Question 1

The compound interest on Rs 30,000 at 7% per annum is Rs 4,347 for a certain period. For how many years was the money invested at this rate?

Accepted Answer

Introduction: This question asks you to determine the time period for a compound interest investment when the principal, rate, and total compound interest are known. Being able to reverse engineer time is very important in aptitude tests involving financial mathematics. Given Data / Assumptions: Principal P = Rs 30,000 Compound interest CI = Rs 4,347 Annual interest rate r = 7% per annum Compounded annually We need to find number of years n Concept / Approach: First, compute the final amount A using: A = P + CIThen apply the compound interest formula: A = P * (1 + r)^nRearrange to get: (1 + r)^n = A / PWe then look for n such that the power of (1 + r) matches this ratio. Often n turns out to be a small integer in exam problems. Step-by-Step Solution: Step 1: Compute the final amount.A = P + CI = 30,000 + 4,347 = Rs 34,347Step 2: Use the compound interest formula.A = P * (1 + r)^n34,347 = 30,000 * (1 + 0.07)^n34,347 = 30,000 * (1.07)^nStep 3: Form the ratio.(1.07)^n = 34,347 / 30,000(1.07)^n ≈ 1.1449Step 4: Compare powers of 1.07.(1.07)^1 = 1.07 (too small)(1.07)^2 = 1.1449 (exact match)Thus, n = 2 years. Verification / Alternative check: Let us verify by forward calculation for 2 years: A = 30,000 * (1.07)^2 = 30,000 * 1.1449 = Rs 34,347CI = A - P = 34,347 - 30,000 = Rs 4,347Since this matches the given compound interest, the period is confirmed as 2 years. Why Other Options Are Wrong: 3 years or more: (1.07)^3 and higher powers would give an amount significantly larger than Rs 34,347. 6 or 5 years: These would result in a much higher factor than 1.1449, creating a much bigger compound interest than Rs 4,347. Common Pitfalls: Some students try to assume simple interest and use linear relationships, which does not work for compound interest. Others do not compute the ratio correctly, or they round intermediate values too aggressively, leading to a wrong n. Always calculate A / P carefully and compare with standard powers of (1 + r). Final Answer: The money was invested for 2 years at 7% compound interest per annum.

Question 2

A certain amount is invested at compound interest. You are asked: "What is the rate of compound interest per annum?" I. The principal was invested for 4 years. II. The compound interest earned over that period was Rs 1,491. Based on these two statem…

Accepted Answer

Introduction: This is a classic data sufficiency style question about compound interest. Rather than asking you to compute a numerical value directly, it tests whether the given pieces of information are enough to determine the unknown rate of compound interest. Such problems are common in management and banking entrance exams. Given Data / Assumptions: We have an unknown principal P invested at compound interest. We need to determine the rate of compound interest per annum. Statement I: The principal was invested for 4 years. Statement II: The total compound interest earned over the investment period was Rs 1,491. No information about the value of the principal P itself is provided. Concept / Approach: For compound interest, the amount after n years is given by: A = P * (1 + r)^nThe compound interest is: CI = A - PTo find the rate r, we must know both the principal P and either the final amount A or the interest CI for a specified time period n. If P is unknown, we get only one equation with two unknowns, which is not enough to solve uniquely. Step-by-Step Analysis of Statements: Step 1: Consider Statement I alone.I: Time n = 4 years.No information about CI or P is given. With only time, there is no way to compute r.Statement I alone is not sufficient.Step 2: Consider Statement II alone.II: CI = Rs 1,491 over the entire period.We still do not know P or the time. Without at least one of these, CI alone does not allow us to compute r.Statement II alone is not sufficient.Step 3: Combine Statements I and II.From I and II together, we know:Time n = 4 years, CI = Rs 1,491But both P and r are unknown. Using the formula:CI = P * ((1 + r)^4 - 1)This is one equation with two unknowns (P and r). Infinitely many combinations of P and r can satisfy this equation, so r cannot be determined uniquely. Verification / Alternative check: Assume two different principal values, say P1 and P2, and see that you can adjust r1 and r2 respectively to produce the same CI of Rs 1,491 over 4 years. This shows that the given information does not pin down one unique rate, confirming that the data is insufficient even together. Why Other Options Are Wrong: Option a: Time alone without CI or P cannot provide r. Option b: CI alone without P and time cannot provide r. Option c: Even combining both statements leaves two unknowns in one equation. Option d: Clearly false because neither statement alone is sufficient. Common Pitfalls: Many students rush to apply formulas without checking how many unknowns are involved. In data sufficiency questions, always check whether you have enough independent equations to solve for all unknowns uniquely. If not, the data is insufficient, even if it feels like a lot of information is given. Final Answer: The correct conclusion is that even both statements I and II together are not sufficient to determine the rate of compound interest.

Question 3

If the simple interest on a certain sum of money for 2 years at 5% per annum is Rs 50, then what is the compound interest on the same principal at the same rate for the same time?

Accepted Answer

Introduction: This question connects simple interest and compound interest for the same principal, rate, and time period. It first gives simple interest and then asks for compound interest, which is a very common pattern in competitive exams to test understanding of both concepts together. Given Data / Assumptions: Simple interest SI = Rs 50 Time t = 2 years Rate r = 5% per annum Same principal, rate, and time are used for both SI and CI We need compound interest CI for 2 years at 5% Concept / Approach: First, find the principal from the simple interest formula: SI = P * r * t / 100Then use the compound interest amount formula: A = P * (1 + r/100)^tand compute CI as: CI = A - P Step-by-Step Solution: Step 1: Use simple interest to find principal.SI = P * r * t / 10050 = P * 5 * 2 / 10050 = P * 10 / 10050 = 0.10 * PP = 50 / 0.10 = Rs 500Step 2: Compute amount with compound interest.r = 5% per annum, t = 2 yearsA = 500 * (1 + 5 / 100)^2A = 500 * (1.05)^2(1.05)^2 = 1.1025A = 500 * 1.1025 = Rs 551.25Step 3: Find compound interest.CI = A - P = 551.25 - 500 = Rs 51.25 Verification / Alternative check: We can also compare SI and CI. For two years, SI is Rs 50, which is Rs 25 per year. With CI, the second year earns interest on principal plus first year interest. That extra interest is about 5% of Rs 25, which is Rs 1.25. So CI should be Rs 50 + Rs 1.25 = Rs 51.25, matching our calculation. Why Other Options Are Wrong: Rs 61.25: Overestimates the compound interest by adding more than the reasonable extra interest. Rs 51: Underestimates slightly and does not account fully for compounding. Rs 42: Too low; even simple interest is Rs 50. Rs 55: Does not match the precise compounding calculation. Common Pitfalls: Students may forget to convert the rate to decimal form or miscalculate (1.05)^2. Others might wrongly use the simple interest formula for compound interest. Always remember that compound interest for two years is slightly higher than simple interest due to interest on interest in the second year. Final Answer: The compound interest on the sum for 2 years at 5% per annum is Rs 51.25.

Question 4

The difference between the simple interest and the compound interest on Rs 1,200 for 1 year at 10% per annum, when interest is reckoned half yearly, is equal to how many rupees?

Accepted Answer

Introduction: This question compares simple interest and compound interest when interest is reckoned half yearly. It tests your ability to handle different compounding frequencies and then compute the numerical difference between the two types of interest for the same principal, rate, and time. Given Data / Assumptions: Principal P = Rs 1,200 Annual rate r = 10% per annum Time t = 1 year Simple interest is calculated yearly at 10% Compound interest is calculated with half yearly compounding at the same nominal rate Concept / Approach: Simple interest uses the formula: SI = P * r * t / 100For half yearly compound interest, the rate per half year is r / 2 and the number of periods is 2 in one year. The amount is: A = P * (1 + r_half/100)^2The compound interest is A - P. The required answer is CI - SI. Step-by-Step Solution: Step 1: Compute simple interest for 1 year at 10%.SI = 1,200 * 10 * 1 / 100SI = 1,200 * 0.10 = Rs 120Step 2: Compute compound interest with half yearly compounding.Rate per half year = 10 / 2 = 5% = 0.05Number of half years n = 2A = 1,200 * (1 + 0.05)^2(1.05)^2 = 1.1025A = 1,200 * 1.1025 = Rs 1,323CI = A - P = 1,323 - 1,200 = Rs 123Step 3: Find the difference between CI and SI.Difference = CI - SI = 123 - 120 = Rs 3 Verification / Alternative check: The extra amount due to half yearly compounding, compared to yearly simple interest, should be small because the rate and time are not large. A difference of Rs 3 on Rs 1,200 is 0.25%, which is reasonable for the compounding effect over one year. Why Other Options Are Wrong: Rs 2, Rs 4, Rs 5, Rs 6: These values do not match the exact computation based on the correct formulas. They may arise from incorrect exponent, wrong rate per period, or arithmetic mistakes. Common Pitfalls: Candidates may mistakenly apply 10% twice instead of splitting into 5% per half year, or they might forget to subtract principal to find CI. Also, some simply compare interest amounts without respecting the compounding periods, which leads to incorrect differences. Final Answer: The difference between compound interest and simple interest in this case is Rs 3.

Question 5

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

Accepted Answer

Introduction: This problem asks you to find the principal when you are given the difference between compound interest and simple interest for 2 years at the same rate. This is a standard shortcut type question where a known formula for the difference between CI and SI over 2 years can be used. Given Data / Assumptions: Difference between CI and SI for 2 years = Rs 41 Annual rate of interest r = 5% per annum Time t = 2 years We need to find the principal P Concept / Approach: For 2 years at rate r per annum, the difference between compound interest and simple interest on principal P is given by: Difference = P * (r/100)^2This comes from the extra interest earned in compound interest due to interest on interest in the second year. Using this direct formula allows quick calculation of the principal. Step-by-Step Solution: Step 1: Write the formula for difference.Difference = P * (r/100)^2Here, Difference = 41, r = 5%Step 2: Substitute the values.41 = P * (5/100)^2(5/100)^2 = 25 / 10,000 = 1 / 400So 41 = P * (1 / 400)Step 3: Solve for P.P = 41 * 400P = Rs 16,400 Verification / Alternative check: We can check quickly by recomputing the difference. With P = Rs 16,400 and r = 5%, Difference = 16,400 * (5/100)^2Difference = 16,400 * 1/400 = Rs 41This exactly matches the given difference, confirming that the principal is correct. Why Other Options Are Wrong: Rs 7,200, Rs 9,600, Rs 8,400, Rs 12,300: For each of these values, P * (5/100)^2 would not equal Rs 41. They are distractors that reflect typical calculation mistakes or wrong use of the formula. Common Pitfalls: Some students try to compute SI and CI separately using long methods instead of using the direct formula. Others misremember the formula, sometimes writing P * r^2 / 200 instead of P * (r^2 / 10,000). Always remember that (r/100)^2 is used for 2 years when comparing CI and SI. Final Answer: The principal amount (sum of money) is Rs 16,400.

Question 6

An amount of money was lent for 3 years at a certain rate of interest. What will be the difference between the simple interest and the compound interest earned on it at the same rate and for the same period? I. The rate of interest was 8% per annum.…

Accepted Answer

Introduction: This question is a data sufficiency problem related to the difference between simple and compound interest over 3 years. Instead of computing the numeric answer directly, you must decide whether the given statements provide enough information to determine that difference uniquely. Given Data / Assumptions: We have some unknown principal P lent for 3 years. We want the difference between CI and SI after 3 years at the same rate. Statement I: Rate of interest r = 8% per annum. Statement II: Total simple interest for 3 years = Rs 1,200. Concept / Approach: For time t = 3 years and rate r, simple interest is: SI = P * r * t / 100Compound interest over 3 years is related to P and r by: CI = P * ((1 + r/100)^3 - 1)The difference between CI and SI is a function of both P and r. To determine this difference, we must know both P and r. Therefore, we will analyze whether the statements together or individually give us both these values. Step-by-Step Analysis of Statements: Step 1: Statement I alone.Statement I gives only r = 8% per annum.Principal P is unknown, so we cannot compute SI or CI numerically, hence the difference cannot be found.Statement I alone is not sufficient.Step 2: Statement II alone.Statement II gives SI = Rs 1,200 for 3 years.We know the product P * r but not their individual values, because:1,200 = P * r * 3 / 100So P * r is fixed, but P and r individually are not determined.Statement II alone is not sufficient.Step 3: Combine Statements I and II.From I: r = 8%From II: 1,200 = P * 8 * 3 / 1001,200 = P * 24 / 1001,200 = 0.24 * PSo P = 1,200 / 0.24 = Rs 5,000Now both P and r are known.With P and r known, we can compute SI and CI for 3 years and hence their difference. Therefore, both statements together are sufficient to answer the question. Verification / Alternative check: With P = Rs 5,000 and r = 8%, SI for 3 years is already given as Rs 1,200. CI can be computed using: CI = 5,000 * ((1.08)^3 - 1)The difference CI - SI would be a single definite number, showing that the data is sufficient. Why Other Options Are Wrong: Option a and b: Each statement alone clearly lacks either P or r. Option c and e: They claim that even together the statements are not sufficient, which is incorrect because together they uniquely determine both P and r. Common Pitfalls: Students sometimes think that knowing SI for 3 years is enough to compute CI, forgetting that the rate must also be known. In data sufficiency questions, always identify how many unknowns exist and whether the combined information gives you enough independent equations to solve for those unknowns exactly. Final Answer: Both statements I and II together are necessary and sufficient to determine the required difference between simple interest and compound interest.

Question 7

You deposit 6,500 dollars into an account that pays 8% annual interest compounded monthly. How much money will be in the account (total balance) after 7 years?

Accepted Answer

Introduction: This problem involves compound interest with monthly compounding. It tests your ability to correctly convert an annual nominal rate to a monthly rate, count the total number of months, and apply the compound interest formula over many compounding periods. Given Data / Assumptions: Initial deposit (principal) P = 6,500 dollars Annual interest rate r = 8% per annum Interest is compounded monthly Investment period t = 7 years We need the final account balance after 7 years Concept / Approach: For monthly compounding, the periodic rate is the annual rate divided by 12, and the number of periods is 12 times the number of years. The compound amount is found by: A = P * (1 + r_monthly)^nwhere: r_monthly = r_annual / 12n = 12 * t Step-by-Step Solution: Step 1: Convert annual rate to monthly rate.r_annual = 8% = 0.08r_monthly = 0.08 / 12 ≈ 0.0066667Step 2: Compute number of compounding periods.t = 7 years, so n = 12 * 7 = 84 monthsStep 3: Apply the compound interest formula.A = 6,500 * (1 + 0.08 / 12)^84A ≈ 6,500 * (1.0066667)^84Using accurate calculation, A ≈ 11,358.24 dollars Verification / Alternative check: We can approximate the effective annual rate. For 8% compounded monthly, the effective annual rate is roughly: (1 + 0.08 / 12)^12 - 1 ≈ 0.083This is about 8.3% per year. Over 7 years, the growth factor is around (1.083)^7 ≈ 1.746, and 6,500 * 1.746 ≈ 11,349, which is close to 11,358.24, confirming our detailed computation. Why Other Options Are Wrong: 12,334 and 12,386 dollars: These exceed the correct amount and correspond to higher assumed effective rates. 15,789 dollars: Much too high, more than doubling the money at 8% over 7 years. 10,950 dollars: Too low, underestimates the impact of monthly compounding. Common Pitfalls: Some students mistakenly use 8% divided by 7 years, or they use annual compounding instead of monthly compounding. Others forget to multiply years by 12 to find the number of monthly periods. Always ensure that rate and time are expressed in consistent units when using the compound interest formula. Final Answer: The account balance after 7 years will be approximately 11,358.24 dollars.

Question 8

How much money must you deposit today in an account that pays 9% annual interest compounded monthly in order to have 12,000 dollars in the account after 6 years?

Accepted Answer

Introduction: This is a present value problem under compound interest with monthly compounding. Instead of asking for the final amount from a known principal, it asks you to find how much needs to be deposited today to reach a target future value. Such questions are very important in financial planning and time value of money topics. Given Data / Assumptions: Future value (target amount) A = 12,000 dollars Annual nominal interest rate r = 9% per annum Interest is compounded monthly Investment period t = 6 years We must find present value or principal P Concept / Approach: For monthly compounding, we use: A = P * (1 + r_monthly)^nwhere: r_monthly = r_annual / 12n = 12 * tTo find present value P, we rearrange the formula: P = A / (1 + r_monthly)^n Step-by-Step Solution: Step 1: Convert annual rate to monthly.r_annual = 9% = 0.09r_monthly = 0.09 / 12 = 0.0075Step 2: Compute the number of months.t = 6 years, so n = 6 * 12 = 72 monthsStep 3: Use the present value formula.P = 12,000 / (1 + 0.0075)^72P = 12,000 / (1.0075)^72With accurate calculation, this gives: P ≈ 7,007.08 dollars Verification / Alternative check: We can quickly check using an approximate effective annual rate. The effective annual rate is about: (1 + 0.09 / 12)^12 - 1 ≈ 0.0938 or 9.38%Over 6 years, the rough growth factor is (1.0938)^6 ≈ 1.712. Dividing 12,000 by 1.712 gives approximately 7,010 dollars, which is close to the more precise value 7,007.08, confirming that our calculation is reasonable. Why Other Options Are Wrong: 9,007 dollars and 8,007 dollars: These are too high as present values and would grow to more than 12,000 at 9% compounded monthly. 4,007 dollars: Far too low, would not grow to the target. 6,950 dollars: Slightly below the correct present value, reflecting a rounding or rate error. Common Pitfalls: Students sometimes mistakenly use annual compounding for a monthly compounding problem, or they forget to convert the rate to a monthly rate and time to months. Another error is to multiply by the future factor instead of dividing by it when finding present value. Always rearrange the compound interest formula carefully when solving for P. Final Answer: You must deposit approximately 7,007.08 dollars today to have 12,000 dollars after 6 years at 9% interest compounded monthly.

Question 9

You deposit 300 dollars in a savings account that pays 4% simple annual interest. What will be your account balance after 9 months?

Accepted Answer

Introduction: This is a straightforward simple interest question involving a time period less than a full year. The aim is to find the final account balance, not only the interest, after 9 months on a 300 dollar deposit at 4% simple annual interest. Given Data / Assumptions: Principal P = 300 dollars Annual simple interest rate r = 4% per annum Time t = 9 months Interest type = simple interest We need final balance after 9 months Concept / Approach: Simple interest is calculated using: SI = P * r * t / 100where t is in years. For 9 months, we convert the time to years by taking t = 9 / 12. The final amount is then: A = P + SI Step-by-Step Solution: Step 1: Convert time to years.t = 9 months = 9 / 12 years = 3 / 4 years = 0.75 yearsStep 2: Compute simple interest.SI = P * r * t / 100SI = 300 * 4 * 0.75 / 100SI = 300 * 3 / 100SI = 900 / 100 = 9 dollarsStep 3: Compute final balance.A = P + SI = 300 + 9 = 309 dollars Verification / Alternative check: Another way is to compute annual interest first and then take three quarters of it. Annual interest at 4% on 300 dollars: Annual interest = 300 * 4 / 100 = 12 dollarsFor 9 months, interest is 3/4 of 12, which is 9 dollars. Adding this to the principal again gives a final balance of 309 dollars. Why Other Options Are Wrong: 409, 509, 609 dollars: These assume very large interest or compound growth not supported by simple interest at 4%. 315 dollars: Would correspond to an interest of 15 dollars, which is more than the correct annual interest for 12 months. Common Pitfalls: Students may forget to convert 9 months into years or may mistakenly use 9 as if it were years in the formula. Some also confuse simple and compound interest and unnecessarily apply powers, which is not required here. Final Answer: The account balance after 9 months will be 309 dollars.

Question 10

Simon deposits 400 dollars in an account that pays 3% interest compounded annually. What will be the balance of Simon's account at the end of 2 years?

Accepted Answer

Introduction: This question deals with compound interest on a small deposit over a short period. It tests basic understanding of the compound interest formula with annual compounding and how to compute the final account balance after a given number of years. Given Data / Assumptions: Principal P = 400 dollars Annual interest rate r = 3% per annum Compounding frequency = annually Time t = 2 years We need the final balance after 2 years Concept / Approach: For annual compounding, the amount after t years is: A = P * (1 + r/100)^tHere, r = 3 and t = 2. We will substitute and compute the result, then compare with the options to identify the correct choice. Step-by-Step Solution: Step 1: Convert rate to decimal form.r = 3% per annum, so 1 + r/100 = 1 + 0.03 = 1.03Step 2: Use the compound interest formula.A = 400 * (1.03)^2(1.03)^2 = 1.0609Step 3: Compute the final amount.A = 400 * 1.0609 = 424.36 dollarsThus, the balance in Simon's account after 2 years is 424.36 dollars. Verification / Alternative check: We can verify this year by year: End of year 1: 400 + 3% of 400 = 400 + 12 = 412 dollarsEnd of year 2: 412 + 3% of 412 = 412 + 12.36 = 424.36 dollarsThis matches our previous calculation using the formula, so the answer is confirmed. Why Other Options Are Wrong: 524.56 and 545.36 dollars: These values are too high and imply a much higher rate or longer time. 456.36 dollars: Slightly larger than the correct value, likely from miscalculating interest in the second year. 430 dollars: Rounded or estimated value that does not match the precise computation. Common Pitfalls: Students may treat the problem as simple interest and add the same 12 dollars each year without considering that the second year interest is calculated on the increased amount. Another mistake is incorrect squaring of 1.03. Always perform the compounding accurately. Final Answer: The balance of Simon's account at the end of 2 years is 424.36 dollars.

Question 11

Jackie deposits 325 dollars in an account that pays 4.1% interest compounded annually. How much money will Jackie have in her account after 3 years?

Accepted Answer

Introduction: This question is another example of compound interest with annual compounding, but with a rate that includes a decimal (4.1%). It demonstrates how to handle non integer rates and exponentiation over 3 years to find the final account balance. Given Data / Assumptions: Principal P = 325 dollars Annual interest rate r = 4.1% per annum Compounded annually Time t = 3 years We need the total amount after 3 years Concept / Approach: With annual compounding, we use: A = P * (1 + r/100)^tHere, r = 4.1 and t = 3. The interest is added once at the end of each year, and the amount grows multiplicatively. Step-by-Step Solution: Step 1: Convert the rate to growth factor.1 + r/100 = 1 + 4.1 / 100 = 1 + 0.041 = 1.041Step 2: Compute the cube of 1.041.(1.041)^3 ≈ 1.128275Step 3: Apply the formula.A = 325 * (1.041)^3A ≈ 325 * 1.128275 ≈ 366.64 dollarsThus, Jackie will have about 366.64 dollars after 3 years. Verification / Alternative check: We can approximate year by year: End of year 1: 325 + 4.1% of 325 ≈ 325 + 13.33 = 338.33 dollarsEnd of year 2: 338.33 + 4.1% of 338.33 ≈ 338.33 + 13.87 ≈ 352.20 dollarsEnd of year 3: 352.20 + 4.1% of 352.20 ≈ 352.20 + 14.44 ≈ 366.64 dollarsThis sequential calculation matches the formula based result, confirming the answer. Why Other Options Are Wrong: 346.64 and 356.64 dollars: These values are lower than the correct compound amount and reflect underestimation of interest in later years. 376.64 dollars: Slightly higher than correct, from over rounding or increasing interest at each step too much. 360 dollars: Rough estimate without exact compounding. Common Pitfalls: Errors often come from incorrect exponentiation of 1.041 or using simple interest instead of compounding. Some students mistakenly multiply 4.1% by 3 and add that, which would give only simple interest and a different result. Final Answer: Jackie will have approximately 366.64 dollars in her account after 3 years.

Question 12

At what annual compound interest rate will an investment of 20,000 dollars grow to 100,000 dollars in 15 years?

Accepted Answer

Introduction: This is a growth rate problem using compound interest over a long period. You are given the initial amount, final amount, and time, and asked to determine the annual interest rate that achieves this growth. Such questions are closely related to exponential growth and logarithms, and they help in understanding long term investment planning. Given Data / Assumptions: Principal P = 20,000 dollars Final amount A = 100,000 dollars Time t = 15 years Interest compounded annually at rate r We need to find r (per annum in percent) Concept / Approach: The compound interest amount formula is: A = P * (1 + r)^tHere, A / P is the overall growth factor. We rearrange to solve for r: (1 + r)^t = A / P1 + r = (A / P)^(1 / t)r = (A / P)^(1 / t) - 1We will then convert r to a percentage. Step-by-Step Solution: Step 1: Compute the growth factor.A / P = 100,000 / 20,000 = 5So (1 + r)^15 = 5Step 2: Solve for 1 + r.1 + r = 5^(1 / 15)Using calculation, 5^(1 / 15) is approximately 1.1133. Step 3: Find r.r ≈ 1.1133 - 1 = 0.1133r ≈ 11.33% Verification / Alternative check: We can validate by forward computation with r = 11.33% as an approximate rate: Growth factor per year ≈ 1.1133Over 15 years, (1.1133)^15 ≈ 5Thus, 20,000 * 5 ≈ 100,000 dollars, which matches the required final amount. The small rounding in r ensures that the final value is extremely close to 100,000. Why Other Options Are Wrong: 9.33% and 10.33%: These lower rates would produce smaller growth factors. Over 15 years, the final amount would be less than 100,000. 12.33%: This higher rate would make the investment grow to significantly more than 100,000 in 15 years. 8.33%: Far too small and would not grow the investment by a factor of 5. Common Pitfalls: Many learners attempt to use simple interest logic or try to average rates instead of using exponential relationships. Another challenge is handling the 15th root correctly; rough estimation or use of logarithms is required. In exam settings, options are usually spaced such that a reasonable approximation, like 11.33%, can be identified even with limited calculations. Final Answer: The required annual compound interest rate is approximately 11.33% per annum.

Question 13

Calculate the effective annual interest rate corresponding to a nominal annual rate of 13% when interest is compounded quarterly.

Accepted Answer

Introduction: This question asks for the effective annual interest rate when a nominal rate is compounded quarterly. This is a core idea in finance: nominal rates are quoted, but effective rates show the real growth over a year after accounting for compounding frequency. Given Data / Assumptions: Nominal annual interest rate r_nominal = 13% per annum Compounding frequency = quarterly (4 times per year) We need the effective annual rate r_effective, which gives the same annual growth with one compounding per year. Concept / Approach: The effective annual rate for a nominal rate r_nominal compounded m times per year is given by: r_effective = (1 + r_nominal / m)^m - 1Here r_nominal must be expressed as a decimal. For quarterly compounding, m = 4. Finally, convert r_effective back to percentage form. Step-by-Step Solution: Step 1: Convert nominal rate to decimal.r_nominal = 13% = 0.13Step 2: Identify compounding periods.m = 4 (quarterly)Step 3: Apply the effective rate formula.r_effective = (1 + 0.13 / 4)^4 - 1r_effective = (1 + 0.0325)^4 - 1r_effective = (1.0325)^4 - 1Calculating (1.0325)^4 gives approximately 1.1365. r_effective ≈ 1.1365 - 1 = 0.1365r_effective ≈ 13.65% Verification / Alternative check: A rough approximation method is to note that the effective rate will be slightly higher than 13% due to compounding. Since 0.13 / 4 = 3.25% per quarter, the total simple interest equivalent would be 13%, and compounding this slightly increases it to about 13.6%. This checks well with our more exact computation of 13.65%. Why Other Options Are Wrong: 14.665%, 15.65%, 16.65%: These are significantly higher than the correct effective rate and would correspond to larger growth than 13% nominal quarterly. 13.00%: This would be the rate with no compounding effect, which is not the case here. Common Pitfalls: A common mistake is to assume that the effective rate is the same as the nominal rate or to multiply 13% by 4, which is completely incorrect. Some also forget to convert the percentage to decimal before applying the formula. Always remember to use the expression (1 + r/m)^m - 1 for effective rate calculations. Final Answer: The effective annual interest rate corresponding to 13% nominal compounded quarterly is approximately 13.65%.

Question 14

Calculate the effective annual interest rate corresponding to a nominal annual rate of 10% when interest is compounded continuously.

Accepted Answer

Introduction: This question explores the concept of continuous compounding, which is a theoretical limit where interest is added at every instant. You are asked to find the effective annual rate corresponding to a nominal rate of 10% compounded continuously, a concept widely used in higher finance and mathematical modeling. Given Data / Assumptions: Nominal annual interest rate r_nominal = 10% per annum Compounding type = continuously compounded interest We need the effective annual rate r_effective that gives the same growth if compounded once per year. Concept / Approach: For continuous compounding, the amount after time t years is given by: A = P * e^(r_nominal * t)For one year (t = 1), this becomes: A = P * e^(r_nominal)Thus, the effective annual rate is: r_effective = e^(r_nominal) - 1where r_nominal is expressed as a decimal, and e is the base of natural logarithms, approximately 2.71828. Step-by-Step Solution: Step 1: Convert the nominal rate to decimal.r_nominal = 10% = 0.10Step 2: Compute the growth factor for one year.Growth factor = e^(0.10)This value is approximately 1.10517. Step 3: Determine the effective rate.r_effective = 1.10517 - 1 = 0.10517r_effective ≈ 10.52% Verification / Alternative check: We know that for annual compounding at 10%, the effective rate is exactly 10%. For more frequent compounding, such as quarterly or monthly, the effective rate becomes slightly higher than 10%. Continuous compounding is the theoretical maximum frequency, so the effective rate must be somewhat above 10%, and a value near 10.5% fits this pattern well. Why Other Options Are Wrong: 9.52%: This is less than the nominal rate and cannot result from compounding. 11.52% and 12.52%: These are too high for a 10% nominal rate, even with continuous compounding. 10.00%: This corresponds to annual compounding, not continuous compounding. Common Pitfalls: A key mistake is to use the standard discrete compounding formula (1 + r/m)^m for continuous compounding. Another issue is forgetting to convert the percentage to decimal before using the exponential function. Always use the formula A = P * e^(r * t) for continuous compounding, and then subtract 1 from the growth factor to obtain the effective rate. Final Answer: The effective annual interest rate corresponding to a 10% nominal rate with continuous compounding is approximately 10.52%.

Question 15

An amount of Rs. 5,000 is invested at a fixed rate of 8% per annum, compounded annually. What will be the value of this investment after 5 years?

Accepted Answer

Introduction / Context: This aptitude question tests understanding of compound interest, which is a core concept in time value of money and investment planning. You are given a principal amount, an annual interest rate, a compounding pattern, and a time period, and you must compute the future value of the investment after several years. Given Data / Assumptions: Principal amount P = Rs. 5,000. Rate of interest r = 8% per annum. Interest is compounded annually. Time period n = 5 years. No additional deposits or withdrawals during the 5 years. Concept / Approach: For compound interest with annual compounding, the amount at the end of n years is given by: A = P * (1 + r)^n Here r is written in decimal form, so 8% becomes 0.08. The idea is that each year interest is added to the new balance, and then the next year interest is calculated on this increased amount. Step-by-Step Solution: Step 1: Convert the rate to decimal form: r = 8% = 0.08. Step 2: Substitute into the formula A = 5000 * (1 + 0.08)^5. Step 3: Compute the growth factor: 1 + 0.08 = 1.08. Step 4: Raise this to the power 5: 1.08^5 ≈ 1.469328. Step 5: Multiply by the principal: A ≈ 5000 * 1.469328 ≈ 7,346.64. Thus, the investment grows to approximately Rs. 7,346.64 after 5 years. Verification / Alternative Check: You can verify by compounding year by year. After each year multiply the previous balance by 1.08 and continue for 5 steps. The final figure will again be very close to Rs. 7,346.64, confirming the calculation. Why Other Options Are Wrong: Option B (Rs. 6,802.43) is closer to what you get after only 4 years of compounding, not 5 years. Option C (Rs. 7,934.38) is higher than the correct 5 year value and resembles a value for a higher rate or longer period. Option D (Rs. 6,381.41) corresponds to a lower interest rate scenario such as 5% per annum, not 8%. Option E states None of these, which is incorrect because we have a matching correct amount in option A. Common Pitfalls: Learners often confuse simple interest with compound interest and may use A = P + P * r * n, which underestimates the true value. Another error is forgetting to convert the percentage to decimal or raising to the wrong power for the number of years. Final Answer: The value of the investment after 5 years at 8% per annum compounded annually is Rs. 7,346.64.

Question 16

An investment of 13,200 dollars grows to 22,680.06 dollars in 8 years when interest is compounded annually. What is the annual rate of interest?

Accepted Answer

Introduction / Context: This problem checks your ability to work backwards with the compound interest formula. Instead of finding the future value from a known rate, you are given the principal and maturity amount and must determine the unknown annual interest rate that produced this growth. Given Data / Assumptions: Principal P = 13,200 dollars. Future value A = 22,680.06 dollars. Time period n = 8 years. Interest is compounded yearly. We assume a constant rate r throughout the 8 years. Concept / Approach: For annual compounding the relationship between principal, rate, time and amount is: A = P * (1 + r)^n Here A and P are known, and n is known, so we solve for r by rearranging the formula: (1 + r)^n = A / P 1 + r = (A / P)^(1 / n) r = (A / P)^(1 / n) - 1 Step-by-Step Solution: Step 1: Compute the ratio A / P = 22680.06 / 13200 ≈ 1.718186. Step 2: Take the eighth root: (1 + r) = (1.718186)^(1 / 8). Step 3: Evaluating this gives approximately 1.07. Step 4: Subtract 1 to get r ≈ 0.07. Step 5: Convert to a percentage: r ≈ 7% per annum. Thus, a 7% annual interest rate compounded yearly turns 13,200 dollars into about 22,680.06 dollars in 8 years. Verification / Alternative Check: You can plug r = 7% back into the formula: A = 13200 * (1.07)^8 If you compute this value it closely matches 22,680.06 dollars, which confirms that the rate is correct. Why Other Options Are Wrong: At 5% or 6% per annum the growth factor over 8 years is smaller, so the amount would be less than 22,680.06 dollars. At 8% per annum the amount would exceed 22,680.06 dollars because the growth factor would be larger than needed. The option 4% is far too low and would give much less growth than observed. Common Pitfalls: A common mistake is to divide the total interest by eight and treat that as the rate, which corresponds to simple interest logic. Another error is rounding too early when taking the nth root, which can lead to an approximate rate that does not match any option. Always keep adequate decimal places until the final step. Final Answer: The annual interest rate that produced the given compound amount is 7% per annum.

Question 17

An amount of 3,300 dollars is needed in 4 years at an interest rate of 11% per annum compounded annually. What is the total interest earned over the 4 year period?

Accepted Answer

Introduction / Context: This question combines the concepts of present value, future value, and total interest earned under compound interest. You know the amount required in the future and the interest rate, and you must compute how much of that future amount is interest rather than original principal. Given Data / Assumptions: Future value A = 3,300 dollars (amount needed after 4 years). Annual interest rate r = 11% per annum, compounded yearly. Time period n = 4 years. Interest is compounded annually with no additional deposits during the period. Concept / Approach: First find the present value P that will grow to 3,300 dollars at 11% in 4 years using: A = P * (1 + r)^n Rearrange to find P: P = A / (1 + r)^n The total interest earned is then: Interest = A - P Step-by-Step Solution: Step 1: Convert 11% to decimal form: r = 0.11. Step 2: Compute the discount factor (1 + r)^4 = 1.11^4. Step 3: Calculate P = 3300 / (1.11^4) ≈ 2,173.81 dollars. Step 4: Compute interest earned: Interest = 3300 - 2173.81 ≈ 1,126.19 dollars. Therefore the interest earned over 4 years is approximately 1,126.19 dollars. Verification / Alternative Check: You can grow the present value forward to check. Multiply 2,173.81 dollars by 1.11 four times. The result will be very close to 3,300 dollars, which confirms that the calculations for present value and interest are correct. Why Other Options Are Wrong: Option B (1,050.00 dollars) and option D (1,000.00 dollars) underestimate the interest, which would correspond to a lower effective rate or a shorter time period. Option C (1,237.00 dollars) overestimates the interest compared with the correct compound interest computation. Option E is incorrect because one of the numerical options matches the correct interest value precisely. Common Pitfalls: Students may mistakenly calculate simple interest using P * r * n instead of compound interest, or they may treat 3,300 dollars as the principal. It is crucial to distinguish between future value and present value and to remember that interest is the difference between them. Final Answer: The total compound interest earned over the 4 year period is 1,126.19 dollars.

Question 18

Sharon Stone deposits 2,000 dollars at the end of each year into an account that earns 10% interest compounded annually. After 25 years, how much total interest has she earned on her deposits?

Accepted Answer

Introduction / Context: This question involves an ordinary annuity, where equal deposits are made at the end of each year. The goal is to find how much compound interest Sharon earns over 25 years, not just the future value of the account. This tests understanding of annuity formulas and the separation of contributions from interest income. Given Data / Assumptions: Annual deposit (payment) R = 2,000 dollars. Interest rate r = 10% per annum (0.10 in decimal form). Compounding is annual. Number of years n = 25. Deposits are made at the end of each year (ordinary annuity). Concept / Approach: The future value of an ordinary annuity is given by: FV = R * ((1 + r)^n - 1) / r The total contributions made by Sharon are: Total deposits = R * n The total interest earned is then: Interest earned = FV - Total deposits Step-by-Step Solution: Step 1: Use r = 0.10 and n = 25. Step 2: Compute the annuity factor: ((1 + 0.10)^25 - 1) / 0.10. Step 3: Evaluate (1.10)^25 and subtract 1, then divide by 0.10 to obtain the factor, which is approximately 98.347. Step 4: Compute FV = 2000 * factor ≈ 2000 * 98.347 ≈ 196,694.12 dollars. Step 5: Compute total deposits: 2000 * 25 = 50,000 dollars. Step 6: Interest earned = 196,694.12 - 50,000 ≈ 146,694.12 dollars. Thus Sharon earns around 146,694.12 dollars in interest. Verification / Alternative Check: You can confirm by computing the value of the account each year via a spreadsheet: add 2,000 at year end and multiply by 1.10 for growth. After 25 cycles, subtract the total deposits to see that the interest matches the computed value closely. Why Other Options Are Wrong: Option B (96,694.12 dollars) is too low and would correspond to a shorter investment period or lower rate. Option C (120,000.00 dollars) and option D (180,000.00 dollars) do not match the precise annuity calculation; they are rough guesses based on doubling or tripling the principal contributions. Option E is incorrect because the correct interest amount appears as option A. Common Pitfalls: Learners may mistakenly treat this as a single lump sum investment instead of an annuity, or they might forget to subtract the total deposits and give the future value instead of the interest. Another common error is assuming simple interest, which ignores compounding. Final Answer: The total compound interest earned over the 25 years is 146,694.12 dollars.

Question 19

Sharon Stone deposits 500 dollars at the beginning of each 3 month period into an account earning 10% interest compounded quarterly. How much money will she have in the account after 25 years?

Accepted Answer

Introduction / Context: Here we have an annuity due, which means deposits are made at the beginning of each compounding period. The account compounds quarterly at a nominal rate of 10% per year. You must compute the total amount in the account after 25 years, taking into account both the frequent deposits and compound interest. Given Data / Assumptions: Regular deposit R = 500 dollars. Deposits are made every 3 months, that is quarterly. Nominal annual interest rate = 10% compounded quarterly. Quarterly rate i = 10% / 4 = 2.5% = 0.025. Number of years = 25, so number of quarters n = 25 * 4 = 100. Deposits occur at the beginning of each quarter (annuity due). Concept / Approach: The future value of an annuity due is: FV = R * ((1 + i)^n - 1) / i * (1 + i) The factor (1 + i) at the end converts an ordinary annuity to an annuity due because each payment has one extra period to earn interest. Step-by-Step Solution: Step 1: Quarterly rate i = 0.10 / 4 = 0.025. Step 2: Number of periods n = 25 * 4 = 100. Step 3: Compute the ordinary annuity factor: ((1 + 0.025)^100 - 1) / 0.025. Step 4: Multiply by (1 + i) to convert to annuity due. Step 5: Multiply the overall factor by R = 500 dollars. This calculation gives approximately 221,681.19 dollars as the accumulated amount after 25 years. Verification / Alternative Check: As a reasonableness check, the total deposits equal 500 * 100 = 50,000 dollars. Since the interest rate is high and the time period is long, the final amount must be several times larger than 50,000. An amount near 221,000 dollars is therefore plausible, while much smaller numbers would be unrealistic. Why Other Options Are Wrong: Option B (196,694.12 dollars) corresponds roughly to a smaller regular payment or an ordinary annuity rather than an annuity due. Option C (150,000.00 dollars) underestimates the effect of compounding over 25 years. Option D (226,681.19 dollars) is close but higher than the precise computed value and would arise from slightly different parameters. Option E is incorrect because there is a specific correct numerical value among the options. Common Pitfalls: A major mistake is to treat the cash flows as an ordinary annuity and omit the extra (1 + i) factor. Another is to use the annual rate directly without converting to a quarterly rate, which would severely distort the result. Careful attention to compounding frequency and timing of deposits is essential. Final Answer: After 25 years Sharon will have approximately 221,681.19 dollars in the account.

Question 20

What single lump sum deposited today would allow withdrawals of 2,000 dollars per year for 7 years at 5% interest compounded annually?

Accepted Answer

Introduction / Context: This problem is about finding the present value of an ordinary annuity. Instead of knowing the deposit and computing future value, you are given a desired series of withdrawals and an interest rate, and you must find how much money must be set aside today to fund those withdrawals. Given Data / Assumptions: Yearly withdrawal (payment) R = 2,000 dollars. Interest rate r = 5% per annum, compounded annually. Number of years n = 7. Withdrawals occur at the end of each year (ordinary annuity). No additional deposits are made after the initial lump sum. Concept / Approach: The present value of an ordinary annuity is given by: PV = R * (1 - (1 + r)^(-n)) / r This formula discounts each future payment back to the present, summing the series into one equivalent lump sum today. Step-by-Step Solution: Step 1: Convert the rate to decimal form: r = 0.05. Step 2: Plug in values: PV = 2000 * (1 - (1.05)^(-7)) / 0.05. Step 3: Compute (1.05)^(-7). First find (1.05)^7 and then take its reciprocal. Step 4: After evaluation, the factor (1 - (1.05)^(-7)) / 0.05 is about 5.786. Step 5: Multiply by the payment: PV ≈ 2000 * 5.786 ≈ 11,572.75 dollars. So approximately 11,572.75 dollars is required today to support the withdrawals. Verification / Alternative Check: You can verify by taking the found present value and compounding it year by year while subtracting the 2,000 dollar payment at year end. After the seventh withdrawal the remaining balance should be very close to zero, confirming that the calculated present value is accurate. Why Other Options Are Wrong: Options B and D (12,000 and 13,000 dollars) ignore the effect of interest and are simple guesses based on total withdrawals. Option C (10,500 dollars) is too low and would not generate enough interest to cover all payments. Option E (9,800 dollars) is even lower and clearly insufficient. Common Pitfalls: One common error is to multiply the annual payment by the number of years and forget interest altogether. Another is to confuse present value with future value and apply the wrong formula. Always note whether payments are at the beginning or end of periods and match that with the correct annuity expression. Final Answer: The required lump sum deposit today is approximately 11,572.75 dollars.

Compound Interest Questions