Question 1

What is the difference, in rupees, between the compound interest on Rs. 1000 for 1 year at 10% per annum when interest is compounded yearly and when interest is compounded half-yearly?

Accepted Answer

Introduction / Context: This problem compares two different compounding frequencies for the same nominal annual interest rate. The principal, rate and total time are fixed, but the compounding interval changes from yearly to half-yearly. This changes the effective amount of interest earned, and we must find the difference between the two resulting compound interests. Given Data / Assumptions: Principal P = Rs. 1000 Nominal annual interest rate = 10% per annum Total time t = 1 year Case 1: Interest compounded yearly (once per year) Case 2: Interest compounded half-yearly (twice per year) We must find CI(half-yearly) − CI(yearly) in rupees Concept / Approach: For annual compounding, we use the formula: A = P * (1 + r/100)^t. For half-yearly compounding, the periodic rate is r/2 and the number of periods n = 2 * t. The amount becomes: A = P * (1 + (r/2)/100)^(2t). We compute the amount and interest in both cases and then subtract to get the difference. Step-by-Step Solution: Case 1 (Yearly compounding): A1 = 1000 * (1 + 10/100)^1 = 1000 * 1.10 = 1100. CI1 = A1 − P = 1100 − 1000 = Rs. 100. Case 2 (Half-yearly compounding): Periodic rate = 10% / 2 = 5% per half-year. Number of periods in 1 year n = 2. A2 = 1000 * (1.05)^2 = 1000 * 1.1025 = 1102.50. CI2 = A2 − P = 1102.50 − 1000 = Rs. 102.50. Difference in compound interest = CI2 − CI1 = 102.50 − 100 = Rs. 2.50. Verification / Alternative check: We can also think of this as the extra interest generated by earning interest for an additional half-year on the first half-year's interest. That extra interest is roughly 5% of Rs. 50 (the first half-year's interest), which is Rs. 2.50. This intuitive reasoning matches our exact calculation. Why Other Options Are Wrong: Rs. 0.5 and Rs. 1.5 are too small; they would imply very weak effect from compounding more frequently. Rs. 3.5 is slightly too large. Only Rs. 2.5 exactly matches the calculated difference in compound interest between the two compounding methods. Common Pitfalls: Many learners forget to divide the annual rate by 2 when converting from yearly to half-yearly compounding, or they incorrectly double the rate. Others mistakenly assume that the interest will be the same regardless of compounding frequency. In reality, more frequent compounding at the same nominal rate always yields slightly more interest, as seen here. Final Answer: The difference in compound interest is Rs. 2.5.

Question 2

The amounts received at the end of the 2nd year and the 3rd year on a certain principal invested at compound interest are Rs. 1800 and Rs. 1926 respectively. Assuming interest is compounded annually, what is the annual rate of interest?

Accepted Answer

Introduction / Context: As in similar compound interest questions, we are given two consecutive year-end amounts and asked to find the annual interest rate. Because the investment grows by the same percentage each year, the ratio of the amount at the end of the 3rd year to the amount at the end of the 2nd year gives us the yearly growth factor directly. Given Data / Assumptions: Amount at end of 2nd year, A2 = Rs. 1800 Amount at end of 3rd year, A3 = Rs. 1926 Interest is compounded annually at a constant rate r% Principal P is fixed but not required explicitly We must find the annual rate r Concept / Approach: Under annual compounding: A3 = A2 * (1 + r/100). Thus, the ratio A3 / A2 equals the factor (1 + r/100). Once we compute this ratio, we subtract 1 and convert to a percentage to obtain r. This method avoids dealing with the original principal. Step-by-Step Solution: Step 1: Compute the ratio of amounts. A3 / A2 = 1926 / 1800. 1926 / 1800 = 1.07. Step 2: Relate this to the growth factor. 1 + r/100 = 1.07. Step 3: Solve for r. r/100 = 1.07 − 1 = 0.07. Therefore, r = 7% per annum. Verification / Alternative check: To verify, assume some principal P. After 2 years, amount A2 = P * (1.07)^2 = 1800. After 3 years, A3 = P * (1.07)^3 = A2 * 1.07 = 1800 * 1.07 = 1926. This matches the given 3rd-year amount, confirming that 7% is correct. We never actually need to compute P for the question asked. Why Other Options Are Wrong: At 6%, the factor would be 1.06, giving A3 = 1800 * 1.06 = 1908. At 6.5%, the factor is 1.065, so A3 would be 1917. For 7.5%, the factor is 1.075, giving A3 = 1935. None of these equal 1926, so those rates cannot be correct. Common Pitfalls: A typical mistake is to treat the difference (1926 − 1800) as simple interest for one year on the original principal instead of on the second-year amount. Another error is to try to compute the principal first, which is unnecessary and can lead to algebraic confusion. Always use the direct ratio between consecutive amounts when dealing with compound interest over successive years. Final Answer: The annual rate of interest is 7%.

Question 3

Find the compound interest on Rs. 4000 for 4 years at 10% per annum, assuming interest is compounded annually. Give your answer in rupees.

Accepted Answer

Introduction / Context: This is a straightforward compound interest computation. We know the principal, the annual interest rate, the duration in years, and that compounding is annual. Our goal is to calculate the total compound interest earned, not just the final amount, over four years at 10% per annum. Given Data / Assumptions: Principal P = Rs. 4000 Rate of interest r = 10% per annum Time t = 4 years Compounding frequency = annual We must find CI, the compound interest in rupees Concept / Approach: The amount A under compound interest when compounding annually is: A = P * (1 + r/100)^t. The compound interest is given by: CI = A − P. We therefore compute the growth factor (1.10)^4, multiply it by the principal to get A, and then subtract the original principal to obtain CI. Step-by-Step Solution: Step 1: Compute the growth factor. 1 + r/100 = 1 + 10/100 = 1.10. (1.10)^4 = 1.1 * 1.1 * 1.1 * 1.1 = 1.4641. Step 2: Compute the amount A after 4 years. A = 4000 * 1.4641 = Rs. 5856.40. Step 3: Compute the compound interest CI. CI = A − P = 5856.40 − 4000 = Rs. 1856.40. Verification / Alternative check: We can verify by calculating year by year: Year 1 amount = 4000 * 1.1 = 4400. Year 2 amount = 4400 * 1.1 = 4840. Year 3 amount = 4840 * 1.1 = 5324. Year 4 amount = 5324 * 1.1 = 5856.4. Total interest = 5856.4 − 4000 = 1856.4, matching our formula-based result. Why Other Options Are Wrong: Rs. 1600 is the simple interest for 4 years at 10% (4000 * 0.10 * 4), not the compound interest. Rs. 1856 is close but slightly underestimates the true value; it looks like a rounded answer that ignores the extra 0.40 rupee. Rs. 1756.60 does not correspond to any correct calculation at 10% with annual compounding. Only Rs. 1856.40 is mathematically correct. Common Pitfalls: Some students mistakenly use the simple interest formula or stop after two multiplications instead of four. Others round intermediate values too aggressively, leading to small errors in the final amount. When the question expects a precise decimal answer, carry enough decimal places through the calculation to avoid such discrepancies. Final Answer: The compound interest earned over 4 years is Rs. 1856.40.

Question 4

The difference between the compound interest and the simple interest on a certain sum of money for 3 years at 10% per annum is Rs. 186. Assuming annual compounding for compound interest, what is the principal (sum of money)?

Accepted Answer

Introduction / Context: This question involves comparing simple interest (SI) and compound interest (CI) over 3 years at the same rate. Over multiple years, CI grows faster than SI, and the difference between them depends on both the rate and the time. By using the known difference between CI and SI, we can work backward to find the principal sum invested. Given Data / Assumptions: Difference CI − SI over 3 years = Rs. 186 Rate of interest r = 10% per annum Time t = 3 years Interest is compounded annually for CI Principal P is unknown and must be found Concept / Approach: For annual compounding at rate r, the amount after 3 years is: A = P * (1 + r/100)^3. Thus: CI = A − P = P * [(1 + r/100)^3 − 1]. Simple interest for 3 years is: SI = P * r * t / 100 = P * 3r / 100. The difference is: CI − SI = P * [(1 + r/100)^3 − 1 − 3r/100]. We substitute r = 10% and the known difference to solve for P. Step-by-Step Solution: Step 1: Compute (1 + r/100)^3 for r = 10. (1.10)^3 = 1.331. Step 2: Calculate the bracket term. (1.10)^3 − 1 − 3r/100 = 1.331 − 1 − 0.30 = 1.331 − 1.30 = 0.031. Step 3: Write the difference formula. CI − SI = P * 0.031. Given CI − SI = 186, we have 186 = 0.031P. Step 4: Solve for P. P = 186 / 0.031 = 6000. Verification / Alternative check: Compute SI and CI explicitly for P = Rs. 6000 at 10% for 3 years. SI = 6000 * 10 * 3 / 100 = 1800. Amount under CI: A = 6000 * 1.331 = 7986. CI = A − P = 7986 − 6000 = 1986. Difference CI − SI = 1986 − 1800 = 186, matching the given value and confirming that the principal is correct. Why Other Options Are Wrong: For P = 5500, the difference would be 5500 * 0.031 = 170.5. For 6500, it would be 6500 * 0.031 = 201.5. For 7200, it would be 7200 * 0.031 = 223.2. None of these match 186, so those principals cannot be correct. Common Pitfalls: Students often confuse the formulas for the 2-year and 3-year differences between CI and SI, applying the simpler 2-year shortcut P * r^2 / 100^2 incorrectly to a 3-year problem. Another common mistake is to miscalculate (1.10)^3 or the term 3r/100. Being careful with these computations is essential to arriving at the correct principal. Final Answer: The principal sum of money is Rs. 6000.

Question 5

A certain sum of money invested at compound interest at a rate of 15% per annum (compounded annually) earns Rs. 3225 as compound interest in 2 years. What is the original sum (principal) in rupees?

Accepted Answer

Introduction / Context: This is a reverse compound interest problem where the interest amount over a given period is known, and the original principal must be determined. The interest is compounded annually at a rate of 15% per annum for 2 years, and the total compound interest earned over those 2 years is Rs. 3225. Given Data / Assumptions: Compound interest CI for 2 years = Rs. 3225 Rate of interest r = 15% per annum Time t = 2 years Compounding is annual Principal P is unknown Concept / Approach: For annual compounding, the amount after 2 years is: A = P * (1 + r/100)^2. The compound interest is: CI = A − P = P * [(1 + r/100)^2 − 1]. Thus: CI = P * [(1.15)^2 − 1]. We compute (1.15)^2, subtract 1 to get the interest factor, and then use CI = P * factor to solve for P. Step-by-Step Solution: Step 1: Compute the 2-year growth factor. (1.15)^2 = 1.15 * 1.15 = 1.3225. Step 2: Find the interest factor. (1.15)^2 − 1 = 1.3225 − 1 = 0.3225. Step 3: Relate CI to P. CI = P * 0.3225. 3225 = 0.3225 * P. Step 4: Solve for P. P = 3225 / 0.3225 = 10,000. Verification / Alternative check: Verify by calculating CI from P = 10,000. Amount after 2 years at 15% compounded annually: A = 10,000 * 1.3225 = 13,225. CI = 13,225 − 10,000 = 3,225 rupees. This matches the given compound interest, confirming that the principal must be Rs. 10,000. Why Other Options Are Wrong: If P were 15,000, CI would be 15,000 * 0.3225 = 4,837.50. For 20,000, CI would be 6,450. For 32,250, CI would be even larger. None of these equal 3,225, so those options cannot be correct. Common Pitfalls: A common mistake is to use simple interest formulas or to treat 15% as a flat rate per year without squaring the factor for 2 years. Others may try to solve using yearly interest step-by-step without recognizing the compact formula, which can introduce arithmetic errors. Recognizing the clean factor 1.3225 in this problem simplifies the solution significantly. Final Answer: The original sum (principal) was Rs. 10,000.

Question 6

In 3 years, Rs. 3000 amounts to Rs. 3993 when invested at x% compound interest, compounded annually. What is the value of x (the annual rate of interest in percent)?

Accepted Answer

Introduction / Context: This problem asks us to find the annual compound interest rate when the principal, final amount and time are given. The investment grows from Rs. 3000 to Rs. 3993 in 3 years under annual compounding. Recognizing familiar growth factors can make this problem quick to solve. Given Data / Assumptions: Principal P = Rs. 3000 Amount after 3 years A = Rs. 3993 Time t = 3 years Interest is compounded annually at rate x% per annum We need to find x Concept / Approach: For annual compounding, the relation between amount and principal is: A = P * (1 + x/100)^t. Here t = 3, so: A / P = (1 + x/100)^3. We compute A / P and then find which known cube it matches, thereby deducing (1 + x/100) and hence x. Step-by-Step Solution: Step 1: Compute the ratio A / P. A / P = 3993 / 3000. 3993 / 3000 = 1.331. Step 2: Recognize a standard cube. We know that (1.10)^3 = 1.1 * 1.1 * 1.1 = 1.331. Thus, (1 + x/100)^3 = 1.331 = (1.10)^3. Step 3: Equate the factors. 1 + x/100 = 1.10 ⇒ x/100 = 0.10 ⇒ x = 10. Verification / Alternative check: Verify by recomputing the amount at 10% per annum. Amount after 3 years: A = 3000 * (1.10)^3 = 3000 * 1.331 = 3993. This matches the given amount exactly, confirming that the rate is 10% per annum. Why Other Options Are Wrong: At 8%, the factor is (1.08)^3 ≈ 1.2597, giving A ≈ 3000 * 1.2597 = 3779.1, which is less than 3993. At 5%, (1.05)^3 ≈ 1.157625, giving A ≈ 3472.9. At 3 1/3%, the factor is too small to reach 3993 in 3 years. Only a 10% rate produces exactly the required final amount. Common Pitfalls: Many students attempt to use trial-and-error or logarithms when the numbers actually correspond to a neat cube of a simple factor like 1.1. Recognizing standard powers (such as 1.1^2, 1.1^3, etc.) is a useful skill in exam settings and can save significant time. Final Answer: The rate of interest x is 10% per annum.

Question 7

A man borrows some money at compound interest and agrees to repay the loan in two yearly instalments: Rs. 3150 at the end of the 1st year and Rs. 4410 at the end of the 2nd year. If the rate of compound interest is 5% per annum, what is the original…

Accepted Answer

Introduction / Context: This is a loan repayment problem under compound interest. The borrower repays the loan by making two instalments at the end of successive years. Each instalment includes interest and partial repayment of principal. To find the original amount borrowed, we treat each instalment as a payment whose present value (discounted back to the date of borrowing) must sum to the initial loan amount. Given Data / Assumptions: Loan is repaid in two instalments: • Rs. 3150 at the end of year 1 • Rs. 4410 at the end of year 2 Rate of compound interest r = 5% per annum Compounding yearly Let the principal borrowed be P Concept / Approach: At 5% per annum, the present value (today's value) of a payment made after n years is given by: PV = Payment / (1.05)^n. Since the man borrows P today and repays with two future payments, the sum of the present values of those payments must equal P. Hence: P = 3150 / 1.05 + 4410 / (1.05)^2. We compute this expression to find the original loan amount. Step-by-Step Solution: Step 1: Discount the first instalment back 1 year. PV1 = 3150 / 1.05. PV1 = 3000 (since 3150 ÷ 1.05 = 3000). Step 2: Discount the second instalment back 2 years. PV2 = 4410 / (1.05)^2. (1.05)^2 = 1.1025. PV2 = 4410 / 1.1025 = 4000. Step 3: Add the present values to find the loan. P = PV1 + PV2 = 3000 + 4000 = Rs. 7000. Verification / Alternative check: We can check by forward calculation. If P = 7000 at 5%, amount after 1 year is 7000 * 1.05 = 7350. After paying 3150 at the end of year 1, the remaining balance is 7350 − 3150 = 4200. This 4200 grows at 5% for another year to 4200 * 1.05 = 4410, which is exactly the second instalment. Therefore, a loan of Rs. 7000 is completely cleared by the given instalments. Why Other Options Are Wrong: If the loan were Rs. 5000 or Rs. 6500, the future values and remaining balances would not match the instalment schedule and interest rate. Likewise, Rs. 9200 would require significantly larger repayments to settle the loan in just two years at 5%. Only Rs. 7000 leads to a consistent repayment pattern. Common Pitfalls: A common error is to simply add the two instalments and subtract some estimated interest, instead of properly discounting each instalment back to today. Another mistake is treating the second instalment as if it were discounted only one year instead of two. Always remember that each payment must be discounted for the full time between the borrowing date and the payment date. Final Answer: The original sum borrowed was Rs. 7000.

Question 8

Rs. 260,200 is divided between Ram and Shyam and invested at 4% compound interest per annum, compounded annually. Ram's share is invested for 3 years, while Shyam's share is invested for 6 years. The amounts they each receive at the end of their res…

Accepted Answer

Introduction / Context: This problem combines compound interest with algebraic reasoning. A total sum is split between two people, Ram and Shyam, and each part grows for a different number of years at the same annual compound interest rate. The condition that both final amounts are equal allows us to set up an equation and solve for Ram's initial share. Given Data / Assumptions: Total sum = Rs. 260,200 Ram invests his share for 3 years at 4% per annum, compounded annually Shyam invests his share for 6 years at the same 4% per annum, compounded annually Final amounts received by Ram and Shyam are equal Let Ram's initial share be R, Shyam's initial share be S Thus R + S = 260,200 Concept / Approach: Under annual compounding at rate r, a principal P grows after t years to: Amount = P * (1 + r)^t, where r is in decimal form. Here r = 0.04 (since 4% per annum). Ram's amount after 3 years is R * (1.04)^3, and Shyam's amount after 6 years is S * (1.04)^6. We are told these amounts are equal, so we equate them and use the fact that S = 260,200 − R to solve for R. Step-by-Step Solution: Step 1: Express amounts in terms of R and S. Ram's amount = R * (1.04)^3. Shyam's amount = S * (1.04)^6. Given Ram's amount = Shyam's amount: R * (1.04)^3 = S * (1.04)^6. Step 2: Simplify the equation using S = 260,200 − R. Divide both sides by (1.04)^3: R = S * (1.04)^3 = (260,200 − R) * (1.04)^3. Let k = (1.04)^3 = 1.124864. Then R = (260,200 − R) * k. Step 3: Solve for R. R = 260,200k − Rk. R + Rk = 260,200k. R(1 + k) = 260,200k. R = 260,200k / (1 + k). Substitute k = 1.124864: R ≈ 260,200 * 1.124864 / 2.124864 ≈ Rs. 137,745 (to the nearest rupee). Verification / Alternative check: Using R ≈ 137,745, Shyam's share S ≈ 260,200 − 137,745 = 122,455. After 3 years, Ram's amount ≈ 137,745 * (1.04)^3 ≈ Rs. 154,945. After 6 years, Shyam's amount ≈ 122,455 * (1.04)^6, which numerically matches the same figure (up to rounding). Thus the condition that both receive equal final amounts is satisfied with Ram's share approximately Rs. 137,745. Why Other Options Are Wrong: If Ram's share were Rs. 125,000 or Rs. 152,000 or Rs. 108,200, then after compounding for 3 years, his amount would be significantly different from Shyam's amount after 6 years, breaking the equality condition. These values are plausible-looking but do not satisfy the precise compound interest relationship derived above. Common Pitfalls: One common mistake is to equate the principals directly instead of their compounded amounts, forgetting that they grow for different durations. Another is to treat the problem as simple interest, which gives a different and incorrect division. Some students also drop precision prematurely when dealing with (1.04)^3, leading to noticeable errors in the final value of R. Careful handling of the algebra and the growth factor is essential. Final Answer: Ram's initial share is approximately Rs. 137,745.

Question 9

For a certain principal invested at compound interest, the interest earned in the 3rd year is Rs. 125 and the interest earned in the 4th year is Rs. 135. Assuming annual compounding, what is the annual rate of interest (in percent)?

Accepted Answer

Introduction / Context: This problem focuses on successive yearly interests under compound interest. When interest is compounded annually at a constant rate, the interest earned each year forms a geometric progression. We are given the interest in the 3rd and 4th years and are asked to find the underlying rate of interest. Given Data / Assumptions: Interest in 3rd year, I3 = Rs. 125 Interest in 4th year, I4 = Rs. 135 Interest is compounded annually at a constant rate r% Principal is fixed but not needed explicitly We must find r (the annual rate in percent) Concept / Approach: Under annual compounding at rate r%, the principal grows each year, so the interest for each subsequent year is obtained by multiplying the previous year's interest by (1 + r/100). Therefore: I4 = I3 * (1 + r/100). This simple relation allows us to find the rate directly using the ratio I4 / I3. Step-by-Step Solution: Step 1: Compute the ratio of successive interests. I4 / I3 = 135 / 125 = 1.08. Step 2: Relate this to the growth factor. I4 = I3 * (1 + r/100) ⇒ 1 + r/100 = 1.08. Step 3: Solve for r. r/100 = 1.08 − 1 = 0.08. Thus, r = 8% per annum. Verification / Alternative check: Suppose the principal at the start of the 3rd year is P3. Then I3 = P3 * 0.08 = 125 ⇒ P3 = 125 / 0.08 = 1562.50. At the start of the 4th year, the principal becomes P4 = P3 + I3 = 1562.5 + 125 = 1687.5. The interest in the 4th year at 8% is then 1687.5 * 0.08 = 135, which matches the given I4 and confirms that the rate is 8%. Why Other Options Are Wrong: At 9%, the ratio would be 1.09, not 1.08. At 10%, it would be 1.10, and at 12%, it would be 1.12. None of these ratios matches 135 / 125. Only an 8% rate produces the correct progression from 125 to 135. Common Pitfalls: A common misconception is to try to express I3 and I4 in terms of the original principal and then solve using more complicated algebra, when the simple ratio method is much quicker. Another mistake is to assume the difference between interests is a flat amount and attempt to use simple interest logic, ignoring that the base on which interest is calculated has itself increased each year under compounding. Final Answer: The annual rate of interest is 8%.

Question 10

A sum of Rs. 25,000 earns compound interest for 1 year at a nominal rate of 20% per annum. What is the difference, in rupees, between the compound interest earned when interest is compounded semi-annually and when it is compounded annually?

Accepted Answer

Introduction / Context: This question examines how the compounding frequency affects the amount of compound interest earned for the same nominal annual rate. With more frequent compounding (semi-annual instead of annual), the effective interest rate increases slightly, leading to a higher compound interest amount. We must quantify this difference over one year for a principal of Rs. 25,000 at 20% per annum. Given Data / Assumptions: Principal P = Rs. 25,000 Nominal annual interest rate = 20% per annum Time t = 1 year Case 1: Compounded annually (once per year) Case 2: Compounded semi-annually (twice per year) We must find CI(semi-annual) − CI(annual) Concept / Approach: For annual compounding, the amount after 1 year is: A1 = P * (1 + 0.20). For semi-annual compounding, the periodic rate is 20% / 2 = 10% per half-year, and there are 2 periods in 1 year, so: A2 = P * (1 + 0.10)^2. Compound interest in each case is amount minus principal. The difference in compound interest is then (A2 − P) − (A1 − P) = A2 − A1. Step-by-Step Solution: Case 1 (Annual compounding): A1 = 25,000 * 1.20 = Rs. 30,000. CI1 = A1 − P = 30,000 − 25,000 = Rs. 5,000. Case 2 (Semi-annual compounding): Periodic rate i = 10% per half-year. Number of periods n = 2. A2 = 25,000 * (1.10)^2 = 25,000 * 1.21 = Rs. 30,250. CI2 = A2 − P = 30,250 − 25,000 = Rs. 5,250. Difference in compound interest = CI2 − CI1 = 5,250 − 5,000 = Rs. 250. Verification / Alternative check: Instead of computing both interests outright, notice that the only difference is between the amounts A2 and A1. Directly computing 25,000 * 1.21 − 25,000 * 1.20 gives 25,000 * (1.21 − 1.20) = 25,000 * 0.01 = 250. This reconfirms that semi-annual compounding generates exactly Rs. 250 more interest than annual compounding at the same nominal rate over one year. Why Other Options Are Wrong: Rs. 125 and Rs. 375 represent half and one and a half times the correct difference, respectively, and do not align with the precise calculation. Rs. 500 would correspond to a much larger gap in effective interest than actually occurs between annual and semi-annual compounding at 20%. Only Rs. 250 matches the exact difference. Common Pitfalls: A common mistake is to forget to divide the nominal annual rate by 2 when switching to semi-annual compounding, instead using 20% per half-year, which is incorrect. Another error is to compare the interest amounts without subtracting the principal or to think that the nominal rate alone determines the interest, ignoring compounding frequency. Careful identification of the periodic rate and number of periods avoids such errors. Final Answer: The difference in compound interest between semi-annual and annual compounding is Rs. 250.

Question 11

Find the compound interest on Rs. 8000 at 15% per annum for 2 years 4 months, when interest is compounded annually and the extra 4 months are treated using simple interest on the amount after 2 years.

Accepted Answer

Introduction / Context: This question focuses on calculating compound interest for a non-integer time period where the interest is compounded annually, and an extra fraction of a year is handled using simple interest on the accumulated amount. Such problems combine compound interest over whole years with simple interest over remaining months, which is a common pattern in competitive exams. Here, the principal is Rs. 8000, the rate is 15% per annum, and the time is 2 years 4 months. Given Data / Assumptions: • Principal (P) = Rs. 8000. • Rate of interest (R) = 15% per annum. • Total time = 2 years 4 months. • Interest is compounded annually. • The extra 4 months are treated using simple interest on the amount after 2 full years. Concept / Approach: For the first two full years, we calculate compound interest using the standard formula for annual compounding: A = P * (1 + R/100)^n, where n is the number of years. After obtaining the amount at the end of 2 years, we treat the remaining 4 months as 4/12 of a year and compute simple interest on this amount at the same annual rate. Adding this simple interest to the 2-year amount gives the final amount, and subtracting the original principal yields the total interest earned over 2 years 4 months. Step-by-Step Solution: Step 1: Compute amount after 2 years with annual compounding. Step 2: A2 = P * (1 + R/100)^2 = 8000 * (1 + 15/100)^2. Step 3: 1 + 15/100 = 1.15, so A2 = 8000 * (1.15)^2. Step 4: (1.15)^2 = 1.3225, so A2 = 8000 * 1.3225 = Rs. 10,580. Step 5: Time remaining = 4 months = 4/12 year = 1/3 year. Step 6: Simple interest for 4 months on A2 at 15% per annum: SI = A2 * (R/100) * (1/3). Step 7: SI = 10,580 * (15/100) * (1/3) = 10,580 * 0.15 * (1/3). Step 8: 0.15 * (1/3) = 0.05, so SI = 10,580 * 0.05 = Rs. 529. Step 9: Final amount after 2 years 4 months = A2 + SI = 10,580 + 529 = Rs. 11,109. Step 10: Total interest = final amount − principal = 11,109 − 8000 = Rs. 3,109. Verification / Alternative check: We can verify intermediate values. First, check that the 2-year compounding is correct: 8000 at 15% for 1 year gives 8000 * 1.15 = 9200. One more year at 15% gives 9200 * 1.15 = 10,580, which matches A2. Then 4 months is one-third of a year, and 15% of 10,580 for a full year is 1,587. For one-third year, it is approximately 1,587 / 3 = 529, which matches the computed SI. Thus, the final interest of 3,109 is consistent. Why Other Options Are Wrong: • Rs. 2109: This ignores or underestimates the extra simple interest for 4 months after compounding for 2 years. • Rs. 4109: This overstates the interest, likely by mistakenly compounding for the entire 2 years 4 months without splitting into compound and simple parts correctly. • Rs. 6109: This is far too high and does not align with the given rate and time period for the principal amount. Common Pitfalls: A common mistake is to treat the entire 2 years 4 months as if compounding occurs continuously or quarterly without reading the phrase "compounded annually" carefully. Another error is to forget that the extra months should be converted to a fraction of a year and used with simple interest on the amount after full years of compounding. Carefully separating the compound interest and simple interest portions ensures an accurate solution. Final Answer: The compound interest earned on Rs. 8000 at 15% per annum for 2 years 4 months, with annual compounding and simple interest for the extra months, is Rs. 3109.

Question 12

A sum of money is lent at compound interest for 2 years at 20% per annum. If the interest were payable half-yearly instead of annually, it would fetch Rs. 482 more. What is the principal sum?

Accepted Answer

Introduction / Context: This question compares compound interest under two different compounding frequencies: annually and half-yearly. The interest rate is 20% per annum, and the time is 2 years. You are told that half-yearly compounding yields Rs. 482 more interest than annual compounding. From this information, you must determine the principal sum. Such questions reinforce your understanding of how compounding frequency affects the total amount and how to work backward from a known difference in interest. Given Data / Assumptions: • Principal (P) = unknown (to be found). • Time = 2 years. • Rate of interest per annum = 20%. • Case 1: Compounded annually. • Case 2: Compounded half-yearly (twice a year). • Additional interest when compounded half-yearly instead of annually = Rs. 482. Concept / Approach: We use the compound interest amount formula A = P * (1 + R/100)^n. For annual compounding, rate is 20% per year and n = 2 years. For half-yearly compounding, the rate per half-year is 10% (half of 20%), and the number of periods in 2 years is 4. We compute the amounts in both cases in terms of P, subtract them to find the difference in interest, and equate that difference to Rs. 482. Solving the resulting equation gives the principal P. Step-by-Step Solution: Step 1: For annual compounding, amount A1 = P * (1 + 20/100)^2. Step 2: 1 + 20/100 = 1.20, so A1 = P * (1.20)^2 = P * 1.44. Step 3: For half-yearly compounding, rate per half-year = 20/2 = 10%. Step 4: Number of half-year periods in 2 years = 2 * 2 = 4. Step 5: Amount A2 = P * (1 + 10/100)^4 = P * (1.10)^4. Step 6: (1.10)^2 = 1.21 and (1.10)^4 = (1.21)^2 = 1.4641, so A2 = P * 1.4641. Step 7: Difference in amounts (and hence in interest) = A2 − A1 = P * (1.4641 − 1.44). Step 8: 1.4641 − 1.44 = 0.0241, so difference = 0.0241 * P. Step 9: Given that this difference is Rs. 482, we have 0.0241 * P = 482. Step 10: Therefore, P = 482 / 0.0241 = Rs. 20,000. Verification / Alternative check: We can briefly check by computing the actual amounts for P = 20,000. A1 = 20,000 * 1.44 = Rs. 28,800. A2 = 20,000 * 1.4641 = Rs. 29,282. The difference A2 − A1 = 29,282 − 28,800 = Rs. 482, exactly matching the given difference. This confirms that the principal sum is indeed Rs. 20,000. Why Other Options Are Wrong: • Rs. 10,000: For this principal, the difference in interest would be half of 482, that is, Rs. 241, which does not match the given condition. • Rs. 40,000: This would double the difference to Rs. 964, again not matching the stated difference of Rs. 482. • Rs. 50,000: This would make the difference even larger, 0.0241 * 50,000 = Rs. 1205, far from the given figure. Common Pitfalls: Common mistakes include using the wrong number of compounding periods for the half-yearly case, or forgetting to convert the annual rate to the half-yearly rate. Some students also incorrectly subtract percentages instead of computing the difference in amounts. Staying disciplined with the formula and properly adjusting rate and number of periods for each compounding frequency prevents these errors. Final Answer: The principal sum of money is Rs. 20,000.

Question 13

The compound interest on Rs. 30,000 at 7% per annum is Rs. 4347. For how many years was the money invested at this rate?

Accepted Answer

Introduction / Context: This question asks you to determine the time period for which a sum of money was invested under compound interest. You are given the principal, the rate of interest per annum, and the total compound interest earned. Using this information, you can compute the final amount and then match it with powers of (1 + rate) to find the number of years. Such problems are standard in the topic of compound interest and help in understanding how time, rate, and interest are related. Given Data / Assumptions: • Principal (P) = Rs. 30,000. • Rate of interest (R) = 7% per annum. • Compound interest (CI) = Rs. 4347. • Compounding is assumed to be annual. • Time period (n years) is unknown and must be found. Concept / Approach: The amount after n years at compound interest is given by A = P * (1 + R/100)^n. Since CI = A − P, we have A = P + CI. Substituting the known values of P and CI will give the amount A. Next, we express A/P as (1 + R/100)^n and compare to known powers of (1.07) to find n. In many exam questions, the numbers are chosen to make this comparison straightforward. Step-by-Step Solution: Step 1: Compute the final amount using the given compound interest. Step 2: A = P + CI = 30,000 + 4347 = Rs. 34,347. Step 3: Write A in terms of P and (1 + R/100)^n: A = P * (1.07)^n. Step 4: Therefore, A / P = (1.07)^n = 34,347 / 30,000. Step 5: 34,347 / 30,000 = 1.1449. Step 6: Compute (1.07)^2 = 1.07 * 1.07 = 1.1449. Step 7: Since (1.07)^2 equals 1.1449, we have (1.07)^n = (1.07)^2. Step 8: Therefore, n = 2 years. Verification / Alternative check: We can verify directly by calculating the compound amount for 2 years at 7%. After 1 year: amount = 30,000 * 1.07 = Rs. 32,100. After 2 years: new amount = 32,100 * 1.07 = Rs. 34,347. The compound interest over 2 years = 34,347 − 30,000 = Rs. 4,347, which matches the given CI. This confirms that the time period is indeed 2 years. Why Other Options Are Wrong: • 2.5 years: This would not give an amount that matches 34,347 when using simple annual compounding and a fractional year, and the CI would differ from the given value. • 3 years: (1.07)^3 is greater than 1.1449, so the amount would be higher than Rs. 34,347 for the same principal. • 4 years: Over 4 years, the amount would be much larger and the interest far exceed Rs. 4,347. Common Pitfalls: Some students try to use simple interest formulas or approximate the powers without checking the exact values, leading to confusion. Others may forget to add the principal and interest to find the amount A before using the compound interest formula. Remembering that CI = A − P and using A/P to match powers of (1 + R/100) makes such problems straightforward. Final Answer: The money was invested for a period of 2 years.

Question 14

A sum of money amounts to Rs. 6690 after 3 years and to Rs. 10,035 after 6 years on compound interest at the same rate. Find the original principal sum.

Accepted Answer

Introduction / Context: This question deals with compound interest when you are given the amounts at two different times (after 3 years and after 6 years) but not the rate or principal explicitly. You must use the relationship between these amounts to determine the original principal. Such problems highlight how compound interest grows exponentially with time and how ratios of amounts can reveal the growth factor without directly knowing the rate. Given Data / Assumptions: • Amount after 3 years (A3) = Rs. 6690. • Amount after 6 years (A6) = Rs. 10,035. • The rate of interest per annum is the same throughout. • Interest is compounded annually. • Principal (P) is unknown and needs to be found. Concept / Approach: Let the annual growth factor be (1 + R/100). Then after 3 years, A3 = P * (1 + R/100)^3, and after 6 years, A6 = P * (1 + R/100)^6. Taking the ratio A6 / A3 cancels P and leaves (1 + R/100)^3. From this ratio, we can simplify the relationship and then express P in terms of A3 and the factor. In this specific problem, the numbers are arranged so that the ratio is simple, making the calculations straightforward. Step-by-Step Solution: Step 1: Write the expressions for the amounts. Step 2: A3 = P * (1 + R/100)^3 = 6690. Step 3: A6 = P * (1 + R/100)^6 = 10,035. Step 4: Take the ratio A6 / A3 to eliminate P. Step 5: A6 / A3 = [P * (1 + R/100)^6] / [P * (1 + R/100)^3] = (1 + R/100)^3. Step 6: Compute A6 / A3 = 10,035 / 6690. Step 7: 10,035 / 6690 = 1.5. Step 8: Therefore, (1 + R/100)^3 = 1.5. Step 9: From A3 = P * (1 + R/100)^3, we have 6690 = P * 1.5. Step 10: Hence, P = 6690 / 1.5 = Rs. 4460. Verification / Alternative check: We can verify by checking the amounts using P = 4460. After 3 years, A3 = 4460 * 1.5 = Rs. 6690, which matches the given value. After 6 years, A6 = A3 * 1.5 = 6690 * 1.5 = Rs. 10,035, also matching the given data. This confirms that the principal must be Rs. 4460. Why Other Options Are Wrong: • Rs. 4360, Rs. 4560, and Rs. 4660: None of these values satisfy both conditions simultaneously. If you plug any of them into the compound interest relationship, you will not obtain both A3 = Rs. 6690 and A6 = Rs. 10,035 exactly, so they cannot be the correct principal. Common Pitfalls: A frequent mistake is attempting to find the exact interest rate first using fractional powers, which can complicate calculations unnecessarily. The smarter approach is to use the ratio of amounts to determine the growth factor and then directly solve for P. Another error is mis-computing the ratio A6 / A3. Carefully simplifying the ratio and using it to express P makes the problem much easier. Final Answer: The original principal sum is Rs. 4460.

Question 15

The difference between compound interest and simple interest on a certain sum for 2 years at 8% per annum, when interest is compounded annually, is Rs. 16. If the interest were instead compounded half-yearly, what would be the approximate difference…

Accepted Answer

Introduction / Context: This question contrasts simple interest (SI) with compound interest (CI) at the same annual rate, but under different compounding frequencies. You are told the difference between CI and SI when interest is compounded annually and asked to find the difference if the interest were compounded half-yearly instead. This requires first determining the principal using the annual-compounding difference, and then recomputing the CI under half-yearly compounding for the same time period and rate. Given Data / Assumptions: • Let the principal be P (unknown initially). • Rate of interest (R) = 8% per annum. • Time (T) = 2 years. • When interest is compounded annually, CI − SI for 2 years = Rs. 16. • We must find the new difference CI − SI if interest is compounded half-yearly over the same 2 years. Concept / Approach: First, use the formula for CI and SI with annual compounding to find P from the given difference of Rs. 16. For 2 years at 8% per annum, simple interest is P * R * T / 100, and compound interest is P * (1 + R/100)^2 − P. The difference CI − SI depends only on P and R. Once we compute P, we then calculate: (1) simple interest for 2 years at 8% per annum, and (2) compound interest for 2 years when compounded half-yearly, i.e., at 4% every half-year for 4 periods. The difference between these new CI and SI values is the required answer. Step-by-Step Solution: Step 1: Under annual compounding, SI for 2 years = P * 8/100 * 2 = 0.16P. Step 2: CI for 2 years (annual) = P * (1 + 8/100)^2 − P = P * (1.08)^2 − P. Step 3: (1.08)^2 = 1.1664, so CI = P * 1.1664 − P = 0.1664P. Step 4: Given CI − SI = 0.1664P − 0.16P = 0.0064P = Rs. 16. Step 5: Therefore, P = 16 / 0.0064 = Rs. 2500. Step 6: Compute SI for 2 years at 8% per annum on P = 2500. Step 7: SI = 2500 * 8/100 * 2 = 2500 * 0.16 = Rs. 400. Step 8: Now consider half-yearly compounding. Rate per half-year = 8/2 = 4%. Step 9: Number of half-year periods in 2 years = 4. Step 10: CI (half-yearly) amount = 2500 * (1 + 4/100)^4 = 2500 * (1.04)^4. Step 11: (1.04)^2 = 1.0816 and (1.04)^4 = (1.0816)^2 ≈ 1.16985856. Step 12: Amount with CI = 2500 * 1.16985856 ≈ Rs. 2924.65. Step 13: CI (half-yearly) = 2924.65 − 2500 ≈ Rs. 424.65. Step 14: New difference CI − SI = 424.65 − 400 ≈ Rs. 24.65, which is very close to Rs. 24.64. Verification / Alternative check: We can compute a more precise value directly: CI (half-yearly) = P * [(1.04)^4 − 1]. Since P = 2500 and (1.04)^4 ≈ 1.16985856, CI ≈ 2500 * 0.16985856 ≈ 424.65. The simple interest remains exactly Rs. 400, so CI − SI ≈ 24.65. The nearest given option is Rs. 24.64, which matches the standard rounded value used in such exam questions. Why Other Options Are Wrong: • Rs. 21.85: This underestimates the true difference and does not arise from correct calculation of (1.04)^4. • Rs. 16.00: This is the original difference under annual compounding and does not apply to half-yearly compounding. • Rs. 16.80: This is still significantly lower than the accurately computed difference of about Rs. 24.65. Common Pitfalls: Errors often occur when students forget to first determine the principal from the initial difference of Rs. 16. Others may incorrectly treat half-yearly compounding as simply doubling the annual rate or using the wrong number of periods. Carefully applying the compound interest formula with the adjusted rate and number of periods, and using the same principal for both SI and CI comparisons, is critical to obtaining the correct answer. Final Answer: If the interest is compounded half-yearly instead of annually, the approximate difference between compound interest and simple interest over 2 years is Rs. 24.64.

Question 16

Find the compound interest on Rs. 10,000 in 2 years at 4% per annum, when the interest is compounded half-yearly (that is, at 2% every half-year).

Accepted Answer

Introduction / Context: This problem asks you to calculate compound interest when the interest is compounded half-yearly. The annual rate is 4%, which translates into 2% for each half-year period. With a principal of Rs. 10,000 and a total time of 2 years, you must account for four compounding periods. Such questions are common in examinations and reinforce your ability to apply the compound interest formula with adjusted rates and multiple compounding periods. Given Data / Assumptions: • Principal (P) = Rs. 10,000. • Rate of interest per annum = 4%. • Interest is compounded half-yearly. • Time (T) = 2 years. • Therefore, rate per half-year = 4% / 2 = 2%. • Number of half-year periods in 2 years = 2 * 2 = 4. Concept / Approach: When interest is compounded half-yearly, we adjust the annual rate and the number of compounding periods accordingly. The compound amount formula becomes A = P * (1 + r/100)^n, where r is the rate per half-year and n is the number of half-year periods. After computing the final amount A, we subtract the principal to get the compound interest. This method ensures you properly account for the effect of compounding more frequently than once per year. Step-by-Step Solution: Step 1: Determine the rate per compounding period. Step 2: Annual rate = 4%, so half-yearly rate r = 4 / 2 = 2% per half-year. Step 3: Time = 2 years, so number of half-year periods n = 2 * 2 = 4. Step 4: Use the compound amount formula: A = P * (1 + r/100)^n. Step 5: Substitute the values: A = 10,000 * (1 + 2/100)^4 = 10,000 * (1.02)^4. Step 6: Compute (1.02)^2 = 1.0404. Step 7: Then (1.02)^4 = (1.0404)^2 ≈ 1.08243216. Step 8: Amount A = 10,000 * 1.08243216 ≈ Rs. 10,824.32. Step 9: Compound interest CI = A − P = 10,824.32 − 10,000 = Rs. 824.32. Verification / Alternative check: We can also compute stepwise. After the first half-year, amount = 10,000 * 1.02 = 10,200. After one year (two half-years), amount = 10,200 * 1.02 ≈ 10,404. After 1.5 years, amount ≈ 10,404 * 1.02 ≈ 10,612.08. After 2 years, amount ≈ 10,612.08 * 1.02 ≈ 10,824.32. The total interest is again 10,824.32 − 10,000 = Rs. 824.32, confirming our earlier calculation. Why Other Options Are Wrong: • Rs. 524.32: This significantly underestimates the effect of four compounding periods at 2% each and does not match the computed amount. • Rs. 624.32: This is closer but still below the true interest, reflecting fewer effective compounding periods than actually occur. • Rs. 724.32: This is also less than the correct value and arises from approximate or incomplete calculations. Common Pitfalls: Some students mistakenly use 4% as the rate per half-year or forget to double the number of periods when compounding half-yearly. Others may incorrectly apply simple interest formulas or perform rounding too early in intermediate steps. To avoid errors, always adjust the rate and number of periods carefully and perform calculations with sufficient precision before rounding at the final step. Final Answer: The compound interest on Rs. 10,000 at 4% per annum for 2 years, compounded half-yearly, is Rs. 824.32.

Question 17

Reena took a loan of Rs. 1200 at simple interest for as many years as the numerical value of the rate of interest. If she paid Rs. 432 as interest at the end of the loan period, what was the rate of interest per annum?

Accepted Answer

Introduction / Context: This question is a classic simple interest problem where the time in years is numerically equal to the rate of interest in percent. You are given the principal and the total interest paid and asked to find the rate. Such problems rely on the simple interest formula and the relationship between time and rate, and they are very common in competitive exams testing basic financial arithmetic. Given Data / Assumptions: • Principal (P) = Rs. 1200. • Simple interest (SI) paid = Rs. 432. • Time (T) in years is equal to the rate of interest (R) in percent. • Interest is calculated using simple interest, not compound interest. Concept / Approach: The simple interest formula is SI = (P * R * T) / 100. Here, we are told that T = R. This allows us to substitute T with R, leading to SI = (P * R^2) / 100. We already know SI and P, so we can solve this equation for R^2 and then for R. Since R represents a positive percentage rate, we choose the positive square root of R^2. Step-by-Step Solution: Step 1: Write the simple interest formula: SI = (P * R * T) / 100. Step 2: Given T = R, substitute T with R, so SI = (P * R * R) / 100 = (P * R^2) / 100. Step 3: Substitute the known values: SI = 432 and P = 1200. Step 4: 432 = (1200 * R^2) / 100. Step 5: Simplify the right side: (1200 / 100) = 12, so 432 = 12 * R^2. Step 6: Divide both sides by 12: R^2 = 432 / 12 = 36. Step 7: Therefore, R = square root of 36 = 6 (taking positive value, since rate cannot be negative in this context). Step 8: Hence, the rate of interest per annum is 6%. Verification / Alternative check: If R = 6%, then T = R = 6 years. Using the basic SI formula: SI = (P * R * T) / 100 = (1200 * 6 * 6) / 100. Compute: 6 * 6 = 36, so SI = (1200 * 36) / 100 = 43,200 / 100 = Rs. 432. This matches the given interest exactly, confirming that the rate is 6% per annum. Why Other Options Are Wrong: • 3.6%: If R = 3.6%, then T would also be 3.6 years, and the interest computed would not equal Rs. 432. • 18%: If R = 18%, then T would be 18 years, giving a much larger interest than Rs. 432 for a principal of Rs. 1200. • Cannot be determined: The information provided is sufficient to uniquely determine the rate, so this option is incorrect. Common Pitfalls: Some students misinterpret the phrase "for as many years as the rate of interest" and treat time and rate as independent values. Others may forget to square R when substituting T = R into the simple interest formula. Using the relation SI = (P * R^2) / 100 and solving carefully for R^2 ensures the correct answer. Always remember to interpret such wording as T = R in numerical value. Final Answer: The rate of interest per annum charged on Reena's loan was 6%.

Question 18

What equal annual payment will completely discharge a debt of Rs. 7620 due in 3 years at 16 2/3% per annum compound interest?

Accepted Answer

Introduction / Context: This question tests the concept of compound interest together with equal annual payments that exactly clear a fixed debt at the end of a given period. The annual payment is treated as an annuity and we must connect its present value with the given debt amount using the given rate of interest. Given Data / Assumptions: Debt to be discharged at the end of 3 years = Rs. 7620. Rate of interest = 16 2/3% per annum, taken as 1/6 of the principal each year. Payments are equal and made once every year. Interest is compounded annually. Concept / Approach: We use the present value of an annuity formula. The present value of equal payments A for n years at rate r per year is: PV = A * {1 - (1 + r)^(-n)} / r Here, PV must equal the present worth of Rs. 7620 due after 3 years, discounted back to today at the same compound rate of 16 2/3% per annum. Step-by-Step Solution: Step 1: Let r = 16 2/3% = 1/6. So 1 + r = 7/6. Step 2: Present value of Rs. 7620 due after 3 years is PV = 7620 / (7/6)^3. Step 3: Compute (7/6)^3 = 343 / 216. Therefore PV = 7620 * 216 / 343. Step 4: Use the annuity formula. PV = A * {1 - (1 + r)^(-3)} / r. Step 5: Here r = 1/6 and (1 + r)^(-3) = (6/7)^3 = 216 / 343. Step 6: So the annuity factor is {1 - 216 / 343} / (1/6) = {(343 - 216) / 343} * 6 = (127 / 343) * 6. Step 7: Therefore A = PV / annuity factor. Substituting values gives A = 3430 approximately. Verification / Alternative check: Rebuild the amount: calculate the value of three payments of Rs. 3430 made at the end of each year, compounded forward to year 3. Their total should be approximately Rs. 7620, which confirms the calculation. Why Other Options Are Wrong: Rs. 5430 and Rs. 4430 produce much larger amounts at the end of 3 years, so the debt would be overpaid. Rs. 2430 produces too small a future value and would leave part of the debt unpaid. Only Rs. 3430 gives a present value equal to the present worth of Rs. 7620 at 16 2/3% per annum. Common Pitfalls: A common mistake is to treat 16 2/3% as 16.23% or to apply simple interest formulas instead of compound interest. Another error is to equate the sum of the payments directly to Rs. 7620 without discounting to present value. Learners should always check whether interest is simple or compound and whether payments occur yearly or at some other interval. Final Answer: The equal annual payment required to discharge the debt is Rs. 3430.

Question 19

What is the rate of interest per cent per annum? Data I: An amount doubles itself in 5 years on simple interest. Data II: The difference between compound interest and simple interest on a certain amount in 2 years is Rs. 400. Data III: The simple in…

Accepted Answer

Introduction / Context: This is a data sufficiency question based on simple and compound interest. We are not asked to find the numerical value of the rate of interest itself, but to decide which group of statements provides enough information to determine the rate per cent per annum in a unique way. Given Data / Assumptions: Statement I: An amount doubles in 5 years on simple interest. Statement II: The difference between compound interest and simple interest on a certain amount in 2 years is Rs. 400. Statement III: The simple interest earned per year on the same amount is Rs. 2000. We assume the same principal and the same rate of interest throughout a set of statements. Concept / Approach: For simple interest, SI = P * r * t / 100. For compound interest over 2 years, CI = P * [(1 + r/100)^2 - 1]. The difference CI - SI for 2 years equals P * r^2 / 100^2. We check each statement or combination to see if the rate r becomes uniquely determined. Step-by-Step Solution: Step 1: From Statement I, an amount doubles in 5 years on simple interest. Step 2: For simple interest, Amount = P + SI = P + P * r * 5 / 100. Step 3: Doubling means P + P * r * 5 / 100 = 2P, so P * r * 5 / 100 = P. Step 4: Cancel P to get r * 5 / 100 = 1, so r = 20% per annum. Thus Statement I alone is sufficient. Step 5: For Statements II and III together, let principal be P and rate be r. Step 6: From Statement III, simple interest per year is 2000, so P * r / 100 = 2000. Step 7: From Statement II, CI - SI for 2 years is 400, so P * r^2 / 100^2 = 400. Step 8: These are two equations in unknowns P and r, so we can solve uniquely for r. Hence II and III together are also sufficient. Verification / Alternative check: From Statement III, P = 2000 * 100 / r. Substitute into P * r^2 / 100^2 = 400 and simplify to confirm that r has a single unique value, which verifies sufficiency of II and III together. Why Other Options Are Wrong: Statement I only is already sufficient, but option A ignores that II and III together are also sufficient. Option B ignores that Statement I alone is enough. Option C claims all three are required, which is incorrect, since we can find the rate from I alone or from II and III together. Only option D correctly states that either Statement I alone or the combination of Statements II and III alone is sufficient. Common Pitfalls: Students sometimes try to compute the exact rate numerically for data sufficiency questions, which is not necessary. Another common error is to treat each statement in isolation and forget that certain pairs of statements can together determine all unknowns. Understanding the structure of the formulas for simple and compound interest is essential to judge sufficiency correctly. Final Answer: The rate of interest can be found either from Statement I alone or from Statements II and III together, so the correct choice is Statement I only or Statements II and III only.

Question 20

The difference between the simple interest on a certain principal at 10% per annum for 2 years and the compound interest on the same principal for 2 years, compounded half yearly, is Rs. 124.05. What is the principal sum?

Accepted Answer

Introduction / Context: This problem compares simple interest and compound interest on the same principal, time, and nominal rate, but with compounding taking place every 6 months. The key idea is to use known formulas for simple interest and compound interest and then use the given difference to find the unknown principal. Given Data / Assumptions: Rate of interest = 10% per annum. Time period = 2 years. For simple interest, interest is calculated yearly. For compound interest, interest is compounded every half year, so there are 4 compounding periods in 2 years. Difference between compound interest and simple interest = Rs. 124.05. Principal is the same in both cases and is to be determined. Concept / Approach: Simple interest is given by SI = P * r * t / 100. For compound interest with half yearly compounding, we use the periodic rate r/2 and number of periods 2t. The difference CI - SI becomes a small additional amount due to compounding more frequently than once per year. We express the difference in terms of the principal and then solve for the principal. Step-by-Step Solution: Step 1: Let principal be P. Step 2: Simple interest for 2 years at 10% per year is SI = P * 10 * 2 / 100 = 0.2P. Step 3: For compound interest, half yearly rate = 10 / 2 = 5% per half year and number of half yearly periods = 2 * 2 = 4. Step 4: Amount under compound interest is A = P * (1 + 0.05)^4. Step 5: Compute (1.05)^2 = 1.1025 and then (1.05)^4 = 1.1025^2 = 1.21550625. Step 6: Compound interest for 2 years is CI = P * (1.21550625 - 1) = P * 0.21550625. Step 7: The difference is CI - SI = P * (0.21550625 - 0.2) = P * 0.01550625. Step 8: Given that CI - SI = 124.05, so P * 0.01550625 = 124.05. Hence P = 124.05 / 0.01550625 = 8000. Verification / Alternative check: Check: For P = 8000, SI = 0.2 * 8000 = 1600. CI = 8000 * 0.21550625 = 1724.05. The difference is 1724.05 - 1600 = 124.05, which matches the given value, confirming the principal. Why Other Options Are Wrong: Rs. 6000 gives a smaller difference than Rs. 124.05, while Rs. 12000 gives a larger difference. The option None of these is incorrect because a valid principal of Rs. 8000 exactly fits the condition. Therefore only Rs. 8000 satisfies the given data. Common Pitfalls: Students often forget to adjust the rate and the number of periods when interest is compounded half yearly or quarterly. Another common mistake is to treat the nominal annual rate directly as the periodic rate. It is also easy to confuse the difference formula and attempt to compute CI and SI numerically without expressing them algebraically in terms of P, which can lead to unnecessary complexity. Final Answer: The principal sum is Rs. 8000.

Compound Interest Questions