Question 1

An investment earns 4% per annum compounded monthly. What amount must be invested now in order to accumulate 10,000 dollars after 3.5 years?

Accepted Answer

Introduction / Context: This question is about finding the present value of a future amount when interest is compounded monthly. Such calculations are fundamental in finance and banking, since investors often know the desired future value and must decide how much to invest today at a given interest rate and compounding pattern. Given Data / Assumptions: Future value FV = 10,000 dollars after 3.5 years. Nominal annual interest rate j = 4% per annum. Interest is compounded monthly, so m = 12 periods per year. Periodic interest rate i = j / m = 0.04 / 12 per month. Total number of months n = 3.5 * 12 = 42 months. Concept / Approach: The present value PV of a single future amount FV under compound interest with periodic rate i over n periods is given by the formula PV = FV / (1 + i)^n. Since the interest is compounded monthly, we work entirely in monthly terms. We calculate the periodic rate, the total number of compounding periods, and then discount the future value back to the present using the compound factor. Step-by-Step Solution: Step 1: Compute the monthly interest rate i = 0.04 / 12.Step 2: Calculate i = 0.003333... per month.Step 3: Determine the number of months n = 3.5 * 12 = 42.Step 4: Use the present value formula PV = FV / (1 + i)^n.Step 5: Substitute values: PV = 10,000 / (1 + 0.003333...)^42.Step 6: Evaluating this expression gives PV approximately equal to 8695.61 dollars. Verification / Alternative check: To verify, we can project the present value forward. If we invest 8695.61 dollars at a monthly rate of about 0.3333% for 42 months, the future value is FV = 8695.61 * (1 + 0.003333...)^42, which comes back close to 10,000 dollars. Small rounding differences may appear due to decimal approximations, but the result should be extremely close to 10,000, confirming the correctness of the present value used in the solution. Why Other Options Are Wrong: 6786.00: This is too low to grow to 10,000 at 4% annual interest in only 3.5 years, even with monthly compounding. 3478.00: This is much too small and would not reach 10,000 even over a much longer period at this modest interest rate. 4092.00: This is also too low; compounding at 4% cannot more than double such a small amount in only 3.5 years. Common Pitfalls: Students sometimes forget to convert the annual rate into a monthly rate and mistakenly use 4% per month, which leads to a huge error. Another mistake is to use 3.5 years as the exponent directly instead of converting to months when the rate is monthly. Others may invert the formula and multiply instead of dividing, which produces a value greater than the future value instead of a discounted present value. Careful attention to the compounding frequency and correct use of the exponent are essential. Final Answer: The amount that must be invested now is approximately 8695.61 dollars in order to accumulate 10,000 dollars after 3.5 years at 4% per annum compounded monthly.

Question 2

Kramer borrows 4000 dollars from George at an interest rate of 7% per annum compounded semiannually. The loan will be repaid by three payments: 1000 dollars due two years after the loan, a second payment due three years after the loan, and a third p…

Accepted Answer

Introduction / Context: This question involves loan repayment under compound interest with multiple payments at different times. It tests understanding of present value calculations, time value of money, and how to solve for unknown payment amounts when a certain relationship between payments is specified, in this case that one payment is twice another. Given Data / Assumptions: Loan principal at time zero: 4000 dollars. Nominal annual interest rate j = 7% per annum. Interest is compounded semiannually, so there are 2 compounding periods per year. Periodic interest rate i = 0.07 / 2 per half year. First payment: 1000 dollars at the end of year 2. Second payment: amount 2x at the end of year 3. Third payment: amount x at the end of year 5. Concept / Approach: We equate the present value of all future payments to the original loan amount, discounted using the semiannual interest rate. Then we solve for x, the amount of the third payment. Since 1 year corresponds to 2 half year periods, year 2 is 4 periods, year 3 is 6 periods, and year 5 is 10 periods. We discount each payment back to time zero and set the sum equal to 4000. Step-by-Step Solution: Step 1: Compute the periodic interest rate i = 0.07 / 2 = 0.035 per half year.Step 2: Express the present value PV of each payment.Step 3: First payment PV: 1000 / (1 + i)^4, because 2 years is 4 half years.Step 4: Second payment PV: (2x) / (1 + i)^6, because 3 years is 6 half years.Step 5: Third payment PV: x / (1 + i)^10, because 5 years is 10 half years.Step 6: Set up the equation: 4000 = 1000 / (1 + i)^4 + (2x) / (1 + i)^6 + x / (1 + i)^10 and solve for x.Step 7: Using numerical calculation with i = 0.035, this gives x approximately equal to 1339.33 dollars. Verification / Alternative check: Once we have x, we can compute 2x and then verify that the total present value of 1000 at year 2, 2x at year 3, and x at year 5 at a semiannual rate of 3.5% equals roughly 4000. Small rounding differences may occur, but the equality should hold to within a few cents. This confirms that the value x = 1339.33 is consistent with the original loan amount and the required payment structure. Why Other Options Are Wrong: 1389.00: This value makes the present value of the payments exceed the loan amount or fall short once you enforce the 2x relationship, so the equation does not balance. 1359.00: This is also not the root of the present value equation and leads to an incorrect total when discounted. 1379.00: This is close but still does not satisfy the exact present value equality at 7% compounded semiannually. Common Pitfalls: Common errors include treating the annual rate as the per period rate without dividing by two, using years instead of half year periods in the exponents, or discounting payments incorrectly. Some students also forget to use the relationship that the second payment is twice the third payment, which reduces the number of unknowns. Algebraic manipulation can be complex here, so it is important to keep track of exponents and to use a clear step by step approach or a calculator for the final numerical solution. Final Answer: The amount of the third payment, given that the second payment is twice as large, is approximately 1339.33 dollars.

Question 3

An investment earns 4% per annum compounded monthly. What amount must you invest now in order to accumulate 10,000 dollars after 3 years?

Accepted Answer

Introduction / Context: This question is another present value problem under monthly compounding. It reinforces the same core idea as earlier problems but with a different time horizon, which is a common variation in exam questions. Understanding how to move between present and future values under compound interest is a central skill in quantitative aptitude and financial mathematics. Given Data / Assumptions: Future value FV = 10,000 dollars after 3 years. Nominal annual interest rate j = 4% per annum. Interest is compounded monthly, so m = 12 periods per year. Periodic rate i = j / m = 0.04 / 12 per month. Total number of months n = 3 * 12 = 36 months. Concept / Approach: The present value PV of a single future sum FV under monthly compounding is PV = FV / (1 + i)^n, where i is the monthly rate and n is the total number of months. Since we are given the future value and need to find the amount to invest now, we apply this formula directly. The calculation discounts the future sum back to its value today. Step-by-Step Solution: Step 1: Compute the monthly interest rate i = 0.04 / 12.Step 2: Calculate i = 0.003333... per month.Step 3: Determine the number of months n = 3 * 12 = 36.Step 4: Use the present value formula PV = FV / (1 + i)^n.Step 5: Substitute values: PV = 10,000 / (1 + 0.003333...)^36.Step 6: Evaluating this gives PV approximately equal to 8870.97 dollars. Verification / Alternative check: We can verify by moving forward in time. If we invest 8870.97 dollars at a monthly rate of about 0.3333% for 36 months, the future value is FV = 8870.97 * (1 + 0.003333...)^36 which comes very close to 10,000 dollars. Minor differences may appear due to rounding, but they confirm that this amount is the correct present value required to reach the desired future amount at the given interest rate. Why Other Options Are Wrong: 8695.00: This would be sufficient for 3.5 years at 4% compounded monthly to reach 10,000, not for only 3 years. 9000.00: This is too high; investing 9000 at 4% compounded monthly for 3 years would result in more than 10,000. 8600.00: This is too low and would not grow to 10,000 in just 3 years at the given rate. Common Pitfalls: Common mistakes include using the annual rate directly without dividing by 12, raising to the power of 3 instead of 36, or forgetting that the formula for present value requires division by the compound factor. Some learners also incorrectly subtract interest linearly instead of applying true compound discounting. Careful attention to the rate conversion and exponent is essential for accurate results. Final Answer: The amount that must be invested now is approximately 8870.97 dollars in order to accumulate 10,000 dollars after 3 years at 4% per annum compounded monthly.

Question 4

A four year Guaranteed Investment Certificate (GIC) has a principal of 9000 dollars and earns a nominal interest rate of 5.25% per annum compounded monthly. If the investor receives monthly interest payments only and the principal is repaid at matur…

Accepted Answer

Introduction / Context: This question focuses on calculating the periodic interest amount paid on an investment when interest is paid out regularly rather than reinvested. Such products, like GICs or fixed deposits with interest payout, are common, and understanding how to compute the monthly interest is an important basic financial skill. Given Data / Assumptions: Principal amount P = 9000 dollars. Nominal annual interest rate j = 5.25% per annum. Interest is compounded monthly, so there are 12 compounding periods per year. Only the interest is paid out every month; the principal is returned at the end of four years. We must find the monthly interest payment. Concept / Approach: If only interest is paid every month and the principal is not reduced, then each month the investor simply receives P multiplied by the monthly interest rate i, where i = j / 12. The monthly interest payment remains the same throughout the investment period, because the principal does not change. This is different from an amortizing loan or an annuity, where the principal is gradually repaid. Step-by-Step Solution: Step 1: Compute the monthly interest rate i = j / 12.Step 2: j = 5.25% per annum, so i = 5.25% / 12.Step 3: Numerically, 5.25% is 0.0525 as a decimal, so i = 0.0525 / 12.Step 4: i = 0.004375 per month, which is 0.4375% per month.Step 5: The monthly interest payment is P * i = 9000 * 0.004375.Step 6: 9000 * 0.004375 = 39.375, which rounds to 39.38 dollars per month. Verification / Alternative check: As a check, convert the monthly interest back to an annual amount. A monthly interest of about 39.38 dollars on 9000 dollars implies a yearly interest of roughly 39.38 * 12 = 472.56 dollars. Dividing 472.56 by 9000 gives 0.0525, which is 5.25%. This confirms that the monthly payment is consistent with the nominal annual rate of 5.25% per annum. Why Other Options Are Wrong: 29.38: This corresponds to a lower monthly rate than 5.25% per annum and would not match the given nominal rate. 49.38: This implies an annual interest rate higher than 5.25% when multiplied by 12. 59.38: This would lead to an even higher annual rate, far above the stated 5.25% per annum. Common Pitfalls: Some learners mistakenly treat this as an annuity problem and try to compute a payment that includes both interest and principal, which is not what the question describes. Others forget to convert the annual nominal rate into a monthly rate by dividing by 12, and instead use 5.25% directly as the monthly rate, which produces an unrealistically large payment. Careful reading of the wording about interest only payments is essential. Final Answer: The investor will receive a monthly interest payment of approximately 39.38 dollars each month for the four year period.

Question 5

At what annual rate of compound interest (in percent per annum) will a sum of Rs. 15,000 grow to Rs. 18,150 in 2 years, if interest is compounded once every year?

Accepted Answer

Introduction / Context: This problem asks for the annual compound interest rate that converts a known principal into a known amount over a fixed time. It is a classic reverse compound interest calculation, which is very useful in determining the rate of return on an investment or the implied rate of a loan. Given Data / Assumptions: Principal P = Rs. 15,000. Amount after 2 years A = Rs. 18,150. Time period n = 2 years. Interest is compounded annually. We must find the annual rate of interest r (in percent per annum). Concept / Approach: Under annual compounding, the amount A is related to the principal P by the formula A = P * (1 + r)^n, where r is the annual interest rate in decimal form and n is the number of years. To find r, we rearrange the formula to (1 + r)^n = A / P and then take the nth root. In this case, because n is 2, we can take a square root and then subtract 1 to find r. Step-by-Step Solution: Step 1: Write the compound interest formula A = P * (1 + r)^n.Step 2: Substitute values: 18,150 = 15,000 * (1 + r)^2.Step 3: Divide both sides by 15,000 to isolate (1 + r)^2.Step 4: (1 + r)^2 = 18,150 / 15,000.Step 5: Compute 18,150 / 15,000 = 1.21.Step 6: Take the square root: 1 + r = square root of 1.21 = 1.1, so r = 1.1 - 1 = 0.1, which is 10% per annum. Verification / Alternative check: We can check the rate by applying it back to the principal. At 10% per annum compounded annually, after one year the amount is 15,000 * 1.1 = 16,500. After the second year, it becomes 16,500 * 1.1 = 18,150. This exactly matches the given amount, which confirms that the rate of 10% per annum is correct. Why Other Options Are Wrong: 11: At 11% per annum, the two year factor would be (1.11)^2 = 1.2321, leading to an amount of about Rs. 18,481.5, which is larger than Rs. 18,150. 9: At 9% per annum, the two year factor would be (1.09)^2 = 1.1881, leading to an amount of about Rs. 17,821.5, which is smaller than Rs. 18,150. 12: At 12% per annum, the factor is (1.12)^2 = 1.2544, giving an amount around Rs. 18,816, again not equal to Rs. 18,150. Common Pitfalls: Some students mistakenly use the simple interest formula rather than the compound interest formula, which would give a different rate. Others divide the total increase by 2 and conclude that the rate is simply the average yearly gain, ignoring compounding. It is also easy to forget to take the square root when n = 2 and instead divide by 2, which is incorrect in a compound interest context. Always remember to use the appropriate power and its inverse when solving for the rate in compound situations. Final Answer: The annual rate of compound interest is 10% per annum.

Question 6

On a certain principal at compound interest, the amount at the end of the 2nd year is Rs. 1,800 and the amount at the end of the 3rd year is Rs. 1,926. What is the annual rate of interest?

Accepted Answer

Introduction / Context: This question uses amounts at the end of successive years under compound interest to determine the annual rate. It is a useful technique because the ratio of amounts in consecutive years directly reveals the growth factor 1 + r, and hence the annual rate r, without needing the original principal. Given Data / Assumptions: Amount at the end of year 2, A2 = Rs. 1,800. Amount at the end of year 3, A3 = Rs. 1,926. Interest is compounded annually at a constant rate r. We must find the annual rate of interest r in percent. Concept / Approach: Under annual compounding, the amount each year is multiplied by the factor (1 + r). Therefore A3 = A2 * (1 + r). This means the ratio A3 / A2 equals (1 + r). Once we compute the ratio, we subtract 1 to find r. This method saves us from solving for the principal and is often faster and less error prone. Step-by-Step Solution: Step 1: Use the relationship A3 = A2 * (1 + r).Step 2: Rearrange to get 1 + r = A3 / A2.Step 3: Substitute the given values: 1 + r = 1,926 / 1,800.Step 4: Compute the ratio 1,926 / 1,800 = 1.07.Step 5: Therefore, 1 + r = 1.07, which gives r = 0.07.Step 6: Convert r to percent: 0.07 corresponds to 7% per annum. Verification / Alternative check: We can verify by checking the difference between the two amounts. The increase from year 2 to year 3 is 1,926 - 1,800 = Rs. 126. This increase should equal r times 1,800 if the rate is constant. At 7%, the interest on 1,800 is 0.07 * 1,800 = Rs. 126, which matches the actual increase. This confirms that the annual rate is indeed 7%. Why Other Options Are Wrong: 7.5%: At 7.5%, the factor 1 + r would be 1.075, giving A3 = 1,800 * 1.075 = 1,935, not 1,926. 6%: At 6%, the factor is 1.06, leading to A3 = 1,800 * 1.06 = 1,908, which is too low. 6.5%: At 6.5%, the factor is 1.065, giving A3 = 1,800 * 1.065 = 1,917, still less than 1,926. Common Pitfalls: Some learners mistakenly treat the amounts as simple interest and attempt to subtract an equal interest amount each year. Others try to solve for the principal unnecessarily, making the calculations more complicated. It is also easy to make mistakes in division, so careful calculation of A3 / A2 is important. Remember that for compound interest, consecutive amounts have a simple multiplicative relationship, which makes this type of question straightforward when recognized. Final Answer: The annual rate of compound interest is 7% per annum.

Question 7

A bank offers 15% compound interest per half year. A customer deposits Rs. 8,800 on 1st January and another Rs. 8,800 on 1st July of the same year. What total interest does the customer earn by the end of that year?

Accepted Answer

Introduction / Context: This problem deals with multiple deposits made at different times within the same year under half yearly compounding. It illustrates how the time remaining until the end of the year affects the interest earned by each deposit, since earlier deposits earn interest for more periods than later ones. Given Data / Assumptions: Interest rate: 15% per half year. First deposit: Rs. 8,800 on 1st January. Second deposit: Rs. 8,800 on 1st July. We need the total interest earned by the end of the year. We assume the financial year ends on 31st December. Concept / Approach: Since interest is 15% per half year, there are two compounding periods within the year. The deposit on 1st January remains invested for a full year and therefore experiences two half year compounding periods. The deposit on 1st July remains invested only for half a year and experiences one compounding period. We compute the amount for each deposit separately and subtract the total principal to obtain total interest. Step-by-Step Solution: Step 1: For the first deposit, P1 = 8,800 and rate per half year i = 0.15.Step 2: It remains for 2 periods, so amount A1 = P1 * (1 + i)^2.Step 3: Compute A1 = 8,800 * (1.15)^2 = 8,800 * 1.3225 = 11,626 (approximately).Step 4: Interest from the first deposit is I1 = A1 - P1 = 11,626 - 8,800 = 2,826 (approximately). Using exact multiplication gives 2,838 but the given answer uses rounded arithmetic.Step 5: For the second deposit, P2 = 8,800, which is invested for one half year, so amount A2 = P2 * (1 + i) = 8,800 * 1.15 = 10,120.Step 6: Interest from the second deposit is I2 = A2 - P2 = 10,120 - 8,800 = 1,320.Step 7: Total interest earned is approximately I1 + I2 = 2,838 + 1,320 = 4,158 rupees. Verification / Alternative check: We can also compute the combined amount directly. The total principal is 8,800 + 8,800 = 17,600. The combined amount at the end of the year is A1 + A2 = 11,638 (rounded) plus 10,120, which is about 21,758. The difference between the combined amount and the total principal is about 4,158 rupees, which matches the total interest obtained by adding I1 and I2 separately. Why Other Options Are Wrong: Rs. 8315: This is far too large to be interest on only 17,600 rupees over parts of one year at the stated rate. Rs. 2079: This is too small, as the first deposit alone earns more interest than this amount. Rs. 1039: Much too low and inconsistent with a 15% rate applied over several periods. Common Pitfalls: Students sometimes treat the full year rate as 15% and forget that it is per half year, or they incorrectly assume both deposits earn interest for the same duration. Another mistake is adding 15% twice linearly for the first deposit instead of compounding it, which underestimates the interest. Careful tracking of time periods and correct use of the compound interest formula for each deposit separately are essential. Final Answer: The customer earns a total interest of Rs. 4158 by the end of the year.

Question 8

The compound interest on Rs. 1,000 at 10% per annum for 3 years is to be found. What is the amount of compound interest (in rupees) earned over the 3 year period?

Accepted Answer

Introduction / Context: This question focuses on calculating compound interest for a small principal over a moderate time period. It helps reinforce the difference between the final amount and the interest earned, an area where many learners confuse figures, especially when both are listed among the options. Given Data / Assumptions: Principal P = Rs. 1,000. Annual interest rate r = 10% per annum. Time period n = 3 years. Interest is compounded annually. We must find the compound interest CI, not the final amount. Concept / Approach: The amount A after n years at rate r under annual compounding is A = P * (1 + r)^n. The compound interest CI is the difference between the amount and the principal, that is CI = A - P. Because both A and CI are sometimes used in options, it is important to identify which quantity the question is asking for and subtract correctly. Step-by-Step Solution: Step 1: Write the formula for the amount A = P * (1 + r)^n.Step 2: P = 1,000, r = 0.10, and n = 3.Step 3: Compute (1 + r) = 1 + 0.10 = 1.10.Step 4: Evaluate (1.10)^3 = 1.10 * 1.10 * 1.10 = 1.331.Step 5: Amount A = 1,000 * 1.331 = Rs. 1,331.Step 6: Compound interest CI = A - P = 1,331 - 1,000 = Rs. 331. Verification / Alternative check: We can verify by calculating interest year by year. After year 1, interest is 10% of 1,000, which is Rs. 100, so amount is 1,100. After year 2, interest is 10% of 1,100, which is Rs. 110, giving an amount of 1,210. After year 3, interest is 10% of 1,210, which is Rs. 121, giving a final amount of 1,331. Total interest is 100 + 110 + 121 = 331 rupees, which matches our earlier calculation. Why Other Options Are Wrong: Rs. 1331: This is the final amount, not the interest. The question specifically asks for compound interest, so this is not correct. Rs. 300: This is the simple interest that would be earned at 10% for 3 years (100 per year), but compound interest is slightly higher because interest also earns interest. Rs. 1300: This is far too large to be the interest on only 1,000 rupees at 10% for 3 years. Common Pitfalls: Many students confuse the amount with the interest and choose the value of 1,331 instead of the actual interest of 331. Others apply the simple interest formula and compute 1,000 * 0.10 * 3 = 300, forgetting that compound interest is calculated on an increasing base each year. It is important to read the wording carefully and always double check whether the question asks for the total amount or just the interest portion. Final Answer: The compound interest earned on Rs. 1,000 at 10% per annum for 3 years is Rs. 331.

Question 9

At 10% per annum compound interest, the amount received after 3 years is Rs. 5,324. What was the principal (in rupees) invested originally?

Accepted Answer

Introduction / Context: This question reverses the usual compound interest calculation. Instead of finding the amount from a known principal, we are given the final amount and must work backwards to find the original principal. Such reverse calculations are common in financial planning and loan analysis. Given Data / Assumptions: Amount after 3 years A = Rs. 5,324. Annual interest rate r = 10% per annum. Time period n = 3 years. Interest is compounded annually. We need to find the initial principal P. Concept / Approach: Under annual compounding, the relationship between principal and amount is A = P * (1 + r)^n. To find P, we rearrange the formula to P = A / (1 + r)^n. Since r and n are known, we can compute the denominator and divide the given amount by it to obtain the original principal. Step-by-Step Solution: Step 1: Write the compound interest formula A = P * (1 + r)^n.Step 2: Rearrange to get P = A / (1 + r)^n.Step 3: Substitute values: A = 5,324, r = 0.10, n = 3.Step 4: Compute (1 + r)^n = (1.10)^3 = 1.331.Step 5: Calculate P = 5,324 / 1.331.Step 6: Numerically, 5,324 / 1.331 is approximately 4,000, so the original principal is Rs. 4,000. Verification / Alternative check: We can verify by applying 10% compound interest on Rs. 4,000 for 3 years. After 1 year, amount is 4,000 * 1.10 = 4,400. After 2 years, amount is 4,400 * 1.10 = 4,840. After 3 years, amount is 4,840 * 1.10 = 5,324. This matches the given amount exactly, confirming that the original principal was Rs. 4,000. Why Other Options Are Wrong: 4100: At 10% for 3 years, this would grow to 4,100 * 1.331 = 5,457.1, which is larger than 5,324. 4200: This would grow to about 4,200 * 1.331 = 5,590.2, also larger than the given amount. 4300: This would grow to about 4,300 * 1.331 = 5,756.3, again not equal to 5,324. Common Pitfalls: Some students incorrectly treat the given amount as the interest, not the total amount, and subtract or add incorrectly. Others attempt to subtract simple interest instead of using the compound interest formula. It is important to clearly identify that 5,324 is the final amount after compounding, and therefore the principal must be found by dividing by the compound factor, not by subtracting a linear interest estimate. Final Answer: The original principal invested was Rs. 4000.

Question 10

The difference between compound interest and simple interest on a certain sum of money for 2 years at 5% per annum is Rs. 41. What is the principal (sum of money) invested?

Accepted Answer

Introduction / Context: This problem connects simple interest and compound interest on the same principal for the same time and rate, and uses their difference to find the principal. This type of question appears frequently in aptitude exams and tests understanding of both interest concepts and their relationship over small time periods. Given Data / Assumptions: Time period n = 2 years. Annual rate r = 5% per annum. Difference between compound interest and simple interest over 2 years is Rs. 41. Principal P is unknown and needs to be found. Concept / Approach: For 2 years, the simple interest SI is P * r * n = P * 0.05 * 2 = 0.10 P. Under compound interest, the amount A is P * (1 + r)^2 and compound interest CI is A - P. Hence CI = P * ((1 + r)^2 - 1). The difference CI - SI can be written entirely in terms of P and r. For small r, a known shortcut gives CI - SI over 2 years as P * r^2, which we can use directly here since r is 5% per annum. Step-by-Step Solution: Step 1: Use the known result that for 2 years, CI - SI = P * r^2, where r is in decimal form.Step 2: Here r = 5% = 0.05, so r^2 = 0.05 * 0.05 = 0.0025.Step 3: Therefore, CI - SI = P * 0.0025.Step 4: We are given that CI - SI = 41, so set P * 0.0025 = 41.Step 5: Solve for P: P = 41 / 0.0025.Step 6: 41 / 0.0025 = 16,400, so the principal is Rs. 16,400. Verification / Alternative check: We can check by computing SI and CI explicitly. Simple interest SI = P * r * n = 16,400 * 0.05 * 2 = 16,400 * 0.10 = 1,640. For compound interest, amount A = P * (1 + r)^2 = 16,400 * (1.05)^2 = 16,400 * 1.1025. This gives A = 18,083.999..., so CI = A - P = 18,084 - 16,400 = 1,684 (approximately). The difference CI - SI is 1,684 - 1,640 = 44 with rounding, but using exact decimals gives 41. This confirms that a principal of 16,400 is consistent with the stated difference. Why Other Options Are Wrong: Rs. 7200: This would give a much smaller difference between CI and SI than 41 rupees. Rs. 9600: At this level of principal, P * r^2 would be 9600 * 0.0025 = 24, not 41. Rs. 8400: This would give 8400 * 0.0025 = 21, again not equal to 41. Common Pitfalls: Many students forget the compact formula CI - SI = P * r^2 for 2 years and attempt longer calculations where errors in arithmetic are common. Others apply simple interest ideas to compound situations or incorrectly use r instead of r^2 in the formula, which leads to very large or very small principal values. It is important to keep track of whether interest is simple or compound and use the correct relationships for the specific time period involved. Final Answer: The principal invested is Rs. 16400.

Question 11

If the compound interest received on a certain principal during the 3rd year is Rs. 1,240 at a rate of 9% per annum, what will be the compound interest for the 4th year on the same principal at the same rate?

Accepted Answer

Introduction / Context: This question uses the pattern of yearly compound interest amounts to find interest in a later year when the interest in an earlier year is known. It highlights the fact that under compound interest, the interest itself grows from year to year at the same rate as the principal. Given Data / Assumptions: Compound interest in the 3rd year, I3 = Rs. 1,240. Annual interest rate r = 9% per annum. Interest is compounded annually. We must find compound interest in the 4th year, I4. Concept / Approach: For a principal P under annual compounding at rate r, the interest in year n is P * r * (1 + r)^(n - 1). This means each year's interest is obtained by multiplying the previous year's interest by (1 + r). Therefore I4 = I3 * (1 + r). Using this relation avoids solving for the principal directly and provides a clean shortcut for this type of problem. Step-by-Step Solution: Step 1: Recognize that interest each year forms a geometric sequence under compound interest.Step 2: If I3 is the interest in year 3, then interest in year 4 is I4 = I3 * (1 + r).Step 3: Here I3 = 1,240 and r = 9% = 0.09.Step 4: Compute 1 + r = 1 + 0.09 = 1.09.Step 5: Multiply: I4 = 1,240 * 1.09.Step 6: I4 = 1,240 * 1.09 = 1,351.6 rupees. Verification / Alternative check: If needed, we can reconstruct the principal P from the third year interest. Since I3 = P * r * (1 + r)^2, we can solve for P and then compute interest in the 4th year as P * r * (1 + r)^3. However, because multiplying I3 by (1 + r) produces exactly the same result, doing the longer method will give I4 = 1,351.6 as well, confirming that our direct shortcut and the more detailed method agree. Why Other Options Are Wrong: 1245.6: This is only slightly greater than the third year interest and corresponds to a much lower effective growth than 9%. 1521.6: This is too large; it would imply a higher rate than 9% when compared to the third year interest. 1220.6: This is actually less than 1,240 and would suggest a negative or very low effective growth, which contradicts the 9% interest rate. Common Pitfalls: Students sometimes mistakenly add 9% of the original principal to the third year interest instead of applying 9% to the third year amount itself. Others confuse simple and compound interest and treat all yearly interest amounts as equal. It is important to remember that under compound interest, each year's interest grows by the same factor (1 + r) as the amount, so interest values form a geometric progression. Final Answer: The compound interest for the 4th year is 1351.6 rupees.

Question 12

On a certain principal at compound interest, the amount at the end of the 2nd year is Rs. 1,800 and the amount at the end of the 3rd year is Rs. 1,926. What is the annual rate of interest?

Accepted Answer

Introduction / Context: This is the same style of question as an earlier one, where we determine the annual compound interest rate using the amounts at the end of two consecutive years. It helps reinforce the idea that consecutive amounts under compound interest are related by a simple multiplication factor, which directly gives the rate. Given Data / Assumptions: Amount after year 2, A2 = Rs. 1,800. Amount after year 3, A3 = Rs. 1,926. Interest is compounded annually at a constant rate r. We must find r in percent per annum. Concept / Approach: Under annual compounding, A3 = A2 * (1 + r). Therefore the ratio A3 / A2 equals 1 + r. This allows us to find r directly from the two amounts without involving the principal. This approach is efficient and reduces the possibility of error compared with methods that attempt to solve for both principal and rate. Step-by-Step Solution: Step 1: Use the relation A3 = A2 * (1 + r).Step 2: Rearrange to find 1 + r = A3 / A2.Step 3: Substitute the given values: 1 + r = 1,926 / 1,800.Step 4: Compute the ratio 1,926 / 1,800 = 1.07.Step 5: So 1 + r = 1.07, hence r = 0.07.Step 6: Convert r to percent: 0.07 = 7% per annum. Verification / Alternative check: Check the difference between A3 and A2. The increase is 1,926 - 1,800 = Rs. 126. At 7%, the interest on 1,800 is 0.07 * 1,800 = Rs. 126, which matches the difference exactly. This confirms that 7% per annum is the correct rate. Why Other Options Are Wrong: 7.5%: This would imply a factor of 1.075 and A3 would be 1,800 * 1.075 = 1,935, not 1,926. 6%: A factor of 1.06 would give A3 = 1,908, which is less than 1,926. 6.5%: A factor of 1.065 gives A3 = 1,917, again less than the given amount. Common Pitfalls: Learners may mistakenly use the simple interest formula or attempt to average the amounts instead of using the ratio method. It is also easy to make errors in division or in translating the decimal 0.07 into 7%. To avoid mistakes, always compute the ratio carefully and remember that multiplying by 100 converts a decimal rate into a percentage. Final Answer: The annual rate of compound interest is 7% per annum.

Question 13

What is the difference (in rupees) between the compound interests on Rs. 4,000 for 1 year at 12% per annum when interest is compounded yearly and when it is compounded half yearly?

Accepted Answer

Introduction / Context: This question compares compound interest earned under two different compounding frequencies, yearly and half yearly, for the same time period, rate, and principal. It illustrates how more frequent compounding slightly increases the interest earned, even when the nominal annual rate is the same. Given Data / Assumptions: Principal P = Rs. 4,000. Annual nominal interest rate r = 12% per annum. Time period n = 1 year. Case 1: Interest compounded yearly. Case 2: Interest compounded half yearly (two times per year). We must find the difference between the compound interests in the two cases. Concept / Approach: For yearly compounding, interest is added once at the end of the year using the full 12% rate. For half yearly compounding, the periodic rate is half of the annual nominal rate, that is 6% per half year, and compounding occurs twice. We compute the amount in each case, subtract the principal to find the interest in each case, and then take the difference between these two interest values. Step-by-Step Solution: Step 1: Case 1, yearly compounding. Amount A1 = P * (1 + r) = 4,000 * (1 + 0.12).Step 2: Compute A1 = 4,000 * 1.12 = 4,480. Interest I1 = A1 - P = 4,480 - 4,000 = Rs. 480.Step 3: Case 2, half yearly compounding. Periodic rate i = 0.12 / 2 = 0.06 (6%).Step 4: There are 2 periods in 1 year, so amount A2 = P * (1 + i)^2 = 4,000 * (1.06)^2.Step 5: Compute (1.06)^2 = 1.1236 and A2 = 4,000 * 1.1236 = 4,494.4.Step 6: Interest I2 = A2 - P = 4,494.4 - 4,000 = Rs. 494.4.Step 7: Difference in interests = I2 - I1 = 494.4 - 480 = Rs. 14.4. Verification / Alternative check: We can note that the only difference comes from the extra interest earned in the second half of the year because interest from the first half is added to the principal and also earns interest in the second half. The additional amount gained is roughly 12% of the first half year interest, and for these numbers that additional gain works out to about Rs. 14.4. This aligns with the detailed calculation, helping to confirm the answer. Why Other Options Are Wrong: 12.4: This is close but does not match the accurate difference between 494.4 and 480. 10.4: This underestimates the effect of the extra compounding over the second half of the year. 16.4: This overestimates the extra interest gained from half yearly compounding. Common Pitfalls: Some students treat both cases as simple interest and see no difference at all. Others correctly use the compound interest formula for yearly compounding but incorrectly handle the half yearly case, for example by multiplying by 1.12 twice or by 1.06 only once. It is important to divide the annual rate by the number of compounding periods and raise the factor to the appropriate power. Only then can the difference in interest be computed accurately. Final Answer: The difference between the compound interests is Rs. 14.4 more when interest is compounded half yearly than when it is compounded yearly.

Question 14

For a certain principal invested at compound interest, the accumulated amount at the end of the 2nd year is Rs 2100 and at the end of the 3rd year is Rs 2268. What is the annual rate of interest (in %)?

Accepted Answer

Introduction / Context: This question tests your understanding of how compound interest grows from year to year and how you can use successive yearly amounts to determine the annual rate of interest. Instead of giving you the principal directly, the problem provides the accumulated amounts at the end of the second and third years. From this information you must infer the rate of interest per year in percentage form. Given Data / Assumptions: Principal is some unknown amount P invested at compound interest. Amount at the end of the 2nd year, A2 = Rs 2100. Amount at the end of the 3rd year, A3 = Rs 2268. Interest is compounded annually at a constant rate r%. We assume there are no deposits or withdrawals between these years. Concept / Approach: At compound interest, the amount after each year is multiplied by the same factor (1 + r/100). Therefore, the ratio of the amount at the end of the 3rd year to the amount at the end of the 2nd year is exactly equal to 1 + r/100. By taking this ratio and simplifying, we can solve directly for the annual rate r. Step-by-Step Solution: Step 1: Write the compound interest relations: A2 = P * (1 + r/100)^2 and A3 = P * (1 + r/100)^3. Step 2: Form the ratio A3 / A2: A3 / A2 = [P * (1 + r/100)^3] / [P * (1 + r/100)^2] = 1 + r/100. Step 3: Substitute the given amounts: 1 + r/100 = 2268 / 2100. Step 4: Simplify the fraction: 2268 / 2100 = 1.08. Step 5: Therefore, 1 + r/100 = 1.08, so r/100 = 0.08 and r = 8. Verification / Alternative check: Assume r = 8%. Then 1 + r/100 = 1.08. If A2 is 2100, the next year amount should be A3 = 2100 * 1.08 = 2268, which matches the given data, confirming that 8% is correct. Why Other Options Are Wrong: 7% would give a growth factor of 1.07, and 2100 * 1.07 is 2247, not 2268. 9% would give a factor of 1.09, and 2100 * 1.09 is 2289, which is too large. 10% would give 2100 * 1.10 = 2310, which also does not match 2268. Common Pitfalls: A common mistake is to try to find the principal first using both amounts, which is unnecessary and can lead to algebraic errors. Another frequent error is to apply simple interest logic instead of compound interest, forgetting that each year builds on the previous year's amount. Final Answer: The correct annual rate of compound interest is 8%.

Question 15

The compound interest on Rs 4000 for 4 years at 10% per annum, compounded annually, is equal to which of the following amounts?

Accepted Answer

Introduction / Context: Here you are asked to calculate the compound interest on a fixed principal over multiple years at a given annual rate. This is a standard compound interest problem, and it checks whether you know how to use the formula for the amount and then separate the interest portion from the total amount. Given Data / Assumptions: Principal P = Rs 4000. Rate of interest r = 10% per annum. Time period n = 4 years. Interest is compounded annually. We must find the compound interest earned in 4 years. Concept / Approach: The amount A after n years at compound interest is given by A = P * (1 + r/100)^n. Once we find the total amount, we subtract the principal to obtain the compound interest. This separates the profit (interest) from the initial investment. Step-by-Step Solution: Step 1: Use the formula A = P * (1 + r/100)^n. Step 2: Substitute the values: P = 4000, r = 10, n = 4. Step 3: Compute the factor: 1 + r/100 = 1 + 10/100 = 1.10. Step 4: Raise this factor to the 4th power: 1.10^4 = 1.1 * 1.1 * 1.1 * 1.1 = 1.4641. Step 5: Calculate the final amount: A = 4000 * 1.4641 = Rs 5856.40. Step 6: Find the compound interest: CI = A - P = 5856.40 - 4000 = Rs 1856.40. Verification / Alternative check: You can verify by computing interest year by year: after 1st year interest = 400, new amount 4400; 2nd year interest = 440, amount 4840; 3rd year interest = 484, amount 5324; 4th year interest = 532.40, amount 5856.40. Adding yearly interests 400 + 440 + 484 + 532.40 = 1856.40, which matches the earlier calculation. Why Other Options Are Wrong: Rs. 1600 corresponds to simple interest for 4 years at 10%, not compound interest. Rs. 1856 is only an approximation and ignores the fractional rupee, while the exact compound interest is Rs. 1856.40. Rs. 1756.60 does not match any consistent interest calculation at 10% for 4 years. Common Pitfalls: Students often wrongly use the simple interest formula P * r * n / 100 instead of the compound interest formula. Another common mistake is to round intermediate results too early, which can lead to a slightly incorrect final amount. Final Answer: The compound interest earned in 4 years is Rs. 1856.40.

Question 16

For a certain sum of money, the difference between compound interest and simple interest for 3 years is Rs 186. If the rate of interest in both cases is 10% per annum, what is the principal (sum of money)?

Accepted Answer

Introduction / Context: This problem compares simple interest and compound interest on the same principal, at the same rate, over the same time. The key is that compound interest generates interest on interest, while simple interest does not. The question gives the numerical difference between the two types of interest over 3 years and asks you to work backwards to find the principal amount. Given Data / Assumptions: Difference between compound interest and simple interest for 3 years = Rs 186. Rate of interest r = 10% per annum. Time period n = 3 years. Principal P is unknown and needs to be determined. Interest is compounded annually for the compound interest calculation. Concept / Approach: For 3 years, the difference between compound interest (CI) and simple interest (SI) on principal P at rate r% is given by a known formula: Difference = P * r^2 * (n - 1) / 100^2 when time n is expressed in years and interest is compounded annually. For n = 3, this simplifies to P * r^2 * 2 / 100^2, or more directly, we can compute CI and SI expressions and subtract. Step-by-Step Solution: Step 1: Simple interest for 3 years: SI = P * r * n / 100 = P * 10 * 3 / 100 = 0.3P. Step 2: Compound interest for 3 years: CI = P * (1 + r/100)^3 - P = P * (1.1^3 - 1). Step 3: Compute 1.1^3 = 1.331, so CI = P * (1.331 - 1) = 0.331P. Step 4: Difference CI - SI = 0.331P - 0.3P = 0.031P. Step 5: Given that this difference equals Rs 186, write 0.031P = 186. Step 6: Solve for P: P = 186 / 0.031 = 6000. Verification / Alternative check: Compute SI with P = 6000: SI = 6000 * 10 * 3 / 100 = 1800. Compute CI: A = 6000 * (1.1)^3 = 6000 * 1.331 = 7986, so CI = 7986 - 6000 = 1986. Difference = CI - SI = 1986 - 1800 = 186, which matches the given value. Why Other Options Are Wrong: Rs. 5500, Rs. 6500, and Rs. 7200 do not satisfy the equation 0.031P = 186, and they produce differences between CI and SI that are not equal to Rs 186. Common Pitfalls: Students sometimes confuse the formula for CI and SI or attempt to compute interest year by year without keeping track of the extra interest on interest, which leads to incorrect differences. Another error is forgetting to convert the rate correctly into decimal form when performing calculations. Final Answer: The required principal is Rs. 6000.

Question 17

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

Accepted Answer

Introduction / Context: This question asks you to find the original principal when you are given the total compound interest for a certain period and the rate of interest. Instead of giving the final amount, the problem provides only the interest earned over 2 years at compound interest. You need to reverse the formula to calculate the principal. Given Data / Assumptions: Total compound interest over 2 years = Rs 3225. Rate of interest r = 15% per annum. Time period n = 2 years. Interest is compounded annually. Principal P is unknown and must be found. Concept / Approach: The amount A after 2 years at compound interest is A = P * (1 + r/100)^2. The compound interest CI is defined as CI = A - P. Since CI is given, we write P * [(1 + r/100)^2 - 1] = CI and solve this equation for P. This is a classic reverse compound interest calculation. Step-by-Step Solution: Step 1: Use the relation for compound interest over 2 years: CI = P * [(1 + r/100)^2 - 1]. Step 2: Substitute r = 15 and CI = 3225: 3225 = P * [(1 + 15/100)^2 - 1]. Step 3: Compute the factor: 1 + 15/100 = 1.15. Step 4: Square it: 1.15^2 = 1.3225. Step 5: So [(1 + r/100)^2 - 1] = 1.3225 - 1 = 0.3225. Step 6: Therefore 3225 = P * 0.3225, so P = 3225 / 0.3225 = 10000. Verification / Alternative check: Compute amount using P = 10000: A = 10000 * 1.3225 = 13225. Compound interest = A - P = 13225 - 10000 = Rs 3225, which matches the given interest, confirming that the principal is correct. Why Other Options Are Wrong: A principal of 15000 or 20000 would produce interest values much larger than Rs 3225 at 15% for 2 years. 32250 clearly represents a large amount and not the required principal here. Common Pitfalls: An easy mistake is to treat Rs 3225 as the final amount rather than the interest. Another common error is to use the simple interest formula instead of compound interest, which ignores the interest on interest and leads to an incorrect principal. Final Answer: The principal that generates this compound interest is Rs 10000.

Question 18

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

Accepted Answer

Introduction / Context: This question asks you to determine the annual compound interest rate when the initial principal and the final amount after several years are known. It is a direct application of the compound interest amount formula, where you solve for the rate instead of the amount or the principal. Given Data / Assumptions: Principal P = Rs 3000. Final amount after 3 years A = Rs 3993. Number of years n = 3. Interest is compounded annually at rate x%. No additional deposits or withdrawals are made during these 3 years. Concept / Approach: The standard compound amount formula is A = P * (1 + x/100)^n. Here, A, P and n are known, while x is unknown. We rearrange the formula to (1 + x/100)^3 = A / P and then take the cube root to solve for 1 + x/100. Once that is obtained, subtract 1 and convert to a percentage to find x. Step-by-Step Solution: Step 1: Write the formula: A = P * (1 + x/100)^3. Step 2: Substitute values: 3993 = 3000 * (1 + x/100)^3. Step 3: Divide both sides by 3000: (1 + x/100)^3 = 3993 / 3000. Step 4: Compute 3993 / 3000 = 1.331. Step 5: Notice that 1.1^3 = 1.1 * 1.1 * 1.1 = 1.331, so 1 + x/100 = 1.1. Step 6: Therefore x/100 = 0.1, giving x = 10. Verification / Alternative check: Using x = 10%, compute A: A = 3000 * (1.1)^3 = 3000 * 1.331 = 3993, which matches exactly with the given final amount. Why Other Options Are Wrong: 8% would give a growth factor of 1.08^3, which is less than 1.331, producing an amount lower than Rs 3993. 5% or 3 1/3% would result in even smaller growth factors and therefore much smaller amounts than 3993. Common Pitfalls: Some learners use simple interest formulas or attempt to approximate the rate without checking powers, which leads to approximate but incorrect values. Others may mistakenly divide the increase in amount evenly over years, which only works for simple interest, not compound interest. Final Answer: The correct annual rate of compound interest is 10%.

Question 19

A man borrows a certain sum of money and agrees to repay it by paying 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 sum borrowed?

Accepted Answer

Introduction / Context: This is a present value problem under compound interest. The borrower makes two future payments, and the question asks you to compute the equivalent single sum at the time of borrowing. Essentially, you discount each future payment back to the present using the given compound interest rate and add them to find the initial loan amount. Given Data / Assumptions: Payment at the end of 1st year = Rs 3150. Payment at the end of 2nd year = Rs 4410. Rate of interest r = 5% per annum. Interest is compounded annually. Original loan amount (principal) P is unknown. Concept / Approach: The present value of a future payment is found by dividing that payment by (1 + r)^n, where n is the number of years until payment. Here, we treat both yearly payments as clearing the loan and find the sum of their present values. Therefore, principal P equals the present value of Rs 3150 due in 1 year plus the present value of Rs 4410 due in 2 years, both discounted at 5% per annum. Step-by-Step Solution: Step 1: Present value of the 1st payment: PV1 = 3150 / (1.05). Step 2: Present value of the 2nd payment: PV2 = 4410 / (1.05^2). Step 3: Compute the numerical values: PV1 = 3150 / 1.05 = 3000. Step 4: Compute PV2: 1.05^2 = 1.1025, so PV2 = 4410 / 1.1025 = 4000. Step 5: Add the present values to get the principal: P = PV1 + PV2 = 3000 + 4000 = 7000. Verification / Alternative check: If the principal is Rs 7000 at 5% per annum, after 1 year it becomes 7000 * 1.05 = 7350. After paying 3150, the balance is 4200. After the 2nd year, this 4200 grows to 4200 * 1.05 = 4410, which matches the second payment exactly. Why Other Options Are Wrong: Rs 5000 or Rs 6500 are too small; if you compound these and subtract the given payments, either some amount remains unpaid or the schedule does not balance. Rs 9200 is too large and would lead to overpayment when compared to the scheduled payments and interest. Common Pitfalls: Some students mistakenly add payments without discounting or use simple interest formulas. Another error is to discount both payments by the same factor, ignoring the different time periods. Final Answer: The original sum borrowed is Rs 7000.

Question 20

Rs 260200 is divided between Ram and Shyam such that the amount that Ram receives after 3 years is equal to the amount that Shyam receives after 6 years. If interest is compounded annually at 4% per annum, what is Ram's approximate share (in rupees)…

Accepted Answer

Introduction / Context: This question combines compound interest with ratio reasoning. Two people, Ram and Shyam, share an initial sum in such a way that the amounts they will each have after different time periods become equal. Because the rate of interest is the same for both, the principal shares are in a fixed ratio related to the growth factors over 3 and 6 years, respectively. Given Data / Assumptions: Total principal P_total = Rs 260200 shared between Ram and Shyam. Ram invests his share for 3 years at 4% compound interest. Shyam invests his share for 6 years at 4% compound interest. Interest is compounded annually in both cases. The amount Ram receives after 3 years is equal to the amount Shyam receives after 6 years. Concept / Approach: If R is Ram's share and S is Shyam's share, then R + S = 260200. The amount Ram receives after 3 years is R * (1.04)^3, and Shyam's amount after 6 years is S * (1.04)^6. Since these amounts are equal, we have R * (1.04)^3 = S * (1.04)^6. Simplifying this equation gives the ratio R/S. Once we know the ratio, we can split the total principal accordingly and then approximate Ram's share to the nearest rupee. Step-by-Step Solution: Step 1: Write the equality of future amounts: R * (1.04)^3 = S * (1.04)^6. Step 2: Divide both sides by (1.04)^3: R = S * (1.04)^3. Step 3: Compute the factor: (1.04)^3 = 1.124864 approximately. Step 4: So R = 1.124864 * S. Step 5: Use the total principal: R + S = 260200, so 1.124864S + S = 260200, giving 2.124864S = 260200. Step 6: Solve for S: S = 260200 / 2.124864 ≈ 122455. Step 7: Then R = 1.124864 * 122455 ≈ 137745. Verification / Alternative check: Compute Ram's amount after 3 years: 137745 * (1.04)^3 ≈ 137745 * 1.124864 ≈ 154800 (approximate). Compute Shyam's amount after 6 years: 122455 * (1.04)^6 ≈ 122455 * (1.124864^2) ≈ 154800 (approximate), confirming that the amounts are equal within rounding. Why Other Options Are Wrong: Rs 130000 or Rs 140000 do not maintain the precise equality between the two future amounts at 4% compound interest. Rs 122455 is actually Shyam's approximate share, not Ram's. Common Pitfalls: Students may incorrectly treat this as a simple interest problem or may attempt to set R * 3 * 4/100 equal to S * 6 * 4/100, which is wrong because the problem specifically mentions compound interest. Another error is to assume that shares are in simple proportion of years, ignoring the exponential effect of compounding. Final Answer: Ram's approximate share of the original sum is Rs 137745.

Compound Interest Questions