Question 1

Kashundra Jones plans a lump sum deposit so that she can withdraw 3,000 dollars at the end of each quarter for 10 years from an account that earns 10% interest per year compounded quarterly. What lump sum amount must she deposit now?

Accepted Answer

Introduction / Context: This is another present value of an ordinary annuity question but with quarterly compounding. Kashundra wants to make regular withdrawals while interest accrues more frequently than annually, so the quarterly rate and the total number of quarters must be used in the calculation. Given Data / Assumptions: Regular withdrawal R = 3,000 dollars at the end of each quarter. Nominal annual interest rate = 10%, compounded quarterly. Quarterly rate i = 10% / 4 = 2.5% = 0.025. Withdrawal horizon = 10 years, so n = 10 * 4 = 40 quarters. Withdrawals are at the end of each quarter (ordinary annuity). Concept / Approach: The present value of an ordinary annuity with periodic rate i and n periods is: PV = R * (1 - (1 + i)^(-n)) / i Here PV is the lump sum required today so that, when the account grows at rate i each quarter, it will exactly fund the planned withdrawals. Step-by-Step Solution: Step 1: Set i = 0.025 and n = 40. Step 2: Compute (1 + i)^(-n) = (1.025)^(-40). Step 3: Evaluate the annuity factor: (1 - (1.025)^(-40)) / 0.025. Step 4: Multiply this factor by R = 3000. Step 5: The result gives PV ≈ 75,308.33 dollars. This is the amount that must be on deposit today to allow the desired withdrawals. Verification / Alternative Check: Total planned withdrawals equal 3,000 * 40 = 120,000 dollars. Since some of this total comes from interest, the initial deposit must be less than 120,000 dollars. A value around 75,000 dollars is reasonable because the account has 10 years of growth with relatively high interest. Why Other Options Are Wrong: Options B, C, and E (70,000; 65,000; 60,000 dollars) are all below the accurately calculated present value and would not produce enough money to fund all payments. Option D (80,000 dollars) is higher than necessary and would leave unused balance if withdrawals proceeded as described. Common Pitfalls: Learners sometimes use the annual rate directly instead of converting to a quarterly rate, which yields a wrong answer. Another mistake is to treat the quarterly withdrawals as yearly withdrawals by using n = 10 instead of 40, which significantly miscalculates the present value. Final Answer: Kashundra must deposit approximately 75,308.33 dollars now.

Question 2

Fifteen equal semiannual payments are made into a sinking fund that earns 7% interest compounded semiannually so that 4,850 dollars will be in the fund at the end. What is the amount of each payment (rounded to the nearest cent)?

Accepted Answer

Introduction / Context: This question deals with a sinking fund, which is a fund created by making regular payments so that a specific future amount is accumulated. The fund earns compound interest at a given periodic rate, and you must determine the size of each equal payment that results in a target future value. Given Data / Assumptions: Target future value FV = 4,850 dollars. Nominal annual interest rate = 7% compounded semiannually. Semiannual rate i = 7% / 2 = 3.5% = 0.035. Number of semiannual payments n = 15. Payments are deposited at the end of each semiannual period (ordinary annuity). Concept / Approach: Future value of an ordinary annuity is: FV = R * ((1 + i)^n - 1) / i Here FV and i and n are known, and we solve for the payment R: R = FV * i / ((1 + i)^n - 1) Step-by-Step Solution: Step 1: Set i = 0.035 and n = 15. Step 2: Compute (1 + i)^n = (1.035)^15. Step 3: Find the numerator FV * i = 4850 * 0.035. Step 4: Compute the denominator (1.035^15 - 1). Step 5: Divide to get R ≈ 251.35 dollars. Thus, each semiannual deposit should be about 251.35 dollars. Verification / Alternative Check: You can calculate the future value using this payment amount: multiply 251.35 by the annuity factor ((1.035)^15 - 1) / 0.035. The result is very close to 4,850 dollars, confirming the correctness of the payment size. Why Other Options Are Wrong: Option B (245.45 dollars) and option C (235.87 dollars) are too small, leading to a future value less than 4,850 dollars. Option D (251.00 dollars) is close but slightly under the more accurate rounded value and would accumulate a smaller amount than required. Option E (260.00 dollars) would produce more than 4,850 dollars in the fund. Common Pitfalls: Students sometimes treat the 7% as the semiannual rate instead of the annual nominal rate, which doubles the correct periodic rate and drastically changes the answer. Another common error is to invert the formula and mistakenly solve for FV instead of R. Final Answer: Each semiannual sinking fund payment should be approximately 251.35 dollars.

Question 3

A retirement benefit of 12,000 dollars is to be paid every 6 months for 25 years at 7% interest compounded semiannually. What present value must be set aside now to fund this end of period retirement benefit?

Accepted Answer

Introduction / Context: This retirement planning problem is a present value of an ordinary annuity question with semiannual payments and semiannual compounding. The aim is to find how much money must be invested now so that it can fund regular retirement payments over a long period at a fixed interest rate. Given Data / Assumptions: Retirement payment R = 12,000 dollars every 6 months. Nominal annual interest rate = 7%, compounded semiannually. Semiannual rate i = 7% / 2 = 3.5% = 0.035. Payment period = 25 years, so number of payments n = 25 * 2 = 50. Payments are made at the end of each 6 month period (ordinary annuity). Concept / Approach: The present value of an ordinary annuity is: PV = R * (1 - (1 + i)^(-n)) / i This formula discounts each future retirement payment back to today at the semiannual rate and sums them into a single equivalent amount now. Step-by-Step Solution: Step 1: Set i = 0.035 and n = 50. Step 2: Compute (1 + i)^(-n) = (1.035)^(-50). Step 3: Evaluate the factor (1 - (1.035)^(-50)) / 0.035. Step 4: Multiply this factor by R = 12,000 dollars. Step 5: The resulting present value is approximately 281,468.06 dollars. Thus around 281,468.06 dollars must be invested today. Verification / Alternative Check: As a rough check, the total of all payments is 12,000 * 50 = 600,000 dollars. Because the money earns interest, the present value must be significantly less than 600,000 dollars. A value around 280,000 dollars is reasonable for 25 years at 7% nominal interest. Why Other Options Are Wrong: Options A, B, and D (245,678; 234,689; 234,578 dollars) are all lower than the computed present value and would not be sufficient to fund the entire stream of payments. Option E (200,000 dollars) is clearly too low for such a large series of payments. Only option C closely matches the correct formula based computation. Common Pitfalls: Some learners mistakenly use 7% as the semiannual rate instead of dividing by 2, which doubles the interest rate per period and depresses the present value too much. Others may treat the payments as annual rather than semiannual, which halves the number of periods and again gives the wrong answer. Final Answer: The present value required to fund the semiannual retirement payments is approximately 281,468.06 dollars.

Question 4

Find the finance charge on a credit account with an average daily balance of 8,431.10 dollars and a monthly interest rate of 1.4%.

Accepted Answer

Introduction / Context: This question focuses on a very simple but practical application of percentage calculations in personal finance. Credit card and loan companies often compute a monthly finance charge by multiplying the average daily balance by a monthly interest rate. Given Data / Assumptions: Average daily balance B = 8,431.10 dollars. Monthly interest rate = 1.4% per month. Finance charge is computed for one month only. No additional fees or penalties are included. Concept / Approach: The finance charge for one month can be found using: Finance charge = Balance * Monthly rate The monthly rate must be expressed as a decimal rather than a percentage when performing the multiplication, so 1.4% becomes 0.014. Step-by-Step Solution: Step 1: Convert the interest rate to decimal form: 1.4% = 0.014. Step 2: Multiply the average daily balance by the decimal rate. Step 3: Finance charge = 8431.10 * 0.014. Step 4: Compute this product to obtain approximately 118.04 dollars. Therefore, the finance charge for the month is 118.04 dollars. Verification / Alternative Check: As a quick check, 1% of 8,431.10 dollars is about 84.31 dollars. 0.4% of the balance is roughly 33.72 dollars. Adding these gives around 118.03 dollars, which is extremely close to the more accurate computed answer of 118.04 dollars. Why Other Options Are Wrong: Option A (98.04 dollars) would correspond to a smaller monthly rate of about 1.16%. Option B (88.04 dollars) and option C (68.04 dollars) correspond to even smaller rates and do not match 1.4%. Option E (128.04 dollars) implies a higher rate than 1.4% per month. Common Pitfalls: Learners occasionally divide by 100 twice, effectively using a rate of 0.014% instead of 1.4%, which yields a very tiny finance charge. Others might confuse annual and monthly rates and divide or multiply by 12 incorrectly. Always match the period of the rate with the period of the balance. Final Answer: The monthly finance charge on the account is 118.04 dollars.

Question 5

A loan of 3,500 dollars is repaid with equal annual payments over 4 years at 9% interest compounded annually. What is the amount of each annual payment?

Accepted Answer

Introduction / Context: This is a standard loan amortization problem. You are given the loan principal, term, and annual interest rate, and you must compute the fixed annual payment that exactly pays off the loan, including both principal and interest, over the specified time period. Given Data / Assumptions: Loan principal PV = 3,500 dollars. Annual interest rate r = 9% per annum. Number of annual payments n = 4. Payments are made at the end of each year. Interest is compounded annually at the same rate. Concept / Approach: The fixed payment for a fully amortizing loan is given by the annuity formula: Payment R = PV * r / (1 - (1 + r)^(-n)) Here PV is the present value of all payments, r is the periodic rate, and n is the total number of payments. Step-by-Step Solution: Step 1: Convert interest rate to decimal: r = 0.09. Step 2: Substitute PV = 3500, r = 0.09, n = 4 into the formula. Step 3: Compute the denominator 1 - (1.09)^(-4). Step 4: Compute 3500 * 0.09 = 315. Step 5: Divide 315 by the denominator to obtain R ≈ 1,080.34 dollars. So the borrower must pay approximately 1,080.34 dollars each year. Verification / Alternative Check: You can verify by constructing a repayment schedule: each year compute interest on the outstanding balance at 9%, subtract it from the payment to find the principal reduction, and update the balance. After the fourth payment, the balance should be near zero, confirming that the payment amount is correct. Why Other Options Are Wrong: Options A, B, C, and E either underestimate or overestimate the yearly payment needed to amortize the 3,500 dollar loan at 9% in 4 years. For instance, 890.60 dollars is too small and would leave a remaining balance after 4 years, while 1,089 dollars is slightly higher than necessary. Common Pitfalls: A common error is to divide the principal by the number of years and then add simple interest, which does not match the compound interest structure of an amortized loan. Another mistake is to forget to use the negative exponent when computing (1 + r)^(-n) in the denominator. Final Answer: Each annual payment required to repay the loan is approximately 1,080.34 dollars.

Question 6

A loan of 4,800 dollars has an APR of 12% with monthly payments over 24 months. What is the total finance charge (total interest paid) on this loan?

Accepted Answer

Introduction / Context: This problem combines loan amortization with the concept of a finance charge. The finance charge on an installment loan is the total interest paid over the life of the loan, which is the difference between the sum of all payments and the original principal. Given Data / Assumptions: Loan principal PV = 4,800 dollars. APR (annual percentage rate) = 12%. Interest is compounded monthly, so monthly rate i = 12% / 12 = 1% = 0.01. Number of monthly payments n = 24. Payments are equal in amount each month. Concept / Approach: First find the fixed monthly payment using the annuity formula: Payment R = PV * i / (1 - (1 + i)^(-n)) Then find the total of all payments and subtract the principal to get the finance charge: Finance charge = R * n - PV Step-by-Step Solution: Step 1: Set i = 0.01 and n = 24. Step 2: Compute the denominator 1 - (1.01)^(-24). Step 3: Compute the monthly payment R = 4800 * 0.01 / denominator ≈ 225.95 dollars. Step 4: Compute the total of all payments: R * n ≈ 225.95 * 24 ≈ 5,422.86 dollars. Step 5: Finance charge = 5,422.86 - 4,800 ≈ 622.86 dollars. Thus the total interest paid over the life of the loan is about 622.86 dollars. Verification / Alternative Check: You can build a month by month amortization table, computing interest each month as 1% of the outstanding balance and subtracting the remainder of the payment from principal. Summing all the interest entries will give a total very close to 622.86 dollars. Why Other Options Are Wrong: Option B (522.80 dollars) and option C (322.80 dollars) are too small and would correspond to either a lower APR or a shorter term. Option D (632.80 dollars) overestimates the finance charge relative to the computed payment. Option E (422.80 dollars) is also inconsistent with the described loan terms. Common Pitfalls: Some learners confuse APR with the monthly rate and use 12% as the monthly rate instead of 1%, which gives absurd results. Others forget that the finance charge is total payments minus principal rather than simply principal times rate times time. Final Answer: The total finance charge paid on this loan is approximately 622.86 dollars.

Question 7

Calculate the periodic interest rate (per quarter) corresponding to a nominal annual rate of 9.0% compounded quarterly.

Accepted Answer

Introduction / Context: This conceptual question checks whether you can convert a nominal annual interest rate with a specified compounding frequency into a periodic rate. Such conversions are common when working with loans, investments, and annuities. Given Data / Assumptions: Nominal annual interest rate = 9.0% per year. Compounding frequency = quarterly (4 times per year). We need the interest rate per quarter. Concept / Approach: When a nominal annual rate r_nom is compounded m times per year, the periodic rate i is: i = r_nom / m Where r_nom is expressed as a percentage or as a decimal consistently. Here quarterly compounding means m = 4. Step-by-Step Solution: Step 1: Take the nominal rate r_nom = 9.0%. Step 2: Determine the number of compounding periods per year, m = 4. Step 3: Compute i = 9.0% / 4. Step 4: i = 2.25% per quarter. Therefore the periodic interest rate that applies each quarter is 2.25%. Verification / Alternative Check: You can verify by multiplying the periodic rate by the number of periods: 2.25% * 4 = 9.0%, which matches the original nominal annual rate. This confirms the calculation is consistent. Why Other Options Are Wrong: Option A (3.45% per quarter) and option C (5.25% per quarter) would produce effective annual rates much higher than 9% when compounded. Option D (6.25% per quarter) is even larger and clearly inconsistent. Option E (1.25% per quarter) is too small because four such periods would only yield a nominal rate of 5% per year. Common Pitfalls: Learners sometimes confuse nominal and effective rates and attempt to use exponentiation when only a simple division is required. Another error is misreading quarterly as meaning every four years rather than four times a year. Final Answer: The periodic interest rate is 2.25% per quarter.

Question 8

Calculate the periodic interest rate (per month) corresponding to a nominal annual rate of 9.5% compounded monthly.

Accepted Answer

Introduction / Context: This problem is a straightforward conversion from a nominal annual rate to a monthly periodic rate. Understanding this conversion is vital when dealing with loans or investments that specify an APR but charge or credit interest monthly. Given Data / Assumptions: Nominal annual interest rate = 9.5% per year. Compounding frequency = monthly (12 times per year). We need the interest rate per month. Concept / Approach: The periodic rate i for a nominal annual rate r_nom compounded m times per year is: i = r_nom / m with r_nom expressed as a percentage. Here m = 12 because interest is compounded monthly. Step-by-Step Solution: Step 1: Take the nominal rate r_nom = 9.5%. Step 2: Determine the number of periods per year, m = 12. Step 3: Compute i = 9.5% / 12. Step 4: i ≈ 0.7916% per month. So the account earns about 0.7916% interest each month. Verification / Alternative Check: Multiplying the periodic rate by 12 gives 0.7916% * 12 ≈ 9.5%, which matches the given nominal annual rate. This simple check confirms the correctness of the periodic rate. Why Other Options Are Wrong: Option B (0.8916%) and option C (0.9916%) would produce nominal annual rates higher than 9.5% when multiplied by 12. Option D (0.6916%) or option E (0.5000%) would give annual nominal rates lower than 9.5% and so do not match the stated APR. Common Pitfalls: Some learners attempt to convert using division by 100 and 12 together incorrectly, or they confuse this with finding an effective annual rate. For questions that explicitly ask for a periodic rate from a nominal rate, a simple division by the number of periods is sufficient. Final Answer: The periodic interest rate is approximately 0.7916% per month.

Question 9

Determine the nominal annual interest rate if the periodic rate is 0.83% per month and interest is compounded monthly.

Accepted Answer

Introduction / Context: This question reverses the usual nominal to periodic conversion. You are given the periodic monthly rate and must find the corresponding nominal annual rate, assuming the same compounding frequency used for quoting the APR. Given Data / Assumptions: Periodic monthly rate i = 0.83% per month. Interest is compounded monthly. We want the nominal annual rate expressed as a percentage per year. Concept / Approach: Nominal annual rate r_nom is obtained by multiplying the periodic rate by the number of periods per year: r_nom = i * m where m = 12 for monthly compounding. Here we are not asked for the effective annual rate, only the nominal rate. Step-by-Step Solution: Step 1: Take the periodic rate i = 0.83% per month. Step 2: Determine the number of months per year, m = 12. Step 3: Compute r_nom = 0.83% * 12. Step 4: r_nom = 9.96% per annum, which is usually rounded to 10% per annum. So the nominal annual rate is approximately 10% when quoted to the nearest whole percent. Verification / Alternative Check: If the nominal rate is 10% compounded monthly, the periodic rate is 10% / 12 ≈ 0.83% per month, which matches the given periodic rate. This confirms that the two descriptions are equivalent. Why Other Options Are Wrong: A 7%, 8%, or 9% annual nominal rate corresponds to monthly rates lower than 0.83%. An 11% annual nominal rate would give a monthly rate of about 0.9167%, which is higher than 0.83%. Only 10% per annum matches the monthly rate of 0.83% given in the question. Common Pitfalls: Students sometimes divide instead of multiply, mistakenly calculating 0.83% / 12, which would give a tiny rate. Others confuse nominal rate with effective annual rate and try to use exponentiation unnecessarily. For nominal rates, simple multiplication by the number of periods is the correct approach. Final Answer: The nominal annual interest rate is approximately 10% per annum.

Question 10

An investor can choose between 10.5% interest compounded monthly and 11% interest compounded annually for a two year investment. Assuming all other factors are equal, which option should the investor prefer?

Accepted Answer

Introduction / Context: This conceptual question compares two different interest structures. The nominal rates are close, but the compounding frequency differs. The key idea is that more frequent compounding can sometimes offset a slightly lower nominal rate, resulting in a higher effective return over the investment period. Given Data / Assumptions: Option 1: 10.5% nominal annual rate, compounded monthly. Option 2: 11% nominal annual rate, compounded annually. Investment horizon = 2 years. Principal is the same in both options. Concept / Approach: We compare effective growth factors for each option over 2 years. For monthly compounding, the periodic rate is the nominal rate divided by 12, and the number of periods is 24. For annual compounding, the periodic rate equals the nominal rate, and the number of periods is 2. Option 1 factor = (1 + 0.105 / 12)^(12 * 2) Option 2 factor = (1 + 0.11)^2 Step-by-Step Solution: Step 1: Compute the monthly rate for option 1: 0.105 / 12 ≈ 0.00875. Step 2: Compute the 2 year factor for option 1: (1 + 0.00875)^24 ≈ 1.23255. Step 3: Compute the 2 year factor for option 2: (1 + 0.11)^2 = 1.11^2 ≈ 1.23210. Step 4: Compare 1.23255 with 1.23210. Since 1.23255 is slightly greater than 1.23210, option 1 yields a higher final amount. Verification / Alternative Check: If you invest 1,000 units of currency, option 1 gives about 1,232.55 after 2 years, while option 2 gives about 1,232.10. The difference is small but real and shows the advantage of more frequent compounding at a slightly lower nominal rate. Why Other Options Are Wrong: Option B is wrong because 11% annually does not quite catch up to the benefit of monthly compounding at 10.5%. Option C is incorrect because the two future values are close but not equal. Option D is wrong because sufficient numerical information is given to compute both growth factors. Option E is clearly false, since both options explicitly describe compound interest structures. Common Pitfalls: Students often compare only the nominal rates and assume 11% is better without considering compounding frequency. Others may incorrectly treat monthly compounding as if it were annual by ignoring the division by 12 or by using simple interest formulas. Effective comparisons must always use the correct compounding periods. Final Answer: The investor should prefer 10.5% compounded monthly, as it produces a slightly higher return over two years.

Question 11

If an investment can earn 4% interest compounded monthly, what lump sum must you invest now in order to accumulate 10,000 dollars after 3 years?

Accepted Answer

Introduction / Context: This problem involves finding the present value of a single future amount when interest is compounded monthly. It tests your ability to handle both discounting and the conversion from an annual nominal rate to a monthly periodic rate. Given Data / Assumptions: Future value FV = 10,000 dollars needed after 3 years. Nominal annual interest rate = 4% per year. Interest is compounded monthly. Monthly rate i = 4% / 12. Number of months n = 3 * 12 = 36. Concept / Approach: The present value of a single future amount when compounding is periodic is: PV = FV / (1 + i)^n where i is the periodic rate and n is the total number of periods. Here, we first find the monthly rate and then apply the formula. Step-by-Step Solution: Step 1: Compute the monthly rate i = 0.04 / 12 ≈ 0.003333. Step 2: Number of months n = 36. Step 3: Compute the growth factor (1 + i)^n = (1.003333)^36. Step 4: Compute PV = 10000 / (1.003333^36). Step 5: Evaluating gives PV ≈ 8,870.97 dollars. Thus, you must invest about 8,870.97 dollars today to have 10,000 dollars after 3 years. Verification / Alternative Check: As a rough check, if you invested 9,000 dollars at 4% compounded annually for 3 years, the future value would be roughly 9,000 * 1.1249 ≈ 10,124 dollars. Since monthly compounding gives slightly more growth, a present value just under 9,000 dollars makes sense for a 10,000 dollar target. Why Other Options Are Wrong: Option B (8,695.61 dollars) and option C (7,695.00 dollars) are too low, meaning they would not grow enough to reach 10,000 dollars in 3 years at 4% monthly compounding. Option D (9,092.00 dollars) and option E (9,500.00 dollars) are higher than necessary; investing these amounts would produce more than 10,000 dollars. Common Pitfalls: Common mistakes include using the annual rate directly instead of dividing by 12, or using the wrong exponent (such as 3 instead of 36). Another error is to use simple interest formulas, which do not capture the effect of monthly compounding. Final Answer: You must invest approximately 8,870.97 dollars now to accumulate 10,000 dollars after 3 years at 4% compounded monthly.

Question 12

Kramer borrowed 4,000 dollars from George at 7% interest compounded semiannually. The loan will be repaid by three payments: 1,000 dollars due 2 years after the loan, and two more payments due 3 and 5 years after the loan, with the second payment be…

Accepted Answer

Introduction / Context: This is a multi step time value of money problem involving unequal payments on a loan with semiannual compounding. You know the timing of the payments, the interest rate, and the relationship between the second and third payments, and you must determine the size of the final payment that makes the present value of all payments equal to the loan amount. Given Data / Assumptions: Loan principal PV = 4,000 dollars at time 0. Nominal annual interest rate = 7% compounded semiannually. Semiannual rate i = 7% / 2 = 3.5% = 0.035. First payment: 1,000 dollars at t = 2 years. Second payment: P2 at t = 3 years. Third payment: P3 at t = 5 years. Relationship: P2 = 2 * P3. Concept / Approach: Convert each payment to its present value at time 0 using the semiannual rate and appropriate number of periods. Then set the sum of these present values equal to the loan principal and solve for the unknown third payment P3. PV = 1000 / (1 + i)^4 + P2 / (1 + i)^6 + P3 / (1 + i)^10 with P2 = 2 * P3. Step-by-Step Solution: Step 1: Determine the number of semiannual periods. Two years correspond to 4 periods, three years to 6 periods, and five years to 10 periods. Step 2: Substitute P2 = 2 * P3 = 2x and P3 = x into the present value equation: 4000 = 1000 / (1.035)^4 + 2x / (1.035)^6 + x / (1.035)^10. Step 3: Compute the discount factors 1 / (1.035)^4, 1 / (1.035)^6, and 1 / (1.035)^10. Step 4: Combine the terms in x and solve the resulting linear equation for x. Step 5: This yields x ≈ 1,339.33 dollars. So the third payment P3 is approximately 1,339.33 dollars, and the second payment P2 is about 2,678.66 dollars. Verification / Alternative Check: Substitute P3 ≈ 1,339.33 and P2 ≈ 2,678.66 back into the present value equation and recompute the total PV. The result will be very close to 4,000 dollars, confirming that these payment amounts are consistent with the loan terms. Why Other Options Are Wrong: Options A, B, and C (1,389.00; 1,359.00; 1,379.00 dollars) are larger than the correct value and would lead to a present value of payments that exceeds 4,000 dollars. Option E (1,300.00 dollars) is smaller than the correct value and would not fully amortize the loan at the given interest rate. Common Pitfalls: Learners may mistakenly use annual periods instead of semiannual periods, which changes the exponents and gives an incorrect result. Others might forget that the second payment is twice the third and treat them as independent unknowns, making the algebra more complicated than necessary. Final Answer: The amount of the third payment that satisfies the loan conditions is approximately 1,339.33 dollars.

Question 13

An investor buys a four-year, monthly-payment Guaranteed Investment Certificate (GIC) with a principal of $9000 that earns a nominal interest rate of 5.25% per annum, compounded monthly. If the investor receives equal monthly interest cheques (and t…

Accepted Answer

Introduction / Context: This question comes from financial mathematics and focuses on a common investment product: a monthly-payment Guaranteed Investment Certificate (GIC). In such a GIC, the investor typically receives regular interest payments while the principal (original deposit) remains intact and is returned at maturity. We must determine the monthly interest payment from a $9000 deposit earning a nominal 5.25% per annum compounded monthly. Given Data / Assumptions: Principal (deposit) P = $9000 Nominal annual interest rate = 5.25% per annum Compounding frequency = monthly (12 times per year) Term = 4 years, but principal is not amortized; it is returned at the end Monthly payment is interest only, based on the monthly periodic rate Concept / Approach: For an interest-only investment with monthly payments, the periodic payment equals the principal multiplied by the periodic interest rate. The nominal annual rate must be converted to a monthly rate by dividing by 12. Since the principal is not being repaid during the term, we do not use annuity formulas. We simply compute the monthly interest and treat that as the periodic payment. Step-by-Step Solution: Step 1: Find the monthly periodic rate. Monthly rate i = 5.25% / 12 = 0.0525 / 12. i = 0.004375 per month (which is 0.4375%). Step 2: Compute the monthly interest payment. Monthly payment = P * i = 9000 * 0.004375. Monthly payment = 39.375 dollars. Step 3: Round to two decimal places. Monthly payment ≈ $39.38. Verification / Alternative check: If the investor receives $39.38 each month for 12 months, the total interest in one year is approximately 39.38 * 12 ≈ 472.56 dollars. The simple interest at 5.25% on $9000 for one year is 9000 * 0.0525 = 472.50 dollars. The slight difference is due to rounding the monthly payment to two decimal places, which confirms that $39.38 is a consistent and accurate monthly interest cheque amount. Why Other Options Are Wrong: A payment of $29.38 or $49.38 would correspond to considerably lower or higher effective rates than 5.25% per annum. A payment of $59.38 would imply an unrealistically high annual return on $9000. Only $39.38 aligns with the correct monthly rate of 0.4375% and produces a reasonable annual total close to 5.25% of the principal. Common Pitfalls: A frequent error is to misinterpret the problem as an amortizing annuity, using formulas that include repayment of principal. Another mistake is to forget to divide the nominal annual rate by 12 to obtain the monthly rate. Always check whether principal is being repaid over time or only at maturity, because that completely changes the calculation method. Final Answer: The investor will receive a fixed monthly interest payment of approximately $39.38.

Question 14

At what annual rate of compound interest, compounded once per year, will a sum of Rs. 10,000 grow to Rs. 12,321 in 2 years?

Accepted Answer

Introduction / Context: This question asks you to determine the annual compound interest rate when the principal, final amount and time are given. Such problems are very common in banking, investment and aptitude tests, and they test your understanding of the reverse use of the compound interest formula. Given Data / Assumptions: Principal P = Rs. 10,000 Amount after 2 years A = Rs. 12,321 Time t = 2 years Compounding frequency = once per year (annual) We must find the annual rate of interest r (in percent) Concept / Approach: For annual compounding, the basic relation is: A = P * (1 + r/100)^t. Rearranging, we have: A / P = (1 + r/100)^t. Since we know A, P and t, we first compute the ratio A / P, then take the square root (because t = 2) to find (1 + r/100), and hence r. Step-by-Step Solution: Step 1: Compute the ratio A / P. A / P = 12,321 / 10,000 = 1.2321. Step 2: Relate this to the rate. (1 + r/100)^2 = 1.2321. Step 3: Take the square root. 1 + r/100 = √1.2321 = 1.11 (since 1.11 * 1.11 = 1.2321). Step 4: Solve for r. r/100 = 1.11 − 1 = 0.11, so r = 11%. Verification / Alternative check: Check the result by recomputing the amount at 11%. After 2 years at 11% compound interest: A = 10,000 * (1.11)^2 = 10,000 * 1.2321 = 12,321. This matches the given final amount exactly, confirming that the rate must be 11% per annum compounded annually. Why Other Options Are Wrong: A rate of 7% or 15% would lead to much smaller or larger final amounts than 12,321, while 22% would grow the principal far too quickly. Only 11% results in the exact given amount, so all other options are mathematically inconsistent with the data. Common Pitfalls: Students sometimes forget that the exponent t must be handled properly and attempt to treat the growth as simple interest. Another mistake is to not recognize convenient squares like 1.2321, which is 1.11 squared, and instead approximate poorly. When the numbers look neat, always check whether they match a known power such as (1.1)^2 or (1.11)^2. Final Answer: The annual compound interest rate is 11 percent.

Question 15

For a certain principal invested at compound interest, the amounts received at the end of the 2nd year and at the end of the 3rd year are Rs. 2100 and Rs. 2268 respectively. What is the annual rate of interest (in percent), assuming yearly compoundi…

Accepted Answer

Introduction / Context: Here we work with successive amounts under compound interest. Instead of being given principal and rate directly, we are told the total amounts at the end of the 2nd and 3rd years. By using the relationship between these amounts, we can find the annual compound interest rate without knowing the principal explicitly. Given Data / Assumptions: Amount at end of 2nd year, A2 = Rs. 2100 Amount at end of 3rd year, A3 = Rs. 2268 Interest is compounded annually at a constant rate r% Principal P is the same throughout but is not directly needed We must find r in percent Concept / Approach: Under annual compounding at rate r%, the amount each year is multiplied by a factor (1 + r/100). Therefore, successive amounts are related as: A3 = A2 * (1 + r/100). Taking the ratio A3 / A2 gives us the single-year growth factor directly. From that factor we can find r. This is often easier than working back to the principal first. Step-by-Step Solution: Step 1: Compute the ratio of the two amounts. A3 / A2 = 2268 / 2100. This simplifies to 1.08. Step 2: Relate this to the annual growth factor. A3 = A2 * (1 + r/100) ⇒ 1 + r/100 = 1.08. Step 3: Solve for r. r/100 = 1.08 − 1 = 0.08. Therefore, r = 8% per annum. Verification / Alternative check: We can check the consistency. If the amount at the end of year 2 is Rs. 2100 and grows at 8%, then: A3 = 2100 * 1.08 = 2268. This matches the given third-year amount exactly, confirming the rate. The value of the principal P does not need to be calculated for this question, which is one of the advantages of using ratios in compound interest problems. Why Other Options Are Wrong: A rate of 7% would give A3 = 2100 * 1.07 = 2247, not 2268. At 9%, A3 would be 2100 * 1.09 = 2289. At 10%, the amount would be even higher. None of these match 2268, so they are incorrect. Common Pitfalls: A common error is to mistakenly treat 2100 and 2268 as interests instead of total amounts, which complicates the analysis unnecessarily. Another pitfall is to attempt to compute the principal first when it is not required, leading to algebraic mistakes. Always look for simple ratios between successive amounts when dealing with compound interest over consecutive years. Final Answer: The rate of interest is 8% per annum.

Question 16

A bank offers 15% compound interest per half-year (that is, 15% every 6 months). A customer deposits Rs. 7200 on 1st January and another Rs. 7200 on 1st July of the same year. Assuming interest is compounded half-yearly at 15% per half-year, what to…

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Introduction / Context: This problem involves compound interest with multiple deposits made at different times in the year. The bank pays 15% interest every half-year, so each 6-month period has its own compounding. We must treat each deposit separately, count how many half-year periods it earns interest for, and then sum the interests. Given Data / Assumptions: Interest rate = 15% per half-year (every 6 months) First deposit: Rs. 7200 on 1st January Second deposit: Rs. 7200 on 1st July We are asked for total interest earned by 31st December of the same year Compounding is done half-yearly, so Jan–Jun and Jul–Dec are the two periods Concept / Approach: We treat each deposit as a separate principal invested for a certain number of half-year periods. The formula for compound amount for n periods at rate i per period is: A = P * (1 + i)^n. Interest for that deposit is A − P. The first deposit earns interest for two half-year periods; the second deposit earns interest for only one half-year period. Total interest is the sum of the interests for both deposits. Step-by-Step Solution: Let i = 15% = 0.15 per half-year. Deposit 1 (1st January): P1 = 7200, n = 2. A1 = 7200 * (1.15)^2 = 7200 * 1.3225 = Rs. 9522 (approximately). Interest on deposit 1, I1 = A1 − P1 = 9522 − 7200 = Rs. 2322. Deposit 2 (1st July): P2 = 7200, n = 1. A2 = 7200 * 1.15 = Rs. 8280. Interest on deposit 2, I2 = A2 − P2 = 8280 − 7200 = Rs. 1080. Total interest I = I1 + I2 = 2322 + 1080 = Rs. 3402. Verification / Alternative check: An alternative is to directly compute the combined amount at the end of the year: A1 + A2 = 9522 + 8280 = 17,802. Total principal deposited is 7200 + 7200 = 14,400. The difference 17,802 − 14,400 = 3,402 rupees. This matches our earlier calculation, confirming the total interest earned. Why Other Options Are Wrong: Rs. 850 and Rs. 1701 are far too small compared with what 15% per half-year would generate on Rs. 14,400 over the year. Rs. 6803 is much too large and would imply unrealistically high effective rates or extended compounding periods. Only Rs. 3402 corresponds accurately to the correct compound interest calculation for the two deposits. Common Pitfalls: Learners often mistakenly treat the entire Rs. 14,400 as if it were deposited at the start of the year, ignoring the later deposit. Others forget that there are two separate half-year periods and that the first deposit earns interest for both, while the second earns for only one. Keeping a clear timeline of deposits and compounding periods helps avoid these errors. Final Answer: The customer earns a total interest of Rs. 3402 by the end of the year.

Question 17

What is the compound interest earned on Rs. 1000 at 10% per annum for 3 years, if the interest is compounded annually?

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Introduction / Context: This is a classic compound interest calculation. We are given the principal, the annual interest rate and the time in years. We must compute the compound interest, not the final amount. Many candidates confuse these two, so it is important to distinguish the interest portion from the total amount accumulated. Given Data / Assumptions: Principal P = Rs. 1000 Rate of interest r = 10% per annum Time t = 3 years Compounding frequency = annually (once a year) We are asked for compound interest (CI), not for the full amount A Concept / Approach: For annual compounding, the amount after t years is: A = P * (1 + r/100)^t. The compound interest is: CI = A − P. We will compute the amount using the factor (1.10)^3, then subtract the principal to obtain the interest earned over 3 years. Step-by-Step Solution: Step 1: Compute the growth factor. 1 + r/100 = 1 + 10/100 = 1.10. Step 2: Raise it to the power of 3 (for 3 years). (1.10)^3 = 1.1 * 1.1 * 1.1 = 1.331. Step 3: Find the amount A. A = 1000 * 1.331 = Rs. 1331. Step 4: Find the compound interest. CI = A − P = 1331 − 1000 = Rs. 331. Verification / Alternative check: We can calculate year by year. Year 1: interest = 1000 * 10% = 100; amount = 1100. Year 2: interest = 1100 * 10% = 110; amount = 1210. Year 3: interest = 1210 * 10% = 121; amount = 1331. Total interest over three years = 100 + 110 + 121 = 331, which confirms our earlier result. Why Other Options Are Wrong: Rs. 1331 is the final amount, not the interest. Rs. 300 would be the simple interest for 3 years (1000 * 10% * 3 / 100), not the compound interest. Rs. 1300 is close to the final amount but not correct for either CI or A. Only Rs. 331 corresponds to the actual compound interest earned. Common Pitfalls: The most common mistake is to give the total amount instead of the interest, or to use the simple interest formula when the question clearly states compound interest. Another error is to forget that each year's interest is calculated on the increased amount, not the original principal alone. Always check whether the question is asking for CI or for the final accumulated amount. Final Answer: The compound interest earned is Rs. 331.

Question 18

The amount received on an investment at 10% compound interest per annum after 3 years is Rs. 10,648. Assuming annual compounding, what was the principal sum (in rupees) originally invested?

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Introduction / Context: In this problem we work backwards from a known maturity amount under compound interest to determine the original principal. The interest rate and time are given, and we know that interest is compounded annually at a fixed rate of 10% per annum. Given Data / Assumptions: Final amount A = Rs. 10,648 Rate of interest r = 10% per annum Time t = 3 years Compounding is annual We must find the principal P Concept / Approach: For annual compounding, the relationship between amount and principal is: A = P * (1 + r/100)^t. Here, r = 10 and t = 3, so: A = P * (1.10)^3. We rearrange this formula to solve for P by dividing the amount by the growth factor (1.10)^3. Step-by-Step Solution: Step 1: Compute the growth factor for 3 years. (1.10)^3 = 1.1 * 1.1 * 1.1 = 1.331. Step 2: Use A = P * 1.331. 10,648 = P * 1.331. Step 3: Solve for P. P = 10,648 / 1.331 = 8000. Verification / Alternative check: Check by recomputing the amount from P = Rs. 8000. After 3 years at 10% compounded annually: A = 8000 * (1.10)^3 = 8000 * 1.331 = 10,648. This matches the given final amount, confirming that the original principal must indeed have been Rs. 8000. Why Other Options Are Wrong: If P were Rs. 9000, then A = 9000 * 1.331 = 11,979, which is too high. For Rs. 8500, A would be 11,303.5; for Rs. 7500, A would be 9,982.5, which is too low. Only Rs. 8000 yields exactly the given maturity amount. Common Pitfalls: Students sometimes forget that the exponent must match the number of years and may accidentally multiply the rate by time instead of raising the factor to a power, effectively using simple interest instead of compound interest. Another mistake is rounding the growth factor too early, which can slightly distort the final answer. When the numbers are chosen cleanly, as here, the exact factor (1.331) leads to an integer principal. Final Answer: The principal originally invested was Rs. 8000.

Question 19

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

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Introduction / Context: This question compares simple interest (SI) and compound interest (CI) for the same principal, rate and time. Over two years, CI exceeds SI by a small amount that depends on the square of the rate. Using a known shortcut formula for the difference between CI and SI for 2 years, we can quickly find the principal without computing both interests separately. Given Data / Assumptions: Difference (CI − SI) over 2 years = Rs. 41 Rate of interest r = 5% per annum Time t = 2 years Interest is compounded annually for CI Principal P is unknown Concept / Approach: For 2 years at r% per annum, the difference between compound interest and simple interest on principal P is given by: CI − SI = P * (r^2 / 100^2). This comes from the fact that compound interest in 2 years equals simple interest in 2 years plus interest on the interest of the first year. Once we plug in the known values of r and CI − SI, we can solve directly for P. Step-by-Step Solution: Step 1: Write the formula. CI − SI = P * r^2 / 100^2. Step 2: Substitute r = 5 and CI − SI = 41. 41 = P * 5^2 / 10,000. 5^2 = 25, so 41 = P * 25 / 10,000. Step 3: Solve for P. P = 41 * 10,000 / 25. P = 410,000 / 25 = 16,400. Verification / Alternative check: We can verify by actually computing SI and CI. Simple interest for 2 years at 5%: SI = P * r * t / 100 = 16,400 * 5 * 2 / 100 = 1,640. Amount under CI after 2 years: A = 16,400 * (1.05)^2 = 16,400 * 1.1025 = 18,081. CI = A − P = 18,081 − 16,400 = 1,681. Difference CI − SI = 1,681 − 1,640 = 41, matching the given value exactly. Why Other Options Are Wrong: For P = 7,200, difference would be 7,200 * 25 / 10,000 = 18, not 41. For 9,600, difference is 24; for 8,400 it is 21. None of these match 41, so those principals cannot be correct. Common Pitfalls: Some learners try to separately calculate CI and SI with an unknown principal, leading to two equations and more algebra than necessary. Others confuse the formula for the 2-year difference and forget to square the rate. Remembering the compact relationship CI − SI = P * r^2 / 100^2 for 2 years makes such problems much faster and less error-prone. Final Answer: The principal (sum of money) is Rs. 16,400.

Question 20

For a certain principal invested at compound interest, the interest earned in the 3rd year is Rs. 12,100 at a fixed annual rate of 9% compounded yearly. What will be the compound interest (in rupees) earned in the 4th year on the same principal at t…

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Introduction / Context: This question uses the relationship between successive yearly interests under compound interest. When a fixed rate is applied and interest is compounded annually, the interest for each year itself grows by the same factor (1 + r) from year to year. We are given the interest in the 3rd year and asked to find the interest in the 4th year. Given Data / Assumptions: Compound interest in 3rd year, I3 = Rs. 12,100 Annual rate of interest r = 9% per annum Interest is compounded once per year We must find compound interest in the 4th year, I4 Concept / Approach: Under annual compounding at a constant rate r, each year's interest equals the previous year's interest multiplied by (1 + r/100). This is because the principal grows every year, and the new interest is computed on that increased principal. Therefore: I4 = I3 * (1 + r/100). We simply multiply the 3rd year interest by (1.09) to get the 4th year interest. Step-by-Step Solution: Step 1: Identify the growth factor for interest. 1 + r/100 = 1 + 9/100 = 1.09. Step 2: Use the relationship between successive year interests. I4 = I3 * 1.09. I4 = 12,100 * 1.09. I4 = 12,100 + (12,100 * 0.09) = 12,100 + 1,089 = 13,189. Verification / Alternative check: We can cross-check by observing that if the principal at the start of the 3rd year is some value P3, then: I3 = P3 * 0.09 = 12,100 ⇒ P3 = 12,100 / 0.09. At the start of the 4th year, the principal becomes P4 = P3 + I3 = P3 * 1.09. The interest in the 4th year would be P4 * 0.09 = P3 * 1.09 * 0.09 = I3 * 1.09, which matches our earlier formula and result. This confirms that multiplying by 1.09 is correct. Why Other Options Are Wrong: Values like Rs. 14,376 or Rs. 15,669 would correspond to much higher effective growth than 9% between the 3rd and 4th years. Rs. 17,080 is even more unrealistic. Only Rs. 13,189 maintains the correct 9% growth from the 3rd-year interest of Rs. 12,100. Common Pitfalls: Some candidates mistakenly add a flat 9% of the original principal rather than 9% of the increased principal, thereby ignoring the compounding effect. Others attempt to reconstruct the entire principal from scratch, which is unnecessary here. Recognizing that yearly interests under compound interest form a geometric progression can save considerable time. Final Answer: The compound interest in the 4th year will be Rs. 13,189.

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