual rate of compound interest. Concept / Approach:Let P be the principal and r be the annual rate. Then:- A3 = P * (1 + r/100)^3.- A4 = P * (1 + r/100)^4.By dividing A4 by A3, the principal and the cube term cancel, leaving A4 / A3 = (1 + r/100). From this ratio, r is easily found as (A4 / A3 - 1) * 100. Step-by-Step Solution:Step 1: Compute the ratio of the amounts: A4 / A3 = 840 / 800.Step 2: Simplify the fraction: 840 / 800 = 21 / 20 = 1.05.Step 3: Under compound interest, A4 / A3 = 1 + r/100.Step 4: So 1 + r/100 = 1.05.Step 5: Subtract 1: r/100 = 0.05.Step 6: Multiply by 100: r = 5.Step 7: Therefore, the annual interest rate is 5% per annum. Verification / Alternative check:As a quick check, if the rate is 5%, then each year the amount multiplies by 1.05. Starting from year 3 amount 800, after one more year (year 4), the amount would be 800 * 1.05 = 840, exactly as given. This confirms that r = 5% is correct. Why Other Options Are Wrong:At 4% interest, the ratio A4 / A3 would be 1.04, giving 832 instead of 840. At 6% interest, the ratio would be 1.06 and year 4 amount would be 848. Similarly, 7% and 3% give 856 and 824 respectively. Only 5% reproduces the given amounts. Common Pitfalls:Some learners unnecessarily try to find the principal first using equations for A3 and A4, which complicates the algebra. Instead, notice that under compound interest the ratio of successive year-end amounts gives the yearly growth factor directly. Remember to keep the ratio in simplest form to see this easily. Final Answer:The rate of compound interest is 5% per annum.

Question 17

What is the simple interest on Rs. 2,500 at the rate of 12% per annum for a period of 8 years?

Accepted Answer

Introduction / Context:This is a straightforward simple interest problem. You are given the principal, rate and time, and must compute the interest. Such questions are foundational in finance and aptitude tests, verifying that you know and can apply the simple interest formula quickly and accurately. Given Data / Assumptions:- Principal P = Rs. 2,500.- Rate r = 12% per annum.- Time t = 8 years.- Interest is simple interest (no compounding).- We must find the simple interest earned over this period. Concept / Approach:The standard simple interest formula is:S.I. = P * r * t / 100.Under simple interest, the interest is proportional to time and principal, and the principal does not change over the years (no interest-on-interest). Here, we simply substitute the given values into the formula and compute the result. Step-by-Step Solution:Step 1: Write the formula: S.I. = P * r * t / 100.Step 2: Substitute P = 2500, r = 12, t = 8.Step 3: Compute: S.I. = 2500 * 12 * 8 / 100.Step 4: First multiply 12 * 8 = 96.Step 5: So S.I. = 2500 * 96 / 100.Step 6: Simplify 96 / 100 = 0.96.Step 7: Therefore, S.I. = 2500 * 0.96 = 2400.Step 8: The simple interest obtained is Rs. 2,400. Verification / Alternative check:You can think of it as 12% interest each year. Per year, interest = 12% of 2500 = 0.12 * 2500 = 300. Over 8 years, total interest = 300 * 8 = 2400. This matches the formula-based calculation and confirms the answer. Why Other Options Are Wrong:Rs. 2,200 and Rs. 2,300 are close guesses but correspond to incorrect use of either rate or time (for example, using 11% or 7.33 years). Rs. 2,500 would imply 12.5% per year or different parameters. Rs. 1,800 comes from fewer years or a lower rate. Only Rs. 2,400 fits the exact formula with the given data. Common Pitfalls:Students sometimes mistakenly use the compound interest perspective here, or they mis-multiply the numbers (especially 12 × 8). Always double-check multiplications and ensure you divide by 100 correctly when applying percentages. Keeping track of units (rupees and years) helps maintain clarity. Final Answer:The simple interest on Rs. 2,500 for 8 years at 12% per annum is Rs. 2,400.

Question 18

Find the compound interest on Rs. 2,680 at 8% per annum for 2 years, if interest is compounded annually.

Accepted Answer

Introduction / Context:This question asks you to calculate the compound interest on a given principal with a moderate rate and a 2-year term. It reinforces the use of the compound interest formula and careful handling of decimals when the amount does not come out as a clean integer. Given Data / Assumptions:- Principal P = Rs. 2,680.- Rate r = 8% per annum.- Time n = 2 years.- Compounding is annual.- We must find the compound interest, not the final amount alone. Concept / Approach:Use the compound amount formula:A = P * (1 + r/100)^n.For r = 8% and n = 2 years, the factor is (1.08)^2. Once we compute the amount A, the compound interest is C.I. = A - P. Because the numbers are not round, we keep a couple of decimal places and then match with the nearest option provided. Step-by-Step Solution:Step 1: Write A = 2680 * (1 + 8/100)^2.Step 2: Simplify: 1 + 8/100 = 1.08.Step 3: Compute (1.08)^2 = 1.1664.Step 4: So A = 2680 * 1.1664.Step 5: Multiply: A ≈ 3125.952.Step 6: Compound interest C.I. = A - P ≈ 3125.952 - 2680 = 445.952.Step 7: Rounding to two decimal places, C.I. ≈ Rs. 445.95. Verification / Alternative check:You can do it year by year: After 1 year, amount = 2680 * 1.08 = 2894.4. After 2 years, amount = 2894.4 * 1.08 ≈ 3125.952, which matches the earlier computation. Subtracting the original 2680 again gives ≈ 445.952, confirming the compound interest value and its rounded form 445.95. Why Other Options Are Wrong:Rs. 548.68 and Rs. 664.21 come from incorrectly compounding for more than 2 years or misusing the rate. Rs. 692.57 arises from multiplying with an incorrect factor. Rs. 512.00 would be closer to 19% interest over 1 year, not 8% over 2 years. Only Rs. 445.95 correctly reflects the compound interest at 8% for 2 years on Rs. 2,680. Common Pitfalls:Errors usually occur in squaring 1.08 or in multiplying 2680 by 1.1664. Some students use simple interest instead and compute 2680 * 0.08 * 2 = 428.8, which is slightly less than the compound interest because it ignores interest-on-interest. Always remember that compound interest introduces a small extra amount compared to simple interest. Final Answer:The compound interest on Rs. 2,680 at 8% per annum for 2 years is approximately Rs. 445.95.

Question 19

The compound interest earned by Sunil on a certain principal amount at the end of 2 years at the rate of 8% per annum (compounded annually) is Rs. 2,828.80. What is the total amount that Sunil receives at the end of 2 years (principal plus interest)?

Accepted Answer

Introduction / Context:This question gives you the compound interest earned in 2 years at a known rate and asks for the final amount (principal plus interest). To answer it, you must first recover the principal from the compound interest data, then add the interest back to get the total amount. It combines understanding of compound interest formulas with basic algebra. Given Data / Assumptions:- Let the principal be P rupees.- Compound interest for 2 years at 8% per annum = Rs. 2,828.80.- Interest is compounded annually.- Rate r = 8% per annum, time t = 2 years.- We must find the total amount A = P + C.I. at the end of 2 years. Concept / Approach:For 2 years of annual compounding at rate r, the amount is A = P * (1 + r/100)^2. The compound interest is C.I. = A - P = P * ((1 + r/100)^2 - 1). For r = 8%, (1.08)^2 = 1.1664, so the interest factor is 0.1664. Thus C.I. = P * 0.1664. We set this equal to 2,828.80 and solve for P. Then, the amount A is P + 2,828.80. Step-by-Step Solution:Step 1: Use C.I. formula for 2 years: C.I. = P * ((1.08)^2 - 1).Step 2: Compute (1.08)^2 = 1.1664, so (1.08)^2 - 1 = 0.1664.Step 3: Therefore, C.I. = P * 0.1664.Step 4: Given C.I. = 2828.80, so P * 0.1664 = 2828.80.Step 5: Solve for P: P = 2828.80 / 0.1664.Step 6: Perform the division: P = 17000.Step 7: Total amount A after 2 years = P + C.I. = 17000 + 2828.80 = 19828.80.Step 8: Hence, Sunil receives Rs. 19,828.80 in total after 2 years. Verification / Alternative check:Check by forward computation: Starting from P = 17,000 at 8% per annum for 2 years, amount A = 17000 * (1.08)^2 = 17000 * 1.1664 = 19828.8. Compound interest = A - P = 19828.8 - 17000 = 2828.8, which matches the given C.I. This confirms that both principal and amount are correct. Why Other Options Are Wrong:Rs. 11,828.80 and Rs. 9,828.80 are far below the principal plus interest total and might come from subtracting instead of adding the interest. Rs. 19,328.80 and Rs. 17,828.80 correspond to incorrect principal values when using the C.I. relationship. Only Rs. 19,828.80 is consistent with the given interest and the 8% compound growth over 2 years. Common Pitfalls:Some learners mistakenly think the given Rs. 2,828.80 is the final amount rather than just the interest, leading to underestimation of the principal. Others forget to square 1.08 for two years or miscompute the decimal 0.1664. Always interpret C.I. as amount minus principal and use exact factors when solving for P and A. Final Answer:The total amount that Sunil receives after 2 years is Rs. 19,828.80.

Question 20

A sum of Rs. 8,000 is deposited in a bank for 3 years at 5% per annum compound interest, with interest compounded annually. What will be the difference between the total compound interest earned for 3 years and the total compound interest earned for…

Accepted Answer

Introduction: This compound interest question tests the idea that interest itself earns further interest. We are asked to compare the compound interest earned over 3 years with that earned over 2 years on the same principal at the same rate, and to find the difference between these two interest amounts. Given Data / Assumptions: Principal P = Rs. 8,000. Rate of interest r = 5% per annum. Interest is compounded annually. We need CI for 3 years and CI for 2 years, and then their difference. Concept / Approach: For annual compounding, the amount after n years is: A = P * (1 + r/100)^n. Compound interest for n years = A − P. The difference between CI for 3 years and CI for 2 years is simply the extra interest earned in the 3rd year on the amount standing after 2 years. Step-by-Step Solution: Step 1: Compute amount after 2 years. A₂ = 8000 * (1.05)^2 = 8000 * 1.1025 = Rs. 8,820. Step 2: Compute amount after 3 years. A₃ = 8000 * (1.05)^3 = 8000 * 1.157625 = Rs. 9,261. Step 3: Find CI for 2 years. CI₂ = A₂ − P = 8,820 − 8,000 = Rs. 820. Step 4: Find CI for 3 years. CI₃ = A₃ − P = 9,261 − 8,000 = Rs. 1,261. Step 5: Difference of interests. Difference = CI₃ − CI₂ = 1,261 − 820 = Rs. 441. Verification / Alternative check: We can also write directly: Difference = P * (1.05)^2 * (0.05) = 8000 * 1.1025 * 0.05. 8000 * 0.05 = 400; 400 * 1.1025 = 441. This matches our earlier answer exactly. Why Other Options Are Wrong: Rs. 387, Rs. 469, Rs. 503 and Rs. 425 all come from incorrect use of the compound interest formula, such as using simple interest logic or miscomputing powers of 1.05. Common Pitfalls: Learners sometimes subtract 2 years from 3 years interest by using simple interest formula P * r * t / 100, which is wrong for compound interest. Others forget that the third year interest is on the increased amount after 2 years, not on the original principal alone. Final Answer: The difference between the 3 year and 2 year compound interest amounts is Rs. 441.

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