Question 1

If the simple interest on a sum of money at 5% per annum for 3 years is Rs. 1200, what is the compound interest on the same sum for the same period at the same rate of interest?

Accepted Answer

Introduction / Context: This question directly compares simple interest and compound interest for the same principal, rate, and time. We know the simple interest and must first find the principal. Then we use the compound interest formula to find the required interest amount for the same period and rate. Given Data / Assumptions: Simple interest for 3 years = Rs. 1200. Rate of interest = 5% per annum. Time period = 3 years. Interest is compounded annually for the compound interest calculation. Concept / Approach: Simple interest is given by SI = P * r * t / 100, which we use to find the principal P. Once P is known, compound interest is computed using the formula A = P * (1 + r / 100)^t, and CI = A - P. The interest rate and time remain unchanged between the two calculations. Step-by-Step Solution: Step 1: Let the principal be P. Simple interest is SI = P * 5 * 3 / 100. Step 2: So SI = 15P / 100 = 0.15P. Step 3: We are given SI = Rs. 1200. Therefore 0.15P = 1200. Step 4: Solve for P: P = 1200 / 0.15 = 8000. Thus the principal is Rs. 8000. Step 5: For compound interest, amount after 3 years is A = P * (1 + 5 / 100)^3 = 8000 * (1.05)^3. Step 6: Compute (1.05)^2 = 1.1025 and multiply by 1.05 again to get (1.05)^3 = 1.157625. Step 7: Hence A = 8000 * 1.157625 = 9261. Step 8: Compound interest CI = A - P = 9261 - 8000 = 1261. Verification / Alternative check: Check that the simple interest on Rs. 8000 at 5% for 3 years is 8000 * 5 * 3 / 100 = 1200, which matches the given value. This confirms that the principal is correct and that the subsequent compound interest calculation is based on the correct principal amount. Why Other Options Are Wrong: Rs. 1271, Rs. 1281, and Rs. 1291 are close to the correct value and may attract those who make rounding errors or miscalculate (1.05)^3. However, the exact computation gives a compound interest of Rs. 1261, so all other stated options are incorrect. Common Pitfalls: One common error is to confuse the simple interest amount with the compound interest and assume they are equal for small rates or periods. Another mistake is to round the value of (1.05)^3 too early, which can shift the answer. It is important to keep enough decimal accuracy or use exact fractional calculations during the intermediate steps. Final Answer: The compound interest is Rs. 1261.

Question 2

A certain sum amounts to Rs. 7350 in 2 years and to Rs. 8575 in 3 years at compound interest. Find the original sum invested.

Accepted Answer

Introduction / Context: This question uses the fact that under compound interest, the amount after each equal time interval is multiplied by the same growth factor. We are given successive amounts at the end of 2 and 3 years and must work backwards to find the initial principal using the ratio of these two amounts. Given Data / Assumptions: Amount after 2 years, A2 = Rs. 7350. Amount after 3 years, A3 = Rs. 8575. Interest is compounded annually at a constant rate. We must find the original principal P. Concept / Approach: In compound interest, A = P * (1 + r / 100)^n. Therefore A3 / A2 = (1 + r / 100)^(3) / (1 + r / 100)^(2) = 1 + r / 100. This ratio lets us find the yearly growth factor. Once we know 1 + r / 100, we can express A2 in terms of P and solve for the principal. Step-by-Step Solution: Step 1: Compute the ratio A3 / A2 = 8575 / 7350. Step 2: Simplify the fraction: divide numerator and denominator by 175 to get 8575 / 7350 = 49 / 42 = 7 / 6. Step 3: Thus 1 + r / 100 = 7 / 6, so r / 100 = 7 / 6 - 1 = 1 / 6. Step 4: Therefore r = 100 / 6 = 16 2/3% per annum. Step 5: Now, A2 = P * (1 + r / 100)^2 = P * (7 / 6)^2 = P * 49 / 36. Step 6: We know A2 = 7350, so 7350 = P * 49 / 36. Step 7: Hence P = 7350 * 36 / 49. Step 8: Compute P = (7350 / 49) * 36 = 150 * 36 = 5400. Verification / Alternative check: Check: For P = 5400 and r = 16 2/3%, A2 = 5400 * (7 / 6)^2 = 5400 * 49 / 36 = 7350 and A3 = 5400 * (7 / 6)^3 = 5400 * 343 / 216 = 8575. Both match the given amounts exactly, confirming that the principal is correct. Why Other Options Are Wrong: Rs. 3400, Rs. 4400, and Rs. 6400 do not produce the given pair of amounts when grown at a single constant compound interest rate. They may give one of the amounts approximately but will fail for the other, so they cannot be the correct principal. Only Rs. 5400 satisfies both conditions exactly. Common Pitfalls: Learners sometimes try to guess the rate first by random trial, which wastes time. Another common error is to assume simple interest when the question clearly states compound interest. Recognizing that the ratio of successive amounts directly gives the growth factor is a key idea for solving such problems efficiently. Final Answer: The original sum invested is Rs. 5400.

Question 3

Find the compound interest on Rs. 16,000 at 20% per annum for 9 months, when interest is compounded quarterly.

Accepted Answer

Introduction / Context: This question involves compound interest over a fractional year with quarterly compounding. We must correctly identify the periodic rate and the number of compounding periods for 9 months, then apply the standard compound interest formula to find the interest amount. Given Data / Assumptions: Principal P = Rs. 16,000. Annual rate of interest r = 20% per annum. Time period = 9 months. Interest is compounded quarterly, that is, every 3 months. Concept / Approach: When interest is compounded quarterly, the annual rate is divided by 4 and the time in years is converted into the equivalent number of quarters. For each quarter, the amount is multiplied by (1 + r / 4 / 100). The general formula is A = P * (1 + r / (4 * 100))^(4t). Once we have the total amount A for 9 months, the compound interest is simply A minus the principal. Step-by-Step Solution: Step 1: Annual rate r = 20%, so quarterly rate = 20 / 4 = 5% per quarter. Step 2: Time = 9 months. Each quarter is 3 months, so number of quarters n = 9 / 3 = 3. Step 3: Amount A after 3 quarters is A = 16000 * (1 + 5 / 100)^3. Step 4: Compute 1 + 5 / 100 = 1.05. Step 5: Compute (1.05)^2 = 1.1025 and then (1.05)^3 = 1.1025 * 1.05 = 1.157625. Step 6: Therefore A = 16000 * 1.157625 = 18522. Step 7: Compound interest CI = A - P = 18522 - 16000 = 2522. Verification / Alternative check: An alternative is to compute the interest accumulated each quarter: first interest is 5% of 16000, second interest is 5% of the new amount after first quarter, and so on. Performing those three successive steps also leads to the same final amount of Rs. 18522 and interest of Rs. 2522, confirming the result. Why Other Options Are Wrong: Rs. 2422, Rs. 2622, and Rs. 2722 correspond to slightly different approximations or to using simple interest or an incorrect number of compounding periods. Only Rs. 2522 matches the precise calculation for quarterly compounding over 9 months at 20% per annum. Common Pitfalls: A very common mistake is to treat 9 months as 0.75 years and use annual compounding directly instead of finding the number of quarters. Another error is to forget that the rate per period must be 20% divided by 4, not 20% itself. Careful handling of time and rate conversions is crucial in compound interest problems with non annual compounding. Final Answer: The compound interest is Rs. 2522.

Question 4

A sum is lent at 20% per annum compound interest. What is the ratio of the increase in the amount during the 4th year to the increase in the amount during the 5th year?

Accepted Answer

Introduction / Context: This problem focuses on how interest grows under compound interest. Instead of asking for a specific amount, it compares the interest earned in the 4th year with the interest earned in the 5th year, both at the same compound interest rate. Understanding how yearly interest forms a geometric progression is the key idea. Given Data / Assumptions: The sum is invested at 20% per annum compound interest. We compare the increase in the amount in the 4th year with that in the 5th year. The principal is not specified, but for ratios the exact value of the principal is not needed. Concept / Approach: Under compound interest at rate r, the amount after n years is A_n = P * (1 + r)^n (where r is expressed as a decimal). The interest earned in year n is A_n - A_(n-1). These yearly interest amounts form a geometric progression with common ratio 1 + r. Therefore, the ratio of interest in consecutive years depends only on 1 + r, not on the principal itself. Step-by-Step Solution: Step 1: Let r = 20% = 0.20, so 1 + r = 1.20. Step 2: Amount at the end of year 3 is A3 = P * (1.20)^3. Step 3: Amount at the end of year 4 is A4 = P * (1.20)^4. Step 4: Interest in the 4th year is I4 = A4 - A3 = P * (1.20)^4 - P * (1.20)^3 = P * (1.20)^3 * (1.20 - 1) = P * (1.20)^3 * 0.20. Step 5: Similarly, amount at the end of year 5 is A5 = P * (1.20)^5. Step 6: Interest in the 5th year is I5 = A5 - A4 = P * (1.20)^5 - P * (1.20)^4 = P * (1.20)^4 * 0.20. Step 7: Therefore I4 : I5 = P * (1.20)^3 * 0.20 : P * (1.20)^4 * 0.20. Step 8: Cancel common factors P and 0.20 to get I4 : I5 = (1.20)^3 : (1.20)^4 = 1 : 1.20. Step 9: The ratio 1 : 1.20 is equal to 5 : 6 when expressed in whole numbers. Verification / Alternative check: Assume a convenient principal, for example P = 100. Compute interest year by year and compare the 4th and 5th year interest values. You will see that the 5th year interest is 1.20 times the 4th year interest, confirming the ratio 5 : 6. Why Other Options Are Wrong: Ratios 4 : 5 and 5 : 4 do not reflect the multiplicative factor 1.20 between consecutive years. The option Cannot be determined is incorrect, because the ratio depends only on the rate and not on the actual principal. Once the rate is known, the ratio can be obtained exactly. Common Pitfalls: One pitfall is to think that interest is the same every year under compound interest, which is false. Another mistake is to try to compute the exact interest values with an arbitrary principal but then mis-handle rounding. Recognizing the geometric progression nature of yearly interest simplifies the problem greatly. Final Answer: The ratio of the increase in amount in the 4th year to that in the 5th year is 5 : 6.

Question 5

The compound interest on a sum of money for 2 years is Rs. 832 and the simple interest on the same sum for the same period is Rs. 800. What is the difference between compound interest and simple interest on the same sum for 3 years at the same rate …

Accepted Answer

Introduction / Context: This problem links simple interest and compound interest for the same principal, time, and rate, and then extends the result to a different time period. We are given the compound and simple interest for 2 years and must use that information to find both the rate and the principal. Then we can calculate the new difference between compound and simple interest for 3 years at the same rate. Given Data / Assumptions: Compound interest for 2 years, CI2 = Rs. 832. Simple interest for 2 years, SI2 = Rs. 800. Time period in both cases = 2 years. Rate and principal are the same for all comparisons. We must find CI3 - SI3 for 3 years. Concept / Approach: For 2 years, SI2 = P * r * 2 / 100 and CI2 = P * [(1 + r / 100)^2 - 1]. Their difference CI2 - SI2 has a compact form: CI2 - SI2 = P * r^2 / 100^2. From the two given interest values we can first find r and P. Then we compute SI3 and CI3 for 3 years and finally find CI3 - SI3. Step-by-Step Solution: Step 1: Let the principal be P and the annual rate be r%. Step 2: Simple interest for 2 years is SI2 = P * r * 2 / 100 = 800. Step 3: Compound interest for 2 years is CI2 = P * [(1 + r / 100)^2 - 1] = 832. Step 4: The difference for 2 years is CI2 - SI2 = 832 - 800 = 32. Step 5: For 2 years, CI2 - SI2 = P * r^2 / 100^2, so P * r^2 / 10000 = 32. Step 6: Also SI2 = 800 = P * r * 2 / 100, so P * r = 800 * 100 / 2 = 40000. Step 7: From P * r^2 = 32 * 10000 = 320000 and P * r = 40000, divide the first by the second to get r = 320000 / 40000 = 8% per annum. Step 8: Now find P from P * r = 40000: P = 40000 / 8 = Rs. 5000. Step 9: For 3 years, simple interest is SI3 = P * r * 3 / 100 = 5000 * 8 * 3 / 100 = Rs. 1200. Step 10: For compound interest over 3 years, A3 = P * (1 + r / 100)^3 = 5000 * (1.08)^3. Step 11: Compute (1.08)^2 = 1.1664 and (1.08)^3 = 1.1664 * 1.08 = 1.259712. Step 12: Therefore A3 = 5000 * 1.259712 = 6298.56 and CI3 = A3 - P = 6298.56 - 5000 = Rs. 1298.56. Step 13: The required difference for 3 years is CI3 - SI3 = 1298.56 - 1200 = Rs. 98.56. Verification / Alternative check: You can check that the 2 year values are consistent: SI2 = 2 * 1200 / 3 = 800 and CI2 can be recalculated for P = 5000 and r = 8% to verify that it equals 832. This confirms that the base parameters are correct and validates the 3 year calculations. Why Other Options Are Wrong: Rs. 48.00 and Rs. 66.56 both underestimate the additional compound interest over 3 years, while a significantly larger value than Rs. 98.56 would not match the precise calculation. The option None of these is invalid because one of the given numerical options matches the exact difference. Only Rs. 98.56 satisfies the detailed computation. Common Pitfalls: Many learners forget the compact form for CI2 - SI2 and instead try to calculate compound and simple interest separately without algebraic simplification, which can lead to errors. Another frequent mistake is to apply the 2 year difference directly to 3 years by simple proportion, which is not valid because compound interest grows non linearly with time. Final Answer: The difference between compound interest and simple interest for 3 years is Rs. 98.56.

Question 6

In what time will Rs. 1000 amount to Rs. 1331 at 10% per annum compounded annually?

Accepted Answer

Introduction / Context: This is a direct application of the compound interest formula to find the time needed for a principal to grow to a given amount at a known annual rate with annual compounding. The numbers are chosen to produce an exact power of a simple factor. Given Data / Assumptions: Principal P = Rs. 1000. Amount A = Rs. 1331. Rate of interest r = 10% per annum. Interest is compounded annually. Time in years t is to be determined. Concept / Approach: The compound interest formula for annual compounding is A = P * (1 + r / 100)^t. We substitute the known values and solve for the exponent t. Because the numbers are clean, we expect (1 + r / 100)^t to match a simple power such as (1.1)^t. Step-by-Step Solution: Step 1: Write the formula A = P * (1 + r / 100)^t. Step 2: Substitute P = 1000, A = 1331, r = 10%. Then 1331 = 1000 * (1 + 10 / 100)^t. Step 3: Simplify 1 + 10 / 100 to get 1.10. Step 4: The equation becomes 1331 = 1000 * (1.10)^t. Step 5: Divide both sides by 1000 to obtain 1331 / 1000 = (1.10)^t. Step 6: Recognize that 1331 / 1000 = 1.331. Step 7: Observe that (1.10)^3 = 1.331, so t = 3 years. Verification / Alternative check: Calculate stepwise: After 1 year amount is 1000 * 1.10 = 1100. After 2 years amount is 1100 * 1.10 = 1210. After 3 years amount is 1210 * 1.10 = 1331. This confirms that the amount of Rs. 1331 is reached at the end of 3 years. Why Other Options Are Wrong: After 1 year the amount is only Rs. 1100, not 1331. After 2 years it is Rs. 1210. Even after 4 years the amount would exceed Rs. 1331. Therefore, only 3 years match the required amount exactly, and the other time choices are not correct. Common Pitfalls: Some learners mistakenly apply simple interest, solving 1000 + 1000 * 10 * t / 100 = 1331, which gives a non integer value and does not match the compound interest setting. Others may try to interpolate between years rather than checking the powers of 1.10 directly. Remember that compound growth is exponential, so exact powers often appear in exam style problems. Final Answer: The required time is 3 years.

Question 7

There is a 60% increase in an amount in 6 years at simple interest. What will be the compound interest on Rs. 12,000 after 3 years at the same rate of interest?

Accepted Answer

Introduction / Context: This question first uses simple interest information to determine the annual rate, and then applies that rate to a different principal under compound interest. It tests the ability to move between simple and compound interest while keeping the rate consistent. Given Data / Assumptions: An unspecified amount increases by 60% in 6 years under simple interest. The same rate of interest is then used under compound interest for a different principal. Principal for the compound interest part is Rs. 12,000. Time for the compound interest part is 3 years. Interest is compounded annually for the second part. Concept / Approach: Under simple interest, a 60% increase over 6 years means total simple interest of 60% of the principal for 6 years. From this, we can find the rate of interest per year. We then apply the compound interest formula A = P * (1 + r / 100)^t for the new principal and time and subtract the principal to get the compound interest. Step-by-Step Solution: Step 1: Under simple interest, a 60% increase over 6 years implies SI = 60% of principal in 6 years. Step 2: Simple interest per year as a percentage is 60% / 6 = 10% per annum. Step 3: Therefore the rate r = 10% per annum. Step 4: Now consider the new principal P = 12000 and time t = 3 years under compound interest at r = 10% per annum. Step 5: Amount A = 12000 * (1 + 10 / 100)^3 = 12000 * (1.10)^3. Step 6: Compute (1.10)^2 = 1.21 and (1.10)^3 = 1.331. Step 7: Hence A = 12000 * 1.331 = 15972. Step 8: Compound interest CI = A - P = 15972 - 12000 = 3972. Verification / Alternative check: A quick check is to compute simple interest on Rs. 12000 at 10% for 3 years, which would be 12000 * 10 * 3 / 100 = 3600. Because compound interest must exceed simple interest for the same rate and period, the answer must be slightly higher than 3600. The value 3972 matches this expectation. Why Other Options Are Wrong: Rs. 2972 is too small and even below the corresponding simple interest. Rs. 4972 and Rs. 5972 are too large for 3 years at 10% and do not result from consistent compounding. Only Rs. 3972 is a correct and reasonable compound interest figure at this rate and period. Common Pitfalls: Students sometimes mistakenly assume that a 60% increase in 6 years under simple interest indicates a compound rate directly, which is not correct. Another error is to misinterpret the 60% as a yearly rate instead of an accumulated rate over 6 years. Careful reading of the wording about increase in amount is essential. Final Answer: The compound interest on Rs. 12,000 after 3 years at the same rate is Rs. 3972.

Question 8

On a certain sum of money, the simple interest for 2 years is Rs. 660, while the compound interest for the same period and at the same annual rate is Rs. 696.30. What is the rate of interest per annum?

Accepted Answer

Introduction / Context: This problem compares simple and compound interest for the same principal, time, and rate, over a 2 year period. The difference between the two interest amounts helps us determine the rate of interest per annum, without directly knowing the principal. Given Data / Assumptions: Simple interest for 2 years, SI2 = Rs. 660. Compound interest for 2 years, CI2 = Rs. 696.30. Time period = 2 years. The principal P and rate r% per annum are the same for both cases. Interest in the compound case is compounded annually. Concept / Approach: For 2 years, simple interest is SI2 = P * r * 2 / 100. Compound interest over 2 years is CI2 = P * [(1 + r / 100)^2 - 1]. The difference D = CI2 - SI2 has a useful simplified form: D = P * r^2 / 100^2. With SI2 and CI2 known, we can first find D, then express P * r and P * r^2 and finally solve for r using algebra. Step-by-Step Solution: Step 1: Compute the difference between compound and simple interest for 2 years: D = CI2 - SI2 = 696.30 - 660 = 36.30. Step 2: Let principal be P and rate be r% per annum. Step 3: Simple interest for 2 years is SI2 = P * r * 2 / 100 = 660. Step 4: So P * r = 660 * 100 / 2 = 33000. Step 5: For 2 years, difference D = P * r^2 / 100^2 = 36.30. Hence P * r^2 = 36.30 * 10000 = 363000. Step 6: Divide P * r^2 by P * r to get r. That is, r = (P * r^2) / (P * r) = 363000 / 33000. Step 7: Simplify r = 363000 / 33000 = 11. Step 8: Therefore the rate of interest is 11% per annum. Verification / Alternative check: You can verify this by choosing P = 3000 (for example) and checking. Simple interest for 2 years at 11% is 3000 * 11 * 2 / 100 = 660. Compound interest can be computed as A = 3000 * (1.11)^2, CI2 = A - 3000, and you will obtain approximately Rs. 696.30, which confirms that r = 11% is correct. Why Other Options Are Wrong: If r = 10% or 12% or 13%, the difference CI2 - SI2 would not equal Rs. 36.30. Each of those rates leads to a different algebraic relationship between P * r and P * r^2. Only r = 11% satisfies both the simple and compound interest conditions at the same time. Common Pitfalls: A common error is to try to guess the rate by trial instead of using the compact formula for the difference between compound and simple interest. Another mistake is to miscalculate the difference or to forget to multiply by 10000 when using the P * r^2 relation, which leads to an incorrect rate. Final Answer: The rate of interest is 11% per annum.

Question 9

The difference between simple interest and compound interest on a certain sum of money at 20% per annum for 2 years is Rs. 56. What is the sum?

Accepted Answer

Introduction / Context: This is a classic question that uses the standard formula for the difference between compound interest and simple interest for 2 years at the same rate and on the same principal. The small difference is used to back calculate the principal. Given Data / Assumptions: Rate of interest r = 20% per annum. Time period t = 2 years. Difference between compound interest and simple interest = Rs. 56. Interest in the compound case is compounded annually. Principal P is the same in both cases and is to be found. Concept / Approach: For 2 years, simple interest is SI2 = P * r * 2 / 100. Compound interest is CI2 = P * [(1 + r / 100)^2 - 1]. The difference D = CI2 - SI2 simplifies to D = P * r^2 / 100^2. Once D and r are known, we can solve this simple equation for P directly. Step-by-Step Solution: Step 1: Note that r = 20%, t = 2 years, and D = 56. Step 2: Use the formula for the difference for 2 years: D = P * r^2 / 100^2. Step 3: Substitute r = 20 to get D = P * 20^2 / 100^2 = P * 400 / 10000. Step 4: Simplify: D = P * 0.04. Step 5: Given D = 56, so 0.04P = 56. Step 6: Solve for P: P = 56 / 0.04 = 1400. Verification / Alternative check: Check by direct calculation. Simple interest for 2 years on Rs. 1400 at 20% is SI2 = 1400 * 20 * 2 / 100 = 560. Compound amount after 2 years is A = 1400 * (1.20)^2 = 1400 * 1.44 = 2016, giving CI2 = 2016 - 1400 = 616. The difference CI2 - SI2 = 616 - 560 = 56, which matches the given value. Why Other Options Are Wrong: The other options produce very different differences. Substituting Rs. 3680, Rs. 2650, or Rs. 1170 into the difference formula does not yield 56. Only the principal of Rs. 1400 is consistent with the given difference at 20% per annum for 2 years. Common Pitfalls: Some students forget the compact difference formula and attempt long calculations of compound and simple interest separately, increasing the chance of arithmetic errors. Another common mistake is to misinterpret the given difference as a percentage rather than a monetary amount. Final Answer: The sum is Rs. 1400.

Question 10

The difference between simple interest and compound interest on Rs. 1200 for one year at 10% per annum, when the compound interest is reckoned half yearly, is equal to how much?

Accepted Answer

Introduction / Context: Here we compare simple interest and compound interest for a short period of one year, but the compound interest is calculated half yearly. Even for a single year, half yearly compounding slightly increases the interest, and we must compute exactly how much more it gives compared with simple interest. Given Data / Assumptions: Principal P = Rs. 1200. Annual rate r = 10% per annum. Time t = 1 year. Compound interest is reckoned half yearly, so two compounding periods in one year. We must find CI - SI for this one year period. Concept / Approach: Simple interest for one year is straightforward: SI = P * r * t / 100. For compound interest with half yearly compounding, we use the periodic rate r / 2 and number of periods 2t. Amount under compound interest is A = P * (1 + r / 2 / 100)^(2t), and CI = A - P. The difference CI - SI is then taken directly. Step-by-Step Solution: Step 1: Simple interest for one year is SI = 1200 * 10 * 1 / 100 = 120. Step 2: For half yearly compounding, periodic rate = 10 / 2 = 5% per half year. Step 3: Number of half yearly periods in one year = 2. Step 4: Amount under compound interest is A = 1200 * (1 + 5 / 100)^2 = 1200 * (1.05)^2. Step 5: Compute (1.05)^2 = 1.1025. Step 6: Thus A = 1200 * 1.1025 = 1323. Step 7: Compound interest CI = A - P = 1323 - 1200 = 123. Step 8: The required difference is CI - SI = 123 - 120 = Rs. 3.00. Verification / Alternative check: You can also compute the interest stepwise: after six months, interest is 5% of 1200 = 60, giving an amount of 1260. After the next six months, interest is 5% of 1260 = 63, so the total interest over the year is 60 + 63 = 123. Subtracting the simple interest of 120 again gives the difference of Rs. 3. Why Other Options Are Wrong: Rs. 2.50, Rs. 4.00, and Rs. 4.50 are typical distractors arising from arithmetic slips such as using the wrong periodic rate or miscalculating the second half yearly interest. Only Rs. 3.00 matches both the direct and stepwise calculations. Common Pitfalls: The main error is to forget that the second half yearly interest is calculated on the increased amount, not on the original principal. Another mistake is to apply the 10% rate twice instead of 5% each half year, which incorrectly doubles the annual rate. Final Answer: The difference between compound interest and simple interest is Rs. 3.00.

Question 11

Find the compound interest on Rs. 7500 at 4% per annum for 2 years, when interest is compounded annually.

Accepted Answer

Introduction / Context: This is a straightforward compound interest calculation for a modest rate and time span. It requires application of the standard compound interest formula for annual compounding and careful handling of the arithmetic. Given Data / Assumptions: Principal P = Rs. 7500. Rate of interest r = 4% per annum. Time period t = 2 years. Interest is compounded annually. Concept / Approach: With annual compounding, the amount after t years is A = P * (1 + r / 100)^t. The compound interest CI is given by CI = A - P. Here we substitute P = 7500, r = 4, t = 2, compute the amount, and subtract the principal to get the interest. Step-by-Step Solution: Step 1: Write the formula A = P * (1 + r / 100)^t. Step 2: Substitute the values P = 7500, r = 4, t = 2 to get A = 7500 * (1.04)^2. Step 3: Compute (1.04)^2 = 1.04 * 1.04 = 1.0816. Step 4: Therefore A = 7500 * 1.0816 = 8112. Step 5: Compound interest CI = A - P = 8112 - 7500 = 612. Verification / Alternative check: Break the process into yearly steps. After 1 year, amount is 7500 * 1.04 = 7800, so interest for the first year is 300. After the second year, interest is 4% of 7800 = 312, giving a new amount of 7800 + 312 = 8112. Total interest over 2 years is 300 + 312 = 612, which matches the previous result. Why Other Options Are Wrong: Rs. 712, Rs. 812, and Rs. 912 result from overestimating the interest or from incorrect multiplication of the interest factor. Only Rs. 612 is consistent with a 4% compound rate over 2 years. Common Pitfalls: A common mistake is to compute simple interest, namely 7500 * 4 * 2 / 100 = 600, and ignore the small extra interest earned on the first year interest. Others may round the intermediate product too early, slightly distorting the result. Final Answer: The compound interest is Rs. 612.

Question 12

The population of a town was 3600 three years ago. It is 4800 at present. Assuming that the rate of growth has been constant and compounding annually, what will be the population three years from now?

Accepted Answer

Introduction / Context: This question treats population growth in terms of compound interest. The population three years ago and now are given, and we are told that the growth rate has been constant and compounding annually. Using this, we can find the yearly growth factor and then apply it to project the population three years into the future. Given Data / Assumptions: Population three years ago, P0 = 3600. Current population, P3 = 4800. Growth rate is constant with annual compounding. We must find population after another 3 years, that is, six years from the starting point. Concept / Approach: If the annual growth factor is k = 1 + r, then population after n years is P_n = P_0 * k^n. We are given P3 and P0, so P3 / P0 = k^3. From this we find k. Then P6 = P0 * k^6. However, since P6 = P3 * k^3, and k^3 is already known as P3 / P0, we can compute P6 directly as P6 = P3 * (P3 / P0). Step-by-Step Solution: Step 1: Let k be the annual growth factor. Then P3 = P0 * k^3. Step 2: Substitute P3 = 4800 and P0 = 3600 to get 4800 = 3600 * k^3. Step 3: Therefore k^3 = 4800 / 3600 = 4 / 3. Step 4: We do not need k itself. We need P6, which is P0 * k^6. Step 5: Since k^6 = (k^3)^2, we have k^6 = (4 / 3)^2 = 16 / 9. Step 6: Compute P6 = P0 * k^6 = 3600 * 16 / 9. Step 7: Simplify: 3600 / 9 = 400, so P6 = 400 * 16 = 6400. Step 8: Therefore the population three years from now will be 6400. Verification / Alternative check: We can also think of it as P6 = P3 * k^3. Since k^3 = 4 / 3 and P3 = 4800, P6 = 4800 * 4 / 3 = 4800 * 4 / 3 = 1600 * 4 = 6400. This matches the previous result. Why Other Options Are Wrong: The values 6000, 6500, and 6600 may seem reasonable but do not correspond to a constant compound growth that takes the town from 3600 to 4800 and then forward another 3 years. Only 6400 satisfies the compound growth conditions exactly. Common Pitfalls: Students sometimes treat the growth as linear and assume a fixed increase of 1200 every three years, which would give incorrect answers. Another error is to attempt to find the rate r explicitly, even though it is not necessary to compute k^3 and k^6 for this type of ratio problem. Final Answer: The population three years from now will be 6400.

Question 13

Divide Rs. 1301 between A and B so that the amount of A after 7 years is equal to the amount of B after 9 years, the rate of interest being 4% per annum compounded annually. What is B's share?

Accepted Answer

Introduction / Context: This question involves distributing a given sum between two people such that their respective amounts grow under compound interest and become equal at different future times. The rate is the same for both, but the compounding periods differ. We must use the compound interest formula on both parts and then solve for the share belonging to one of them. Given Data / Assumptions: Total sum to be divided = Rs. 1301. A receives Rs. x, B receives Rs. 1301 - x initially. Rate of interest r = 4% per annum compounded annually. Time for A = 7 years, time for B = 9 years. The amounts of A and B after their respective periods are equal. We must find B's share. Concept / Approach: Let x be the share of A. A's amount after 7 years is x * (1.04)^7. B's share is 1301 - x, and B's amount after 9 years is (1301 - x) * (1.04)^9. The condition that these amounts are equal gives an equation in x. We solve this equation to find x and then compute B's share as 1301 - x. Step-by-Step Solution: Step 1: Let A's share be x. Then B's share is 1301 - x. Step 2: Amount A receives after 7 years is A7 = x * (1.04)^7. Step 3: Amount B receives after 9 years is B9 = (1301 - x) * (1.04)^9. Step 4: Given that A7 = B9, we have x * (1.04)^7 = (1301 - x) * (1.04)^9. Step 5: Divide both sides by (1.04)^7 to obtain x = (1301 - x) * (1.04)^2. Step 6: Compute (1.04)^2 = 1.0816. Step 7: The equation becomes x = (1301 - x) * 1.0816. Step 8: Expand: x = 1301 * 1.0816 - 1.0816x. Step 9: Bring like terms together: x + 1.0816x = 1301 * 1.0816, so 2.0816x = 1301 * 1.0816. Step 10: Solve for x: x = (1301 * 1.0816) / 2.0816 = 676. Step 11: Therefore A's share is Rs. 676, and B's share is 1301 - 676 = Rs. 625. Verification / Alternative check: Check the equality of future amounts. A7 = 676 * (1.04)^7 and B9 = 625 * (1.04)^9. Since B9 = 625 * (1.04)^9 = 625 * (1.04)^2 * (1.04)^7 = 625 * 1.0816 * (1.04)^7 = 676 * (1.04)^7, both amounts are equal, confirming that the division is correct. Why Other Options Are Wrong: Values such as Rs. 626 and Rs. 627 would not yield an exact equality of the future amounts when subjected to 7 and 9 years of compounding at 4%. Rs. 286 is far from the correct value and clearly cannot satisfy the equality condition. Only Rs. 625 produces the required equality. Common Pitfalls: A frequent mistake is to equate the initial shares instead of the future amounts, or to forget to apply different exponents for 7 and 9 years. Another pitfall is approximating the growth factor too roughly, which can obscure the neat result that A's share becomes a simple integer. Final Answer: B's share is Rs. 625.

Question 14

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

Accepted Answer

Introduction / Context: This question combines compound interest with tax deducted on interest each year. The tax reduces the effective rate of growth of the investment. We must compute the net amount the investor has at the end of 3 years after accounting for both compounding and yearly tax deductions. Given Data / Assumptions: Principal P = Rs. 5000. Nominal rate of interest r = 5% per annum. Time t = 3 years. Interest is compounded yearly. Income tax is charged at 20% on the interest earned each year. Tax is deducted at the end of each year before the next year's interest is calculated. Concept / Approach: Each year, gross interest is P_current * 5%. However, only 80% of this interest is actually credited to the account because 20% is taken away as tax. Thus the effective interest rate credited to the investor is 5% * 80% = 4% per year. We can therefore treat the situation as if the principal earns compound interest at an effective rate of 4% per annum for 3 years. Step-by-Step Solution: Step 1: Compute the effective rate after tax. Gross rate is 5% per year, tax is 20% of interest, so net fraction kept is 1 - 0.20 = 0.80. Step 2: Net effective rate per year = 5% * 0.80 = 4%. Step 3: The investment therefore grows at 4% per annum compounded annually. Step 4: Amount after 3 years is A = 5000 * (1 + 4 / 100)^3 = 5000 * (1.04)^3. Step 5: Compute (1.04)^2 = 1.0816. Step 6: Compute (1.04)^3 = 1.0816 * 1.04 = 1.124864. Step 7: Therefore A = 5000 * 1.124864 = 5624.32. Verification / Alternative check: You can also calculate year by year. First year: interest is 5% of 5000 = 250, tax is 20% of 250 = 50, so net interest credited is 200 and new amount becomes 5200. Second year: interest is 5% of 5200 = 260, tax 52, net interest 208, new amount 5408. Third year: interest is 5% of 5408 = 270.40, tax 54.08, net interest 216.32, final amount 5408 + 216.32 = 5624.32. This matches the earlier result. Why Other Options Are Wrong: The other figures do not correspond to a consistent 5% gross rate with 20% tax on the interest. They either assume the full 5% interest without tax or use an incorrect effective rate. Only Rs. 5624.32 matches the detailed year by year calculation. Common Pitfalls: A typical error is to calculate compound interest at 5% for 3 years and then subtract tax on the total interest, instead of taxing each year's interest separately. Another pitfall is to confuse the tax rate on interest with a tax rate on the principal. Final Answer: The amount at the end of the third year is Rs. 5624.32.

Question 15

The principal that amounts to Rs. 4913 in 3 years at 6 1/4% per annum compound interest, compounded annually, is equal to how much?

Accepted Answer

Introduction / Context: In this question, we are given the final amount after 3 years under compound interest and must find the original principal. The rate is 6 1/4% per annum, which has a convenient fractional form. Recognizing standard numeric patterns makes this question faster to solve. Given Data / Assumptions: Amount after 3 years, A = Rs. 4913. Rate of interest r = 6 1/4% per annum, that is, 6.25%. Time t = 3 years. Interest is compounded annually. Principal P is to be found. Concept / Approach: We use the compound interest formula A = P * (1 + r / 100)^t and rearrange it to P = A / (1 + r / 100)^t. The rate 6.25% can be written as the fraction 1/16, so 1 + r / 100 becomes 17 / 16. Thus A and P are related by a simple rational power, making the arithmetic easier. Step-by-Step Solution: Step 1: Express the rate as a fraction. Since 6 1/4% = 6.25% = 6.25 / 100, we know that 6.25% corresponds to 1 / 16 of the principal. Step 2: Therefore 1 + r / 100 = 1 + 1 / 16 = 17 / 16. Step 3: Using A = P * (17 / 16)^3, we have 4913 = P * (17 / 16)^3. Step 4: Note that 17^3 = 4913 and 16^3 = 4096. Step 5: So (17 / 16)^3 = 4913 / 4096. Step 6: The equation becomes 4913 = P * (4913 / 4096). Step 7: Simplify to get P = 4096. Verification / Alternative check: Check by forward calculation. Starting with P = 4096 at 6.25% per annum, the amount after 1 year is 4096 * 17 / 16 = 4352. The second year amount is 4352 * 17 / 16 = 4624. The third year amount is 4624 * 17 / 16 = 4913. This matches the given amount exactly. Why Other Options Are Wrong: Values such as Rs. 3096, Rs. 4076, and Rs. 4085 do not produce Rs. 4913 in 3 years at 6 1/4% compound interest. Only Rs. 4096 leads to an amount that matches the given figure. Common Pitfalls: Learners may misinterpret the rate 6 1/4% and fail to recognize the nice fractional form 1 / 16, leading to more cumbersome decimal calculations. Others may not notice the cube relationship between 4913 and 4096, which is an intentional pattern in this type of question. Final Answer: The required principal is Rs. 4096.

Question 16

Compound interest on a certain sum of money at 20% per annum for 2 years is Rs. 5995. What is the simple interest on the same money at 8% per annum for 6 years?

Accepted Answer

Introduction / Context: This question first gives us the compound interest on an unknown principal at a relatively high rate, and then asks for the simple interest on the same principal at a different rate and for a different period. We must determine the principal using the compound interest information and then use it to compute the new simple interest. Given Data / Assumptions: Compound interest for 2 years at 20% per annum is Rs. 5995. Interest is compounded annually for this part. The same principal is then invested at 8% per annum simple interest for 6 years. We must find the simple interest for 6 years at 8% per annum on that principal. Concept / Approach: For compound interest, CI2 = P * [(1 + r / 100)^2 - 1]. We use this formula with r = 20% and CI2 = 5995 to find the principal P. Then we apply the simple interest formula SI = P * R * T / 100 with rate 8% and time 6 years to find the required simple interest. Step-by-Step Solution: Step 1: Let principal be P and annual rate r1 = 20% for the compound interest part. Step 2: For 2 years at 20%, amount A2 = P * (1.20)^2 = P * 1.44. Step 3: Compound interest for 2 years is CI2 = A2 - P = P * (1.44 - 1) = 0.44P. Step 4: Given CI2 = 5995, so 0.44P = 5995. Step 5: Solve for P: P = 5995 / 0.44. Step 6: Compute P = 13625. Step 7: Now use simple interest with rate r2 = 8% per annum and time t2 = 6 years. Step 8: Simple interest SI = P * r2 * t2 / 100 = 13625 * 8 * 6 / 100. Step 9: Simplify SI = 13625 * 48 / 100 = 13625 * 0.48 = 6540. Verification / Alternative check: Check the compound interest step: For P = 13625, amount after 2 years at 20% is 13625 * 1.44 = 19620. Compound interest is 19620 - 13625 = 5995, matching the given value. Then recompute the simple interest on 13625 at 8% for 6 years to confirm SI = 6540. Why Other Options Are Wrong: Values such as Rs. 5989, Rs. 7844, and Rs. 6789 do not follow from the correct principal and simple interest formula. They may arise from arithmetic mistakes or from confusing the two interest rates or time periods. Only Rs. 6540 is consistent with both the initial compound interest data and the simple interest calculation. Common Pitfalls: One common mistake is to treat the given compound interest as simple interest when solving for the principal, which leads to an incorrect P. Another error is to reuse the 20% rate instead of the new 8% rate when computing the simple interest. It is important to keep track of which rate and time correspond to each part of the problem. Final Answer: The simple interest on the same money at 8% per annum for 6 years is Rs. 6540.

Question 17

A certain sum 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 percent per annum simple interest. The interest earned from scheme M is twi…

Accepted Answer

Introduction / Context: This problem compares compound interest and simple interest on the same principal over the same time period. The interest from the compound interest scheme is given to be twice the interest from the simple interest scheme, and we are asked to find the unknown simple interest rate k. Such questions test understanding of basic interest formulas and the difference between simple and compound interest growth. Given Data / Assumptions: The same principal amount is invested in both schemes M and N. Scheme M: rate = 20 percent per annum compound interest, compounded annually, for 2 years. Scheme N: rate = k percent per annum simple interest, for 2 years. Interest from scheme M is twice the interest from scheme N. We assume rates remain fixed and there are no additional deposits or withdrawals. Concept / Approach: For compound interest with annual compounding, the interest for 2 years on principal P at rate r percent is: CI = P * [(1 + r/100)^2 - 1] For simple interest on the same principal and time with rate k percent, the interest is: SI = P * k * 2 / 100 We are told that CI from scheme M is twice SI from scheme N. This gives an equation in terms of k, which we solve to find the required rate. Step-by-Step Solution: Let the principal be P. Scheme M: r = 20 percent, time = 2 years. CI from M = P * [(1 + 20/100)^2 - 1] (1 + 20/100) = 1.20, so (1.20)^2 = 1.44 Therefore CI from M = P * (1.44 - 1) = P * 0.44 Scheme N: rate = k percent, time = 2 years, simple interest. SI from N = P * k * 2 / 100 = 0.02 * k * P Given: CI from M = 2 * SI from N So P * 0.44 = 2 * (0.02 * k * P) 0.44 = 0.04 * k k = 0.44 / 0.04 = 11 Verification / Alternative check: To verify, pick a convenient value, for example P = Rs. 100. For scheme M, CI for 2 years at 20 percent is 44 rupees, since 100 grows to 144. For scheme N with k = 11 percent per annum simple interest, the interest in 2 years is: SI = 100 * 11 * 2 / 100 = 22 Twice of 22 is 44, which exactly matches the CI from scheme M. This confirms that k = 11 percent is correct. Why Other Options Are Wrong: A rate of 7 percent or 9 percent would produce significantly lower simple interest and would not give a compound interest that is exactly double. Thirteen percent or fifteen percent would lead to higher simple interest, and twice that value would exceed the compound interest earned at 20 percent for 2 years. Only 11 percent fits the condition that compound interest at 20 percent for 2 years is exactly twice the simple interest at k percent for 2 years on the same principal. Common Pitfalls: One common error is to confuse the formulas for simple and compound interest and treat both as if they grow linearly. Another mistake is to forget that the compounding period is annual and misuse a more complex formula. Students also sometimes cancel the principal incorrectly or make algebraic mistakes when solving for k. Carefully writing down the formulas and simplifying step by step helps avoid these issues. Final Answer: The required simple interest rate k for scheme N is 11 percent per annum.

Question 18

What is the present worth of Rs. 169 due in 2 years at 4 percent per annum compound interest, compounded annually?

Accepted Answer

Introduction / Context: Present worth or present value is a key concept in compound interest problems. Instead of finding the amount after a certain time, we are given the future amount and asked to find how much should be invested today. This question tests the ability to reverse the compound interest process using the appropriate formula and to interpret results accurately in terms of money. Given Data / Assumptions: Future amount due in 2 years = Rs. 169. Rate of interest = 4 percent per annum compound interest. Compounding is done annually. We must find the present worth, that is the principal that would grow to Rs. 169 in 2 years at the given rate. Concept / Approach: The compound amount formula is: A = P * (1 + r/100)^n where A is the amount, P is the principal, r is the rate, and n is the number of years. To find present worth P when A is given, we rearrange the formula: P = A / (1 + r/100)^n We substitute A = 169, r = 4, and n = 2 and simplify to get the present value. Step-by-Step Solution: Given A = Rs. 169 Rate r = 4 percent per annum Time n = 2 years P = A / (1 + r/100)^n 1 + r/100 = 1 + 4/100 = 1.04 (1.04)^2 = 1.04 * 1.04 = 1.0816 P = 169 / 1.0816 Compute: 169 / 1.0816 = 156.25 Therefore, the present worth is Rs. 156.25 Verification / Alternative check: We can verify the result by compounding Rs. 156.25 forward for 2 years at 4 percent. After 1 year: 156.25 * 1.04 = 162.50 After 2 years: 162.50 * 1.04 = 169.00 We reach exactly Rs. 169, which matches the given future amount. This confirms that our present value calculation is correct. Why Other Options Are Wrong: Rs. 150 and Rs. 140 are too low; when compounded at 4 percent for 2 years they will not reach 169. Rs. 125.25 is much smaller and clearly would not grow to 169 at such a modest rate. Rs. 160 is closer, but if we compound 160 for 2 years at 4 percent, we get: 160 * 1.04 * 1.04 = 160 * 1.0816 = 173.06 which is higher than 169. Only Rs. 156.25 compounds exactly to Rs. 169 under the given conditions. Common Pitfalls: A common mistake is to treat the problem like simple interest and divide by (1 + r * n / 100) instead of by the proper compound factor. Some students also forget to square the factor (1.04) for 2 years and only divide by 1.04 once. Rounding too early in intermediate steps can also produce small errors. Using the formula carefully and checking by recomputing the forward amount helps avoid these issues. Final Answer: The present worth of Rs. 169 due in 2 years at 4 percent compound interest is Rs. 156.25.

Question 19

Simple interest on a certain sum of money for 3 years at 8 percent per annum is half the compound interest on Rs. 4000 for 2 years at 10 percent per annum. What is the sum invested at simple interest?

Accepted Answer

Introduction / Context: This question links simple interest on an unknown principal with compound interest on a known principal. The simple interest for 3 years is given to be half of the compound interest on Rs. 4000 for 2 years at 10 percent per annum. We need to find the principal that was placed at simple interest. This problem checks understanding of both simple and compound interest formulas and the ability to connect two different investments through a given condition. Given Data / Assumptions: Simple interest (SI) on an unknown principal P for 3 years at 8 percent per annum. Compound interest (CI) on Rs. 4000 for 2 years at 10 percent per annum. The simple interest is equal to half of this compound interest. Rates are annual and remain constant; compounding is annual. Concept / Approach: First we compute the compound interest on Rs. 4000 for 2 years at 10 percent. Then we take half of that amount as the given simple interest. After that, we use the simple interest formula SI = P * r * t / 100 and solve for P. The key steps are computing CI correctly and then equating it to the expression for SI. Step-by-Step Solution: Step 1: Compute CI on Rs. 4000 at 10 percent for 2 years. Amount A = 4000 * (1 + 10/100)^2 (1 + 10/100) = 1.10, so (1.10)^2 = 1.21 A = 4000 * 1.21 = 4840 CI = A - principal = 4840 - 4000 = 840 Step 2: Simple interest equals half of this CI. Therefore, SI = 840 / 2 = 420 Step 3: Use SI formula with P, rate 8 percent, time 3 years. SI = P * 8 * 3 / 100 = 0.24 * P So 0.24 * P = 420 P = 420 / 0.24 = 1750 Therefore, the principal placed at simple interest is Rs. 1750 Verification / Alternative check: We can verify by computing the simple interest on Rs. 1750 at 8 percent for 3 years. SI = 1750 * 8 * 3 / 100 = 1750 * 24 / 100 = 1750 * 0.24 = 420 We already know half of the compound interest on Rs. 4000 for 2 years at 10 percent is 420, since the full CI is 840. Thus the condition in the question is satisfied, confirming that P = 1750 is correct. Why Other Options Are Wrong: Principals like Rs. 1550, Rs. 1650, or Rs. 1850 would give different simple interest values when substituted into the SI formula and would not yield 420. For example, using 1850 would give SI = 1850 * 0.24 = 444, which is not half of 840. Rs. 2000 would give SI = 480, which is greater than half of the computed compound interest. Only Rs. 1750 leads to the correct simple interest of Rs. 420. Common Pitfalls: A common mistake is to confuse half of the compound amount with half of the compound interest. Students may accidentally take half of 4840 instead of half of 840. Another pitfall is treating CI like SI and incorrectly applying a linear formula over 2 years. Some also mis apply the percentage when solving for P. Writing each step clearly and keeping a separate symbol for the CI amount helps avoid these issues. Final Answer: The sum placed on simple interest is Rs. 1750.

Question 20

Find the compound interest on Rs. 6500 for 4 years if the rate of interest is 10 percent per annum for the first 2 years and 20 percent per annum for the next 2 years, compounded annually in each period.

Accepted Answer

Introduction / Context: This question involves compound interest with a changing rate over different time periods. For the first 2 years the rate is 10 percent per annum, and for the next 2 years it increases to 20 percent per annum. We must compute the total compound interest on Rs. 6500 over the entire 4 year period. Such questions check understanding of how to apply different rates sequentially while compounding the amount forward. Given Data / Assumptions: Principal P = Rs. 6500. Rate for first 2 years = 10 percent per annum. Rate for next 2 years = 20 percent per annum. Compounding is annual in each phase. Total time = 4 years, with different rates applied sequentially. Concept / Approach: When the rate changes after a certain period, we treat the process in stages. First we compute the amount after the initial period using the first rate. Then we use that amount as the new principal for the next period at the new rate. The compound interest is simply the final amount minus the original principal. The compound interest formula A = P * (1 + r/100)^n is applied separately for each stage. Step-by-Step Solution: Stage 1: first 2 years at 10 percent. Initial principal P = 6500 Amount after 2 years A1 = 6500 * (1 + 10/100)^2 (1 + 10/100) = 1.10, so (1.10)^2 = 1.21 A1 = 6500 * 1.21 = 7865 Stage 2: next 2 years at 20 percent. New principal for this stage = A1 = 7865 Amount after next 2 years A2 = 7865 * (1 + 20/100)^2 (1 + 20/100) = 1.20, so (1.20)^2 = 1.44 A2 = 7865 * 1.44 = 11325.6 Total amount after 4 years = Rs. 11325.60 Compound interest = A2 - original principal = 11325.60 - 6500 = 4825.60 Rounding to the nearest rupee gives Rs. 4826 Verification / Alternative check: We can combine the two stages into a single equivalent factor: Overall factor = (1.10)^2 * (1.20)^2 = 1.21 * 1.44 = 1.7424 Amount = 6500 * 1.7424 = 11325.6 This matches our earlier calculation. Subtracting the principal again gives 4825.6, which rounds to Rs. 4826. The consistency between both methods confirms the correctness of the result. Why Other Options Are Wrong: Rs. 3845 and Rs. 4415 are both significantly lower than the correct compound interest and would correspond to smaller effective rates. Rs. 5142 is larger than 4826 and would require a higher overall growth factor than what the given rates provide. Rs. 4600 is somewhat close but does not match the exact detailed calculation. Only Rs. 4826 correctly represents the compound interest produced by 10 percent for 2 years followed by 20 percent for 2 years on Rs. 6500. Common Pitfalls: Some students incorrectly take an average rate, such as 15 percent for all 4 years, instead of treating the 10 percent and 20 percent phases separately. Others forget to use the amount from the first phase as the principal for the second phase and reuse the original principal by mistake. Rounding too early or mishandling decimals in multiplication can also cause small errors. Carefully separating the computation into stages and only rounding at the final step helps maintain accuracy. Final Answer: The compound interest on Rs. 6500 for 4 years, with 10 percent per annum for the first 2 years and 20 percent per annum for the next 2 years, is Rs. 4826 (approximately).

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