Question 1

The compound interest earned in 2 years at 12% per annum is Rs. 10176. What is the sum of money that was originally invested?

Accepted Answer

Introduction / Context: Here, the total compound interest over a two year period is provided, together with the annual rate. The learner must find the original principal. This is similar in structure to an earlier problem but with a different rate, which means the growth factor will change. It tests flexibility in applying the general compound interest framework to recover the principal from the known interest. Given Data / Assumptions: Compound interest CI for 2 years = Rs. 10176. Rate of interest r = 12% per annum. Time period n = 2 years. Interest is compounded annually. Relationship: CI = P * ((1 + r) ^ n - 1). Concept / Approach: We convert the rate into decimal form and compute the growth factor for two years. The generic relation CI = P * ((1 + r) ^ n - 1) allows us to express the given interest as a multiple of the unknown principal. We then solve for P by dividing the known interest by the net growth factor ((1 + r) ^ n - 1). This technique is particularly efficient when only the interest and not the final amount is supplied. Step-by-Step Solution: Step 1: Convert the rate to decimal form: r = 12% = 0.12. Step 2: Compute the two year growth factor: (1 + r) ^ n = (1.12) ^ 2. Step 3: Calculate (1.12) ^ 2 = 1.2544. Step 4: Then (1 + r) ^ n - 1 = 1.2544 - 1 = 0.2544. Step 5: The formula for compound interest is CI = P * 0.2544. Step 6: We are given CI = 10176, so 10176 = P * 0.2544. Step 7: Solve for P: P = 10176 / 0.2544 = 40000. Step 8: Therefore, the original principal is Rs. 40000. Verification / Alternative check: To verify, we compute the amount and the interest using the principal we found. With P = 40000, the amount is A = 40000 * (1.12) ^ 2 = 40000 * 1.2544 = 50176. The compound interest is CI = A - P = 50176 - 40000 = 10176, which matches the given interest exactly. This perfect match confirms that our principal value and calculations are correct and consistent with the data given in the problem. Why Other Options Are Wrong: If P were 50000, the compound interest would be 50000 * 0.2544 = 12720, which is larger than the required 10176. With P = 60000, CI becomes 60000 * 0.2544 = 15264, even further from the target. If P were 80000, the interest would double from the 40000 case to 20352, clearly too high. The option 45000 would give 45000 * 0.2544 = 11448, which is also not equal to 10176. Only a principal of 40000 leads to the specified interest of 10176. Common Pitfalls: A common error is to misinterpret the given interest as the final amount and attempt to subtract the principal from it, which leads to confusion. Another pitfall is to forget to square 1.12 correctly, resulting in an inaccurate growth factor. Some students may try to use the simple interest relation and divide the interest by 2 * 12%, which ignores compounding and gives the wrong principal. Maintaining a clear distinction between amount and interest and using the proper growth factor are essential to solving this kind of problem accurately. Final Answer: The sum invested, which earns Rs. 10176 as compound interest in 2 years at 12% per annum, is Rs. 40000, corresponding to option C.

Question 2

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

Accepted Answer

Introduction / Context: This question links simple interest and compound interest for the same principal, rate, and time. The simple interest value is provided, which can be used to determine the principal. Once the principal is known, the compound interest for the same rate and time can be computed using the standard compound interest formula. This type of problem checks whether a learner can move between simple interest and compound interest calculations smoothly. Given Data / Assumptions: Simple interest SI for 3 years = Rs. 1200. Rate of interest r = 5% per annum. Time period n = 3 years. Interest for the compound interest part is compounded annually. We must find the compound interest CI for the same sum, rate, and time. Concept / Approach: For simple interest, the formula is SI = (P * r * t) / 100. Using this, we can solve for the principal P. Once P is known, we apply the compound interest amount formula A = P * (1 + r) ^ n with r in decimal form and n in years. Then CI = A - P. This approach uses the simple interest information purely as a stepping stone to obtain the principal, after which the problem becomes a standard compound interest calculation. Step-by-Step Solution: Step 1: Use the simple interest formula SI = (P * r * t) / 100. Step 2: Substitute SI = 1200, r = 5, and t = 3: 1200 = (P * 5 * 3) / 100. Step 3: Simplify the denominator: (5 * 3) / 100 = 15 / 100 = 0.15. Step 4: So 1200 = P * 0.15, which gives P = 1200 / 0.15 = 8000. Step 5: Now compute the amount under compound interest with principal 8000, rate 5% per annum, time 3 years: A = 8000 * (1.05) ^ 3. Step 6: Compute (1.05) ^ 2 = 1.1025 and then multiply by 1.05 again to get (1.05) ^ 3 = 1.157625. Step 7: Multiply to get the amount: A = 8000 * 1.157625 = 9261. Step 8: The compound interest is CI = A - P = 9261 - 8000 = 1261. Verification / Alternative check: A quick check is to compare simple interest and compound interest. Simple interest over three years at 5% on 8000 is SI = 8000 * 0.05 * 3 = 1200, which is given. Compound interest should be slightly higher than simple interest for the same rate and period because interest itself earns interest. The difference CI - SI = 1261 - 1200 = 61, which is a modest but positive increase, consistent with expectations. The computed compound interest 1261 is therefore reasonable and fits well with the theory of compounding. Why Other Options Are Wrong: Rs. 1251 is slightly smaller than the correct value and might come from approximating the cube of 1.05 or truncating decimals too early. Rs. 1271 is slightly larger than the true CI and indicates an overestimation of compounding. Rs. 1281 is further away, reflecting more significant error in the exponent or multiplication. Rs. 1240 is only marginally above the simple interest, which underestimates the impact of compounding over three years. Only Rs. 1261 fits the exact computation using the correct formulas. Common Pitfalls: One common mistake is to try to directly adjust the simple interest to get an approximate compound interest value without using the exact formula. Others mistakenly treat the simple interest formula as if it were still valid under compounding, simply increasing the rate or time, which is not correct. Incorrect calculation of (1.05) ^ 3 due to arithmetic slips is another risk. Carefully separating the simple and compound interest stages and performing the exponentiation correctly helps avoid these errors. Final Answer: The compound interest on the same sum at 5% per annum for 3 years, given that the simple interest is Rs. 1200, is Rs. 1261, which corresponds to option B.

Question 3

On a principal of Rs. 30000, the compound interest at 7% per annum is Rs. 4347. For how many years was the money invested, assuming annual compounding?

Accepted Answer

Introduction / Context: Instead of finding the amount or interest given the time, this problem reverses the process and asks for the time period, given the principal, rate, and compound interest. This is a useful test of understanding the compound interest relationship between amount and time. The candidate must recognise that the compound interest implies a specific growth factor over the period and then identify the number of years that correspond to that factor. Given Data / Assumptions: Principal P = Rs. 30000. Compound interest CI = Rs. 4347. Rate r = 7% per annum. Interest is compounded annually. We need to find the time period n in years. Concept / Approach: First we compute the final amount A by adding the compound interest to the principal: A = P + CI. Then we use the compound interest amount formula A = P * (1 + r) ^ n, with r in decimal form. Because the rate and principal are known, we can compute the ratio A / P and see which integer power of (1 + r) equals that ratio. For neat examination values, we usually find that n is a small whole number such as 2 or 3. Step-by-Step Solution: Step 1: Compute the amount: A = P + CI = 30000 + 4347 = 34347. Step 2: Convert the rate to decimal form: r = 7% = 0.07. Step 3: The amount formula is A = P * (1 + r) ^ n. Step 4: Compute the ratio A / P = 34347 / 30000. Step 5: Calculate this ratio: 34347 / 30000 = 1.1449. Step 6: Now compute (1.07) ^ 2 = 1.1449. Step 7: We see that (1 + r) ^ 2 equals exactly A / P, so n = 2 years. Step 8: Therefore, the money was invested for 2 years. Verification / Alternative check: To verify, we can compute the compound interest for 2 years directly. With P = 30000 and r = 7%, the amount after one year is 30000 * 1.07 = 32100. After the second year, the amount is 32100 * 1.07 = 34347. The compound interest is 34347 - 30000 = 4347, which matches the given interest. This confirms that the time period of 2 years is correct. Checking for 3 years or more would produce much larger amounts and interests, which would not match the given data. Why Other Options Are Wrong: If n were 3 years, the growth factor would be (1.07) ^ 3, which is greater than 1.1449, leading to an amount higher than 34347. Four or five years would produce even larger growth factors and amounts, further away from the given condition. One year at 7% would give an amount of only 32100, with interest 2100, far smaller than 4347. Therefore, 2 years is the only period that fits the data exactly. Common Pitfalls: A common mistake is to attempt to solve for n algebraically using logarithms, which is correct in general but unnecessary here and can lead to rounding issues. Another error is confusing simple interest and compound interest and trying to use a linear relation like CI = P * r * n directly for compound interest, which fails when interest is compounded. Some learners also forget to convert the rate into decimal form or miscalculate powers of 1.07. Recognising that examination questions often have neat integer time periods helps guide the solution approach. Final Answer: The period for which Rs. 30000 was invested at 7% per annum to earn compound interest of Rs. 4347 is 2 years, corresponding to option A.

Question 4

For the following data sufficiency question on compound interest, determine which statements give enough information to find the compound amount: I. Rs. 200 was borrowed for 192 months at 6% per annum compounded annually. II. Rs. 200 was borrowed fo…

Accepted Answer

Introduction / Context: This is a data sufficiency question rather than a direct numerical calculation. The focus is on deciding which given statements provide enough information to determine the compound amount. Such questions are designed to test logical reasoning about the information provided rather than detailed arithmetic skills. The key idea is to analyse whether each statement alone, or both together, allow the compound interest formula to be applied without ambiguity. Given Data / Assumptions: We want to determine the compound amount on a borrowed sum. Statement I: Rs. 200 was borrowed for 192 months at 6% per annum compounded annually. Statement II: Rs. 200 was borrowed for 16 years at 6% per annum. We need to see which statements are sufficient to compute the compound amount. We assume standard compound interest with annual compounding unless specified otherwise. Concept / Approach: To calculate a compound amount using the formula A = P * (1 + r) ^ n, we need the principal P, the annual rate r in decimal form, and the number of years n. Data sufficiency problems ask whether we can identify these quantities uniquely from the given statements. Statement I explicitly provides all three with no ambiguity. Statement II provides the principal, a rate per annum, and a time stated in years, but does not explicitly confirm that the interest is compounded annually, which creates uncertainty about the compounding pattern and thus about the exact compound amount. Step-by-Step Solution: Step 1: Examine Statement I. It tells us that P = 200, the duration is 192 months, and the rate is 6% per annum compounded annually. Step 2: Convert the duration in Statement I into years: 192 months equals 192 / 12 = 16 years. Step 3: With Statement I alone, we know P, r = 6% per annum, and n = 16 years, and we also know the compounding frequency is annual. Therefore, we can compute A = 200 * (1.06) ^ 16. Step 4: Now examine Statement II. It says P = 200, time = 16 years at 6% per annum, but it does not explicitly mention that interest is compounded annually. Step 5: Because the compounding frequency is not stated in Statement II, we do not know whether interest is compounded annually, semiannually, quarterly, or in some other way, so the exact compound amount remains uncertain. Step 6: Therefore, Statement II alone is not sufficient to determine the unique compound amount, while Statement I alone is sufficient. Verification / Alternative check: If we attempted to treat Statement II as implying annual compounding by assumption, we could compute an amount, but this would rely on an extra assumption not clearly given in the statement. In data sufficiency questions, we must work only with what is explicitly stated. In contrast, Statement I clearly states that the interest is compounded annually, and the total time in months can be converted to years, so there is no ambiguity. Therefore, we confirm that the correct logical conclusion is that only Statement I alone is sufficient. Why Other Options Are Wrong: Option B claims that II alone is sufficient while I alone is not, which contradicts the clear information about compounding in Statement I. Option C states that either statement alone is sufficient, which wrongly treats Statement II as fully specifying the compounding structure. Option D says that both statements together are not sufficient, but this is incorrect because Statement I by itself already provides enough information. Option E suggests that both together are required, but, again, Statement I alone is already complete, so the additional Statement II is not necessary for computation. Common Pitfalls: Many students casually assume that if a rate is given per annum, then the compounding is automatically annual, which is not always guaranteed. In data sufficiency, such assumptions can lead to incorrect conclusions. Another pitfall is to get distracted by the conversion of months to years and to overlook the important phrase "compounded annually" in Statement I. It is crucial to focus on what is explicitly given and to ask whether the principal, rate, time, and compounding pattern are all clearly determined. Final Answer: Only Statement I by itself provides complete information to compute the compound amount, while Statement II alone does not. Therefore, the correct choice is I alone is sufficient while II alone is not sufficient to answer, which is option A.

Question 5

Find the effective annual rate of interest for an investment that earns 5.5% per year when interest is compounded continuously.

Accepted Answer

Introduction / Context: This problem introduces the idea of continuous compounding and asks for the effective annual rate. Continuous compounding means that interest is added at every instant, rather than daily, monthly, or yearly. The effective annual rate is the equivalent simple annual rate that would produce the same growth over one year as the continuous compounding process. This concept is important in more advanced financial mathematics and calculus based applications. Given Data / Assumptions: Nominal rate r = 5.5% per year, compounded continuously. We want the effective annual rate, that is, the rate which, if applied once per year, would produce the same growth on a principal over one year. The standard continuous compounding formula A = P * e ^ (r * t) applies, where e is the base of natural logarithms. Time period t = 1 year for computing the effective annual rate. Concept / Approach: For continuous compounding, the amount after time t at nominal rate r (in decimal form) is A = P * e ^ (r * t). The effective annual rate R is defined such that A = P * (1 + R) after one year. Therefore, for t = 1, we have P * (1 + R) = P * e ^ r. Cancelling P from both sides gives 1 + R = e ^ r and hence R = e ^ r - 1. We simply plug in r = 0.055 and compute the result to get the effective annual rate in decimal form and then convert it to a percentage. Step-by-Step Solution: Step 1: Convert the nominal rate to decimal form: r = 5.5% = 0.055. Step 2: For continuous compounding, the one year amount is A = P * e ^ r. Step 3: Define the effective annual rate R such that A = P * (1 + R). Step 4: Set P * (1 + R) equal to P * e ^ r. This gives 1 + R = e ^ r. Step 5: Therefore, R = e ^ r - 1 = e ^ 0.055 - 1. Step 6: Compute e ^ 0.055 approximately. It is about 1.05654. Step 7: So R = 1.05654 - 1 = 0.05654, which as a percentage is about 5.654%. Step 8: Rounding to two decimal places, the effective annual rate is approximately 5.65%. Verification / Alternative check: We can compare continuous compounding at 5.5% with ordinary annual compounding at 5.5%. Under annual compounding with nominal and effective rate equal at 5.5%, the one year growth factor is 1.055. Under continuous compounding, we found a growth factor of about 1.05654, which is slightly higher. The difference is modest, about 0.15 percentage points, which makes sense because the nominal rate is not very large. The value 5.65% lies just above 5.5%, so it is a reasonable effective rate under continuous compounding. Why Other Options Are Wrong: The options 5.75%, 5.85%, and 5.95% are increasingly larger than the correct effective rate and would require a growth factor much bigger than e ^ 0.055, which is not consistent with the given nominal rate. The option 5.50% corresponds exactly to the nominal rate and would be correct only if interest were compounded annually rather than continuously. Since continuous compounding gives a slightly higher effective rate, 5.65% is the only option that matches the proper calculation. Common Pitfalls: One common mistake is to confuse the nominal and effective rates and to assume that the effective rate equals the nominal rate, regardless of the compounding method. Others may try to use a discrete compounding formula like (1 + r) ^ n - 1 with n set to a large number and then approximate, which is more complex than necessary. Some candidates also miscalculate e ^ r by using an incorrect approximate value for e or by mixing up logarithms. Remembering the clean relation R = e ^ r - 1 for continuous compounding at time t = 1 helps simplify these problems. Final Answer: The effective annual rate of interest for an investment earning 5.5% per year compounded continuously is approximately 5.65%, which corresponds to option A.

Question 6

Josh borrowed 250 dollars from his mother to buy an electric scooter and agreed to pay her back in 1 year at 3% simple annual interest. How much interest will Josh pay at the end of the year?

Accepted Answer

Introduction / Context: This is a basic simple interest question involving a real world story about borrowing money to buy an electric scooter. The main idea is to apply the simple interest formula correctly using the principal, rate, and time. It is a typical example given to beginners when introducing interest concepts and money related word problems. Given Data / Assumptions: Principal P = 250 dollars. Rate of simple interest r = 3% per year. Time t = 1 year. Interest is simple, not compounded. We want the interest amount I that Josh will pay. Concept / Approach: For simple interest, the formula is I = (P * r * t) / 100 when r is given as a percentage. Because the period is exactly one year, the calculation is especially easy: we simply find 3% of 250 dollars. The result is the amount Josh pays as interest in addition to the original 250 dollars when he repays his mother. Step-by-Step Solution: Step 1: Identify the principal P = 250. Step 2: Identify the annual simple interest rate r = 3%. Step 3: Identify the time period t = 1 year. Step 4: Use the simple interest formula I = (P * r * t) / 100. Step 5: Substitute values: I = (250 * 3 * 1) / 100. Step 6: Multiply the numerator: 250 * 3 = 750. Step 7: Divide by 100: 750 / 100 = 7.50. Step 8: Therefore, Josh will pay 7.50 dollars as interest. Verification / Alternative check: A quick mental check is that 1% of 250 is 2.50. Therefore, 3% is 3 times 2.50, which equals 7.50. This mental calculation matches the formula based result exactly and confirms that no arithmetic errors were made. Since the time is only one year, there is no need to consider multiple years or compounding in this problem. Why Other Options Are Wrong: The option 8.50 is larger than 7.50 and would correspond to a rate higher than 3% on 250 dollars. The option 9.50 is even larger and suggests an even higher interest rate. The option 10.50 is still greater and could only be justified by an incorrect interpretation of percentage or time. The option 6.50 is lower than the correct value and would correspond to an interest rate of 2.6% rather than 3%. Only 7.50 is exactly 3% of 250, matching the given data. Common Pitfalls: Although this is a simple calculation, students sometimes confuse percent and decimal representations and multiply by 3 instead of by 3 divided by 100. Others may accidentally treat the rate as 0.3 instead of 0.03 when working in decimal form. Some learners also incorrectly apply a compound interest formula when the problem clearly states simple interest, which is unnecessary and can lead to confusion. Keeping the simple interest formula in mind and checking that the percentage is handled correctly prevents these mistakes. Final Answer: Josh will pay an interest of 7.50 dollars on the 250 dollar loan at 3% simple interest for one year, which corresponds to option A.

Question 7

What is the effective annual rate of interest corresponding to a nominal rate of 6% per annum when interest is payable half yearly?

Accepted Answer

Introduction / Context: This question compares a nominal annual rate with its effective annual rate when interest is compounded more than once per year. A nominal rate of 6% per annum payable half yearly means that the interest is applied twice per year at half the nominal rate. The effective annual rate will be slightly higher than 6% because the interest from the first half year also earns interest during the second half year. Given Data / Assumptions: Nominal rate r = 6% per annum. Compounding frequency = twice per year (half yearly). We want the effective annual rate R that is equivalent. Time horizon for the effective rate is one full year. Concept / Approach: When a nominal annual rate r is compounded m times per year, the periodic rate is r / m, and the effective annual rate R is given by R = (1 + r / m) ^ m - 1, expressed in decimal form. We then convert R into a percentage. In this problem, r = 0.06 in decimal and m = 2 for half yearly compounding, so the periodic rate is 0.06 / 2 = 0.03 per half year. We then compute the one year growth factor (1.03) ^ 2 and subtract 1 to find the effective annual rate. Step-by-Step Solution: Step 1: Convert the nominal rate to decimal: r = 6% = 0.06. Step 2: Since interest is payable half yearly, the number of compounding periods per year m = 2. Step 3: The rate per half year is r / m = 0.06 / 2 = 0.03. Step 4: The one year growth factor under half yearly compounding is (1 + 0.03) ^ 2 = 1.03 ^ 2. Step 5: Compute 1.03 ^ 2 = 1.0609. Step 6: The effective annual rate R is then R = 1.0609 - 1 = 0.0609. Step 7: Express R as a percentage: R = 6.09% approximately. Step 8: Therefore, the effective annual rate is about 6.09%. Verification / Alternative check: We can check reasonableness by comparing with annual compounding at 6%. Under annual compounding, the effective annual rate and nominal rate match, so it would be 6%. When compounding more frequently, such as half yearly, the effective rate should be slightly higher than 6% but not dramatically so. The value 6.09% is just a little above 6%, which fits perfectly with this intuition. Other options such as 7% or 5% would be too far from the nominal rate to be right in a half yearly compounding scenario at this moderate rate. Why Other Options Are Wrong: The option 5% is lower than the nominal 6% and would imply that compounding reduces the effective rate, which contradicts basic interest theory. The option 6% assumes that effective rate equals nominal rate without adjusting for compounding frequency, which is incorrect here. The option 7% is significantly higher than the correct effective rate and would require a much higher growth factor than 1.0609. The option 6.50% is still too high and does not match the exact outcome of the calculation using the formula for effective rate with m = 2. Common Pitfalls: Students often confuse nominal and effective rates, especially when different compounding intervals are involved. Some may mistakenly use the formula R = r * m or R = r / m directly, which are incorrect for effective rate calculations. Others forget to convert the rate from percentage to decimal before applying the formula, causing numerical errors. The key is to remember that the effective rate is based on the compound growth factor over the full year, not just a linear scaling of the nominal rate. Final Answer: The effective annual rate of interest corresponding to a nominal rate of 6% per annum payable half yearly is 6.09%, which is option D.

Question 8

Approximately how many years will it take to double an investment at 6% per annum compounded annually?

Accepted Answer

Introduction / Context: This question asks for the time needed to double an investment at a fixed compound interest rate of 6% per annum. It is a classic problem in finance and aptitude exams and is often used to illustrate the power of compounding. The exact calculation involves solving an exponential equation, but there are also helpful rules of thumb, such as the rule of 72, that allow a quick estimate. Given Data / Assumptions: Rate of interest r = 6% per annum. Interest is compounded annually. We want time n in years such that the amount is twice the principal. This means the growth factor over n years must be 2. Concept / Approach: The standard compound interest formula is A = P * (1 + r) ^ n. To double the investment, we set A = 2P. Substituting into the formula gives 2P = P * (1 + r) ^ n. Cancelling P from both sides yields 2 = (1 + r) ^ n. To solve this exactly, we would take logarithms and solve n = log(2) / log(1 + r). For exam purposes and whole number answer options, we calculate an approximate value and then match it to the nearest plausible integer number of years, which will give the correct option. Step-by-Step Solution: Step 1: Convert the rate to decimal form: r = 6% = 0.06. Step 2: Write the doubling condition: 2 = (1.06) ^ n. Step 3: Take logarithms to solve for n: n = log(2) / log(1.06). Step 4: Approximate log(2) as about 0.3010 and log(1.06) as about 0.0253. Step 5: Compute n = 0.3010 / 0.0253, which is approximately 11.9 years. Step 6: Since the options are given in whole years, the closest integer is 12 years. Step 7: Therefore, it will take approximately 12 years to double the investment at 6% compound interest. Verification / Alternative check: A quick rule of thumb is the rule of 72, which estimates the doubling time as 72 divided by the annual rate in percent. Here it gives 72 / 6 = 12 years, the same result as the more exact logarithmic calculation. We can also check powers of 1.06 directly: (1.06) ^ 10 is about 1.791, which is less than 2, and (1.06) ^ 12 is about 2.012, which is very close to exactly double. This confirms that about 12 years is the correct and reasonable answer. Why Other Options Are Wrong: An answer of 10 years would correspond to roughly (1.06) ^ 10, which is less than 2, so the investment would not quite double. Eleven years still gives a factor slightly below 2, while thirteen years would overshoot more than necessary. Fourteen years produces an even larger overshoot and is less efficient than required. Twelve years gives a factor almost exactly equal to 2, making it the only option that matches the mathematical requirement closely. Common Pitfalls: Some learners mistakenly use simple interest for doubling time, using 2 = 1 + r * n, which is not correct under compounding. Others may misapply the rule of 72 by using 6.0 in decimal form instead of percentage, which gives a nonsensical result. Another common error is to rely solely on rough mental estimates without checking powers of 1.06 or using logarithms, which can lead to choosing a nearby but incorrect option. Combining the rule of 72 with a quick check of powers is usually sufficient in exam situations. Final Answer: It will take approximately 12 years for an investment to double at 6% per annum with annual compounding, so the correct option is C.

Question 9

The property taxes of Mr. and Mrs. Espedido amount to 2450 dollars and are due on July 1. If they pay eight months earlier and the city can earn 6% compounded monthly on surplus funds, what amount should the city accept as an early payment?

Accepted Answer

Introduction / Context: This question is about present value, which is the reverse of the usual compound interest calculation. The city is entitled to receive 2450 dollars on July 1, but if the taxes are paid eight months early, the city can invest the money and earn interest. Therefore, the city should accept a smaller amount now that will grow to 2450 dollars in eight months at the given rate. This is a standard time value of money problem involving monthly compounding. Given Data / Assumptions: Future value (tax due on July 1) F = 2450 dollars. Time difference for early payment = 8 months. Nominal annual interest rate r = 6% compounded monthly. We want the present value P that will grow to 2450 in 8 months at this rate. Number of compounding periods n = 8 months. Concept / Approach: With monthly compounding, the monthly rate i is r / 12 in decimal form. The relationship between present value P and future value F after n months is F = P * (1 + i) ^ n. To find P when F is known, we rearrange this equation to P = F / ((1 + i) ^ n). This gives the amount that, if received now and invested at the monthly rate, would accumulate to the required future amount in n months. Step-by-Step Solution: Step 1: Convert the nominal annual rate to monthly rate: r = 6% per year, so i = 0.06 / 12 = 0.005 per month. Step 2: The number of months until the due date is n = 8. Step 3: Use the present value formula P = F / ((1 + i) ^ n). Step 4: Compute the monthly growth factor: 1 + i = 1 + 0.005 = 1.005. Step 5: Compute (1.005) ^ 8, which is approximately 1.04068. Step 6: Now compute P = 2450 / 1.04068, which is about 2354.17. Step 7: Therefore, the city should accept approximately 2354.17 dollars if the taxes are paid eight months early. Verification / Alternative check: To verify, we can check that investing 2354.17 at 0.5% per month for eight months indeed reaches about 2450. The future value is 2354.17 * (1.005) ^ 8, which returns very close to 2450 when calculated. Also, note that the present value must be slightly less than 2450 because receiving money earlier is more valuable than receiving it later at a positive interest rate. The discount of roughly 95.83 dollars over eight months at a moderate rate of 6% per year is reasonable and consistent with the logic of present value. Why Other Options Are Wrong: The value 2354 is close but is a rounded down approximation that does not match the more precise calculation. The option 2376 is too high and would grow to more than 2450 in eight months, putting the city ahead of what is appropriate for an equivalent payment. The option 2389 is also slightly high and does not correspond to the correct growth factor. The option 2400 is very close to the full tax amount but does not adjust sufficiently for the eight months of potential earnings that the city has by receiving the funds early. Common Pitfalls: Students may mistakenly treat this as a future value problem and multiply instead of dividing by the compounding factor, which yields an amount larger than 2450, contradicting the idea of early payment. Others may use simple interest rather than monthly compounding, which leads to a slightly different present value. Forgetting to convert the annual rate to a monthly rate is another frequent problem. The key is to recognise that present value calculations require dividing the future amount by the appropriate growth factor based on both rate and time. Final Answer: The amount the city should accept for the property taxes if paid eight months early, with an earning rate of 6% compounded monthly, is approximately 2354.17 dollars, which is option A.

Question 10

Two payments of 10000 dollars each are to be made one year and four years from now. If money earns 9% interest compounded monthly, what single payment made two years from now would be equivalent to these two scheduled payments?

Accepted Answer

Introduction / Context: This question is an application of the time value of money with multiple cash flows. It asks for a single equivalent payment at a different time that has the same present economic value as two separate payments. Because interest is compounded monthly at 9% per year, we must use an effective monthly rate and carefully shift each cash flow to the specified comparison time. Such problems are common in financial mathematics and engineering economics. Given Data / Assumptions: Two payments of 10000 dollars each. First payment is due one year from now. Second payment is due four years from now. Interest rate r = 9% per year, compounded monthly. We want an equivalent single payment at time t = 2 years. We assume 12 months per year, so the monthly rate is r / 12. Concept / Approach: The key idea is equivalence at a chosen point in time using compound interest to move cash flows forward or backward. For the payment at year 1, we must move it forward by one year (till year 2). For the payment at year 4, we must discount it backward by two years (back from year 4 to year 2). We use the monthly interest rate i = 0.09 / 12 and the fact that one year equals 12 months, while four years equals 48 months. The equivalent payment at year 2 is the sum of the future value of the first payment and the present value at year 2 of the second payment. Step-by-Step Solution: Step 1: Compute the monthly interest rate: i = 0.09 / 12 = 0.0075. Step 2: The first payment of 10000 dollars occurs at the end of year 1, which is 12 months from now. To move it to year 2, we compound it for another 12 months. Step 3: Future value at year 2 of the first payment is FV1 = 10000 * (1 + i) ^ 12 = 10000 * (1.0075) ^ 12. Step 4: The second payment of 10000 dollars occurs at the end of year 4, or 48 months from now. To move it back to year 2, which is 24 months from now, we discount it over 24 months. Step 5: Value at year 2 of the second payment is PV2 = 10000 / (1 + i) ^ 24 = 10000 / (1.0075) ^ 24. Step 6: Numerically, (1.0075) ^ 12 is about 1.0938, so FV1 is about 10000 * 1.0938. Step 7: Similarly, (1.0075) ^ 24 is about 1.1964, so PV2 is about 10000 / 1.1964. Step 8: Calculating more precisely, the combined equivalent at year 2 is approximately 19296 dollars. Step 9: Therefore, the single payment at year 2 that is equivalent to the two scheduled payments is 19296 dollars. Verification / Alternative check: One way to verify is to convert everything to present value at time zero and compare. If X is the equivalent payment at year 2, its present value is X / (1.0075) ^ 24. The present value of the first payment is 10000 / (1.0075) ^ 12, and the present value of the second payment is 10000 / (1.0075) ^ 48. Setting these equal gives X / (1.0075) ^ 24 = 10000 / (1.0075) ^ 12 + 10000 / (1.0075) ^ 48. Multiplying through by (1.0075) ^ 24 yields X = 10000 * (1.0075) ^ 12 + 10000 / (1.0075) ^ 24, which is the same expression used previously for FV1 + PV2. This confirms internal consistency and supports the answer of approximately 19296 dollars. Why Other Options Are Wrong: The value 17296 is smaller than 19296 and would not be enough to match the combined effect of two 10000 dollar payments under the given interest rate. The option 13296 is much too small and fails to reflect the significant time value component of the later payment. The option 15296 is still below the correct equivalent and underestimates the required single payment. The option 20296 is larger than necessary and would create a present value greater than the sum of the present values of the scheduled payments. Only 19296 balances the cash flows correctly at 9% compounded monthly. Common Pitfalls: Students often forget to convert the annual rate to a monthly rate and instead use 9% as if it were a yearly rate for each period. Another frequent mistake is to treat all cash flows with simple interest instead of compound interest, which distorts the equivalence. Some candidates may move all amounts to the wrong comparison time or misuse forward and backward compounding. Careful attention to the direction of movement in time and consistent use of the periodic rate are essential in solving such multi cash flow problems accurately. Final Answer: The single payment two years from now that is equivalent to payments of 10000 dollars in one year and four years at 9% interest compounded monthly is approximately 19296 dollars, which is option C.

Question 11

The population of a rural region is currently 100000 and is expected to fall by 2% per year for the next 10 years. Assuming this decline is compounded annually, what will be the expected population after 10 years?

Accepted Answer

Introduction / Context: This question uses the compound interest framework to model population decline instead of financial growth. A decrease of 2% per year compounded annually means that each year the population becomes 98% of the previous year. Over ten years, this repeated percentage decrease leads to a significantly lower population. Such exponential decay models are common in demography, ecology, and many applied fields. Given Data / Assumptions: Current population P0 = 100000. Annual rate of decline = 2% per year. Time period n = 10 years. Decline is compounded annually. Each year, population is multiplied by 1 - 0.02 = 0.98. Concept / Approach: The same formula used for compound interest growth can be used for compound decrease by using a factor less than 1. The population after n years, Pn, is given by Pn = P0 * (1 - d) ^ n, where d is the rate of decrease in decimal form. Here d = 0.02 and P0 = 100000. After computing the factor (0.98) ^ 10, we multiply this by 100000 to find the expected population after ten years. This demonstrates how percentage changes accumulate multiplicatively over time. Step-by-Step Solution: Step 1: Convert the rate of decline to decimal form: d = 2% = 0.02. Step 2: The yearly multiplier for the population is 1 - d = 1 - 0.02 = 0.98. Step 3: The population after 10 years is P10 = 100000 * (0.98) ^ 10. Step 4: Compute (0.98) ^ 2 = 0.9604. Step 5: Continue multiplying to get (0.98) ^ 10, which is approximately 0.81707. Step 6: Multiply by the initial population: P10 = 100000 * 0.81707 = 81707. Step 7: Therefore, the expected population after 10 years is about 81707. Verification / Alternative check: A rough check can be done by considering that a 2% annual decline over 10 years is roughly a 20% decline if it were linear, so the population might be expected to reduce to about 80000. With compounding, the decline is slightly more than the simple 20% calculation suggests but still in that vicinity. The computed value 81707 is a little more than 80% of 100000 but still close, which is consistent with the fact that compounding each year leads to a total decline slightly greater than 20% but not dramatically more. This confirms that the answer is reasonable. Why Other Options Are Wrong: The option 91707 is too high and corresponds to only an 8% decrease, which does not match a 2% decline compounded over 10 years. The option 61707 is too low and would require a much higher annual decline rate than 2%. The options 71707 and 70707 also imply a far larger overall drop than 2% per year, and they do not match the compound decay calculation. Only 81707 is consistent with multiplying by 0.98 for ten consecutive years. Common Pitfalls: Students sometimes mistakenly subtract 2% per year in a linear fashion and compute population as 100000 - 10 * 2% of 100000 instead of compounding. Others may accidentally use 1.02 rather than 0.98, which would represent growth rather than decline. Another error is to round intermediate results aggressively, resulting in a final answer that deviates significantly from the precise value. Correctly recognising that a decline is modelled by a multiplier less than 1 and consistently applying that multiplier over the full period is crucial. Final Answer: The expected population of the rural region after 10 years, with a 2% annual decrease compounded annually, is approximately 81707, which matches option A.

Question 12

If Rs. 10000 amounts to Rs. 11664 in 2 years at compound interest, compounded annually, what is the annual rate of compound interest?

Accepted Answer

Introduction / Context: This question requires finding the compound interest rate when the principal, final amount, and time are known. It is an inverse compound interest problem where the rate is the unknown. Such problems test the learner's familiarity with growth factors and their ability to recognise common squares or use simple calculations to isolate the rate per annum. Given Data / Assumptions: Principal P = Rs. 10000. Amount after 2 years A = Rs. 11664. Time period n = 2 years. Interest is compounded annually. We need to find the annual rate of compound interest r in percent. Concept / Approach: The compound amount formula for annual compounding is A = P * (1 + r) ^ n, where r is in decimal form. Here n = 2, so A = P * (1 + r) ^ 2. We can solve for the growth factor (1 + r) by dividing A by P and taking the square root, because n = 2. That is, (1 + r) ^ 2 = A / P, so 1 + r = square root of (A / P). Then r = (square root of (A / P)) - 1. Finally, we convert r into a percentage by multiplying by 100. Step-by-Step Solution: Step 1: Compute the ratio A / P = 11664 / 10000 = 1.1664. Step 2: Since A = P * (1 + r) ^ 2, we have (1 + r) ^ 2 = 1.1664. Step 3: Take the square root of 1.1664 to find 1 + r. Step 4: The square root of 1.1664 is 1.08. Step 5: Therefore, 1 + r = 1.08, which implies r = 1.08 - 1 = 0.08. Step 6: Convert r to a percentage: r = 0.08 * 100% = 8%. Step 7: Hence, the annual rate of compound interest is 8%. Verification / Alternative check: We can check this rate by forward calculation. At 8% per annum on 10000, the amount after one year is 10000 * 1.08 = 10800. After two years, the amount becomes 10800 * 1.08 = 11664. This matches the given amount exactly, confirming that 8% is the correct rate. Since the arithmetic uses a neat square number, the result is clean and typical for exam style questions that avoid messy decimals for the rate. Why Other Options Are Wrong: If the rate were 10%, the amount after two years would be 10000 * (1.10) ^ 2 = 12100, which is greater than 11664. For 9%, the amount would be 10000 * (1.09) ^ 2 = 11881, also too large. A rate of 6% would give 10000 * (1.06) ^ 2 = 11236, which is less than 11664. A rate of 7% would give 10000 * (1.07) ^ 2 = 11449, still less than the target amount. Only 8% produces the exact figure of 11664. Common Pitfalls: Learners sometimes mistakenly apply the simple interest formula and conclude that the effective two year rate is 16.64%, then divide by 2 to get about 8.32%, which is not correct for compound interest. Others may incorrectly take half of 1.1664 instead of its square root, confusing the relationship between squares and doubles. Accurate understanding of the exponent in the compound amount formula and careful handling of square roots are essential in this kind of problem. Final Answer: The annual rate of compound interest that turns Rs. 10000 into Rs. 11664 in 2 years is 8%, which corresponds to option C.

Question 13

At what annual rate of compound interest will a sum of Rs. 50000 amount to Rs. 73205 in 2 years, assuming annual compounding?

Accepted Answer

Introduction / Context: In this question, the initial investment, final amount, and time are given, and we are asked to determine the annual compound interest rate. This is another inverse compound interest problem similar to the previous one, but with different numbers. The goal is to identify the growth factor over two years and then derive the rate that produces this factor under annual compounding. Given Data / Assumptions: Principal P = Rs. 50000. Amount after 2 years A = Rs. 73205. Time period n = 2 years. Interest is compounded annually. We must find the annual compound interest rate r in percent. Concept / Approach: With annual compounding for two years, the amount is given by A = P * (1 + r) ^ 2. Dividing A by P gives (1 + r) ^ 2 = A / P. We then take the square root to get 1 + r and subtract 1 to isolate r in decimal form. Finally, we multiply by 100 to express r as a percentage. Because the numbers are chosen for a clean square, we expect a simple integer percentage rate as the final answer. Step-by-Step Solution: Step 1: Compute the ratio A / P = 73205 / 50000. Step 2: 73205 / 50000 equals 1.4641. Step 3: From the amount formula, (1 + r) ^ 2 = 1.4641. Step 4: Take the square root of 1.4641 to find 1 + r. Step 5: The square root of 1.4641 is 1.21. Step 6: Therefore, 1 + r = 1.21, which implies r = 1.21 - 1 = 0.21. Step 7: Convert r to a percentage: r = 0.21 * 100% = 21%. Step 8: Thus, the required annual rate of compound interest is 21 percent. Verification / Alternative check: We can check this by calculating the amount for two years at 21% per annum on 50000. After one year, the amount is 50000 * 1.21 = 60500. After the second year, it becomes 60500 * 1.21 = 73205. This matches the given final amount exactly, confirming that 21% is the correct rate. The neat square 1.21 squared equals 1.4641, which explains why the resulting numbers are so clean and typical of exam style questions. Why Other Options Are Wrong: A rate of 19% would give a two year growth factor of (1.19) ^ 2, which is about 1.4161, resulting in an amount around 70805, smaller than 73205. A rate of 17% would yield (1.17) ^ 2, about 1.3689, giving an amount near 68445. At 15%, the factor (1.15) ^ 2 equals 1.3225, resulting in only 66125. The option 23% would be even higher, giving (1.23) ^ 2 around 1.5129, and the amount would exceed 75000. None of these match the required 73205 except the 21% case. Common Pitfalls: Some learners mistakenly treat the two year growth ratio 1.4641 as the sum 1 + 2r and solve for r linearly, which would suggest r is around 23.2% in a simple interest mindset. This is incorrect for compound interest, where the rate is linked to the square root of the growth factor. Others may confuse square root with half and incorrectly take half of 1.4641 instead of its square root. It is important to remember that for n years with annual compounding, A / P equals (1 + r) raised to the power n, not multiplied by n. Final Answer: The annual compound interest rate that turns Rs. 50000 into Rs. 73205 in 2 years is 21 percent, which corresponds to option A.

Question 14

For a certain principal sum of money, the difference between the simple interest and the compound interest (compounded annually) for 2 years at 5% per annum is Rs 45. What is the value of the principal (the original sum)?

Accepted Answer

Introduction / Context: This question tests the relationship between simple interest (SI) and compound interest (CI) for a fixed period and rate, and how to use the known difference between them to find the original principal. The rate of interest is 5% per annum and the time period is 2 years, with compound interest calculated annually. The key idea is that for small time periods, there is a simple formula for the difference between CI and SI which makes such problems quick to solve in competitive exams. Given Data / Assumptions: The principal (original sum) is P rupees. Rate of interest, r = 5% per annum. Time period, t = 2 years. Interest is compounded annually for CI. Difference between CI for 2 years and SI for 2 years is Rs 45. Concept / Approach: For a given principal P, rate r% per annum and time 2 years, the simple interest is P * r * 2 / 100. The compound interest for 2 years is P * [(1 + r/100)^2 - 1]. For 2 years, the difference between CI and SI can be directly written as P * r^2 / 100^2. Using this shortcut, we equate the difference to the given Rs 45 and solve for P. This avoids computing full SI and CI separately and saves time. Step-by-Step Solution: Let the principal be P rupees. For 2 years at r% per annum, difference between CI and SI is given by: Difference = P * r^2 / 100^2. Here, r = 5, so r^2 = 25. Therefore, Difference = P * 25 / 10000 = P * 0.0025. We are told that this difference is Rs 45, so: P * 0.0025 = 45. Rearranging, P = 45 / 0.0025. Compute: 0.0025 = 25 / 10000, so 45 / 0.0025 = 45 * 400 = 18,000. Thus, the required principal is Rs 18,000. Verification / Alternative check: We can verify by calculating SI and CI directly. Simple interest for 2 years at 5%: SI = P * r * t / 100 = 18,000 * 5 * 2 / 100 = 18,000 * 0.10 = Rs 1,800. Amount with CI after 2 years: A = P * (1 + r/100)^2 = 18,000 * (1.05)^2 = 18,000 * 1.1025 = Rs 19,845. So CI = 19,845 - 18,000 = Rs 1,845. Difference CI - SI = 1,845 - 1,800 = Rs 45, which matches the given value. This confirms our answer is consistent. Why Other Options Are Wrong: Rs 36,000 would double the difference to Rs 90, not Rs 45. Rs 9,000 would give half the correct difference (Rs 22.50). Rs 27,000 would produce a difference of Rs 67.50, not Rs 45. Rs 22,500 also gives a larger difference than Rs 45. Therefore, none of these values satisfy the given condition exactly. Common Pitfalls: A frequent mistake is to try to compute CI and SI fully without using the shortcut formula, which can lead to arithmetic errors. Another error is to confuse the formula for difference between SI and CI for 2 years with that for 3 years. Some students also forget that r must be squared as a whole number of percent, not as a decimal, before dividing by 100^2. Careful substitution and correct handling of percentages are essential to avoid such mistakes. Final Answer: Thus, the principal sum of money is Rs 18,000.

Question 15

In what time (in years) will a principal of Rs 8000 amount to Rs 9261 at 5% per annum compound interest, when interest is compounded annually?

Accepted Answer

Introduction / Context: This problem checks understanding of the basic compound interest formula and how to find the time period when the principal, rate, and final amount are known. The interest is compounded annually at a rate of 5% per annum, and we are given the initial amount and the final amount. Such questions are very common in bank exams and aptitude tests and can often be solved using either the formula or observation of standard powers of common growth factors like 1.05, 1.10 and so on. Given Data / Assumptions: Principal P = Rs 8000. Amount after t years, A = Rs 9261. Rate of interest r = 5% per annum. Interest is compounded annually. We need to find the time t in years. Concept / Approach: Under compound interest with annual compounding, the amount after t years is given by the formula A = P * (1 + r/100)^t. Here P, r and A are known and we need to solve for t. In many exam questions, the ratio A / P is chosen to match a known power of (1 + r/100), which allows us to recognize t quickly without using logarithms. Here, we will compute A / P and compare it with (1.05)^t for small integer values of t. Step-by-Step Solution: We know A = P * (1 + r/100)^t. So (1 + r/100)^t = A / P. Substitute the given values: P = 8000, A = 9261 and r = 5. Compute the ratio: A / P = 9261 / 8000. We have (1 + 5/100) = 1.05, so we need (1.05)^t = 9261 / 8000. Note that 9261 = 21^3 and 8000 = 20^3, so 9261 / 8000 = (21/20)^3. Also, 1.05 = 21/20, hence (1.05)^3 = (21/20)^3 = 9261 / 8000. Therefore, (1.05)^t = (1.05)^3, which implies t = 3 years. Verification / Alternative check: We can check by directly computing the amount after 3 years. Amount A = 8000 * (1.05)^3. First year: 8000 * 1.05 = 8400. Second year: 8400 * 1.05 = 8820. Third year: 8820 * 1.05 = 9261. This matches the given amount exactly, confirming that the required time is 3 years. For other times like 2 or 4 years, the amount would become either too small or too large compared to 9261, which further supports the correctness of our answer. Why Other Options Are Wrong: If t = 2 years, the amount would be 8000 * (1.05)^2 = 8000 * 1.1025 = 8820, which is less than 9261. If t = 3.5 or 4 years, the amount would exceed 9261 because we are compounding at a positive rate of 5%. Similarly, 5 years would produce an even higher amount. Therefore, none of the other time periods match the given final amount for the specified rate and principal. Common Pitfalls: Students sometimes incorrectly apply the simple interest formula instead of the compound interest formula, leading to linear rather than exponential growth. Another frequent mistake is rounding intermediate values too early, which can disturb the recognition of exact ratios like 9261/8000. It is often helpful to look for perfect cubes or squares in such problems, as exam setters intentionally choose convenient numbers to allow shortcut recognition without calculators. Final Answer: The principal of Rs 8000 will amount to Rs 9261 in 3 years at 5% per annum compounded annually.

Question 16

The difference between the simple interest and the compound interest (compounded annually) on a certain sum of money for 3 years at 10% per annum is Rs 93. What is the value of the sum (principal)?

Accepted Answer

Introduction / Context: Here we compare simple interest (SI) and compound interest (CI) on the same sum at the same rate for 3 years. The rate is 10% per annum with annual compounding. We are given the numerical difference between CI and SI and asked to recover the principal. For 3 years, there is a standard expression for the difference between CI and SI, which helps solve the problem quickly without separately computing full SI and CI values from trial and error. Given Data / Assumptions: Principal (sum) = P rupees (unknown). Rate of interest r = 10% per annum. Time period t = 3 years. Interest for CI is compounded annually. Difference between CI and SI for 3 years is Rs 93. Concept / Approach: The simple interest for 3 years is SI = P * r * t / 100. The compound interest for 3 years is CI = P * [(1 + r/100)^3 - 1]. For 3 years, there is a known shortcut for the difference between CI and SI: Difference = P * [ (r^2 / 100^2) * (100/2) + (r^3 / 100^3) ], but this is less convenient in practice. Instead, we can directly write expressions for SI and CI, subtract, and simplify. Alternatively, we can find a simpler expression for the difference using the expansion of (1 + r/100)^3, which is efficient for standard exam rates like 10%. Step-by-Step Solution: Let P be the principal. Simple interest for 3 years at 10% per annum is SI = P * 10 * 3 / 100 = 0.30P. Amount with compound interest after 3 years is A = P * (1.10)^3 = P * 1.331. So CI = A - P = P * 1.331 - P = 0.331P. Difference between CI and SI is CI - SI = 0.331P - 0.30P = 0.031P. We are told this difference is Rs 93, so 0.031P = 93. Therefore, P = 93 / 0.031 = 3000. Hence, the required sum is Rs 3,000. Verification / Alternative check: We can check the answer by plugging back P = 3000. Simple interest: SI = 3000 * 10 * 3 / 100 = 900. Amount with CI: A = 3000 * 1.1^3 = 3000 * 1.331 = 3993. So CI = 3993 - 3000 = 993. Difference CI - SI = 993 - 900 = 93, matching the given difference exactly. This confirms that the derived principal is correct. Any other principal would change both SI and CI proportionally and would not preserve the exact difference of Rs 93. Why Other Options Are Wrong: If P were Rs 30,000, the difference would be ten times larger, Rs 930, not Rs 93. For Rs 3,030, the difference would be slightly above Rs 93. For Rs 5,000 or Rs 2,500, the difference would scale to different values and would not equal Rs 93. Only Rs 3,000 precisely satisfies the equation 0.031P = 93. Common Pitfalls: Some candidates mistakenly use the 2-year shortcut formula for a 3-year problem, which leads to an incorrect factor in the difference expression. Others compute CI and SI separately but make arithmetic mistakes or forget to subtract properly. Working systematically with the ratio 1.1^3 = 1.331 helps keep calculations clean and reduces the chance of errors. Always ensure you subtract SI from CI, not the other way around, since CI is larger for positive interest rates over more than one year. Final Answer: The principal (original sum) is Rs 3,000.

Question 17

A certain sum of money is invested for 2 years at 20% per annum compound interest. If the interest earned is Rs 723 more when interest is compounded half-yearly instead of annually, what is the principal sum?

Accepted Answer

Introduction / Context: This question compares compound interest under two different compounding frequencies: annually and half-yearly, for the same nominal rate and time period. The rate is 20% per annum, and the time is 2 years. We are told that the interest when compounding half-yearly exceeds the interest when compounding annually by Rs 723, and we must use this information to find the original principal. This is a classic comparison problem that tests understanding of effective rates under different compounding intervals. Given Data / Assumptions: Principal = P rupees (unknown). Nominal rate r = 20% per annum. Time t = 2 years. Case 1: Compounded annually at 20%. Case 2: Compounded half-yearly at 20% per annum, i.e., 10% per half-year. Interest in half-yearly case is Rs 723 more than interest in annual case. Concept / Approach: When compounding annually, the amount after 2 years is A1 = P * (1 + 0.20)^2. When compounding half-yearly, the rate per half-year is 20% / 2 = 10% and there are 2 * 2 = 4 compounding periods, so A2 = P * (1 + 0.10)^4. The corresponding interests are I1 = A1 - P and I2 = A2 - P. The question tells us that I2 - I1 = 723. We can factor P out of both expressions and set up an equation in P using the numerical values of (1.10)^4 and (1.20)^2. Step-by-Step Solution: Amount with annual compounding: A1 = P * (1.20)^2 = P * 1.44. So interest I1 = A1 - P = P * 1.44 - P = 0.44P. Amount with half-yearly compounding: A2 = P * (1.10)^4. Compute (1.10)^2 = 1.21 and (1.10)^4 = (1.21)^2 = 1.4641. So interest I2 = A2 - P = P * 1.4641 - P = 0.4641P. Difference in interest: I2 - I1 = 0.4641P - 0.44P = 0.0241P. Given I2 - I1 = 723, so 0.0241P = 723. Thus, P = 723 / 0.0241 = 30,000. Therefore, the principal is Rs 30,000. Verification / Alternative check: Take P = 30,000. Annual compounding for 2 years: A1 = 30,000 * 1.44 = 43,200, so I1 = 43,200 - 30,000 = 13,200. Half-yearly compounding: A2 = 30,000 * 1.4641 = 43,923 (using 1.4641), so I2 = 43,923 - 30,000 = 13,923. Difference I2 - I1 = 13,923 - 13,200 = 723, which matches the condition perfectly. This confirms that our calculated principal is correct. Why Other Options Are Wrong: If P = 20,000, the difference would be 20,000 * 0.0241 = 482, not 723. For P = 15,000, the difference drops further to around 361.5. For P = 25,000 or P = 40,000, the difference scales to about 602.5 or 964 respectively. None of these equal exactly Rs 723, so those options cannot be correct. Common Pitfalls: A common error is to mistakenly use simple interest for one of the cases or to forget that half-yearly compounding changes both the rate and the number of periods. Another pitfall is incorrect rounding of the factor 1.4641, which can slightly disturb the final difference. In exams, where options are well separated, using 1.4641 is accurate enough. Always set up the algebraic equation carefully as I2 - I1 = 723 and avoid mixing up the order of subtraction. Final Answer: The principal sum invested is Rs 30,000.

Question 18

What is the compound interest earned on Rs 80,000 invested for 1 year at 40% per annum, when the interest is compounded quarterly (every 3 months)?

Accepted Answer

Introduction / Context: Here we apply the standard compound interest formula with a nominal annual rate and quarterly compounding. The principal is Rs 80,000, the nominal rate is 40% per annum, and the compounding is done four times a year. The question asks for the compound interest earned in 1 year, not the final amount, so after finding the amount we must subtract the principal to obtain the interest. This is a straightforward calculation that checks a student's comfort with handling non-annual compounding. Given Data / Assumptions: Principal P = Rs 80,000. Nominal annual rate r = 40% per annum. Compounding frequency: quarterly (4 times per year). Time t = 1 year. We must find the compound interest (CI) earned in this period. Concept / Approach: For compound interest with nominal rate r and n compounding periods per year, the amount after t years is A = P * (1 + r/(100n))^(n*t). Here, r = 40, n = 4 and t = 1. This gives a periodic rate of 40/4 = 10% per quarter, and there are 4 quarters in 1 year. Once we compute A, the compound interest is CI = A - P. Because the rate is high, the growth over the year is significant, and accurate multiplication is important to avoid arithmetic mistakes. Step-by-Step Solution: Periodic rate per quarter = 40% / 4 = 10% = 0.10. Number of compounding periods in 1 year = 4 * 1 = 4. Amount A = P * (1 + 0.10)^4 = 80,000 * (1.10)^4. Compute (1.10)^2 = 1.21; then (1.10)^4 = (1.21)^2 = 1.4641. So A = 80,000 * 1.4641 = 117,128 (approximately). Compound interest CI = A - P = 117,128 - 80,000 = 37,128. Thus, the interest earned is Rs 37,128. Verification / Alternative check: We can also build the amount step by step each quarter. After 1st quarter: 80,000 * 1.10 = 88,000. After 2nd quarter: 88,000 * 1.10 = 96,800. After 3rd quarter: 96,800 * 1.10 = 106,480. After 4th quarter: 106,480 * 1.10 = 117,128. Subtracting the original principal 80,000 from 117,128 again yields 37,128. This stepwise approach confirms the result without relying on the shortcut of squaring 1.21. Why Other Options Are Wrong: Rs 28,317 and Rs 18,732 are both significantly lower than the correct interest and would correspond to a much lower effective annual growth than 40% nominal with quarterly compounding. Rs 21,387 is still too small, and Rs 40,512 is too large for one year on Rs 80,000 at the given rate and frequency. Only Rs 37,128 matches the arithmetic of the compound interest formula. Common Pitfalls: Some students mistakenly use simple interest, computing 80,000 * 40/100 = 32,000, which ignores the effect of quarterly compounding. Others incorrectly use 40% directly in the formula as if it were the quarterly rate, which leads to an exaggerated result. Always divide the nominal rate by the number of compounding periods, and be careful to raise the expression to the correct power n * t. Final Answer: The compound interest earned in 1 year is Rs 37,128.

Question 19

The amount received on a certain principal at 10% per annum compound interest after 3 years is Rs 10,648. What was the principal amount (in rupees)?

Accepted Answer

Introduction / Context: This is a standard reverse compound interest question where the final amount, rate, and time are known, and we must determine the original principal. The rate of interest is 10% per annum and the interest is compounded annually for 3 years. Such questions reinforce familiarity with the compound interest formula and the idea of reversing the growth process by dividing by the growth factor instead of multiplying. Given Data / Assumptions: Amount after 3 years, A = Rs 10,648. Rate r = 10% per annum. Time t = 3 years. Interest is compounded annually. Principal P is unknown and needs to be found. Concept / Approach: Under compound interest with annual compounding, the amount after t years is given by A = P * (1 + r/100)^t. When A, r, and t are known, we rearrange to get P = A / (1 + r/100)^t. For common percentages like 10% and small integer time periods like 3 years, it is easy to compute powers like (1.10)^3 exactly. Here, we use the fact that (1.10)^3 = 1.331 to quickly find P and then check against the given amount. Step-by-Step Solution: Use the formula A = P * (1 + r/100)^t. We have A = 10,648, r = 10 and t = 3. So 10,648 = P * (1.10)^3. Compute (1.10)^3 = 1.331. Therefore, P = 10,648 / 1.331. By calculation, 8,000 * 1.331 = 10,648, so P = 8,000. Hence, the principal amount was Rs 8,000. Verification / Alternative check: We can verify by forward calculation. Starting from Rs 8,000 at 10% compound interest: After 1 year, amount = 8,000 * 1.10 = 8,800. After 2 years, amount = 8,800 * 1.10 = 9,680. After 3 years, amount = 9,680 * 1.10 = 10,648. This matches the given amount exactly. Any other principal from the options would not yield exactly 10,648 after 3 years at 10% per annum, confirming that Rs 8,000 is uniquely correct. Why Other Options Are Wrong: If P were Rs 7,500, the amount after 3 years would be 7,500 * 1.331 = 9,982.5, which is lower than 10,648. For Rs 9,000, the amount becomes 9,000 * 1.331 = 11,979, too high. For Rs 8,500, the amount is 8,500 * 1.331 = 11,313.5. For Rs 10,000, we get 13,310. None of these matches the given amount, so they cannot be correct. Common Pitfalls: Some learners attempt to use simple interest formulas, which will not produce the correct exponential growth. Others miscalculate (1.10)^3, often treating it as 1.30 instead of 1.331. It is important to remember that compounding multiplies the factor each year, not just adds the percentage linearly. Careful handling of the power and division step avoids these errors. Final Answer: The principal amount invested was Rs 8,000.

Question 20

A bank offers 15% compound interest per half-year. A customer deposits Rs 7200 on 1st January and another Rs 7200 on 1st July of the same year. What total interest (in rupees) will the customer earn by the end of that year?

Accepted Answer

Introduction / Context: This question involves compound interest with multiple deposits at different times within the same year. The bank offers 15% compound interest per half-year. One deposit is made at the beginning of the year, and the second deposit is made halfway through the year. We must calculate the interest earned on each deposit separately based on how long it stays in the account and then add the interests to get the total interest for the year. This tests the understanding of time-weighted compounding when there are multiple cash flows. Given Data / Assumptions: Bank offers 15% compound interest per half-year (effectively 30% nominal per annum). First deposit: Rs 7,200 on 1st January. Second deposit: Rs 7,200 on 1st July. Interest is compounded every half-year. We want interest earned by 31st December of the same year. Concept / Approach: Since the compounding period is half-yearly, each half-year earns 15%. The first deposit remains invested for two half-year periods (from 1st January to 1st July, then to 31st December). The second deposit is invested for only one half-year (from 1st July to 31st December). For each deposit, we compute its amount using A = P * (1 + 0.15)^n, where n is the number of half-year periods. The interest for that deposit is A - P. Finally, we sum the interests from both deposits to get the total interest for the year. Step-by-Step Solution: For the first deposit: P1 = 7,200, n1 = 2 half-years. Amount on first deposit: A1 = 7,200 * (1.15)^2. Compute (1.15)^2 = 1.3225, so A1 = 7,200 * 1.3225 = 9,522. Interest on first deposit: I1 = A1 - P1 = 9,522 - 7,200 = 2,322. For the second deposit: P2 = 7,200, n2 = 1 half-year. Amount on second deposit: A2 = 7,200 * 1.15 = 8,280. Interest on second deposit: I2 = A2 - P2 = 8,280 - 7,200 = 1,080. Total interest earned in the year: I = I1 + I2 = 2,322 + 1,080 = 3,402. Verification / Alternative check: Another way to think about it is to treat interest separately: for the first deposit, total gain factor over the year is 1.3225, giving 32.25% overall on 7,200, which equals 2,322. For the second deposit, the money is in for only one half-year at 15%, giving 7,200 * 0.15 = 1,080. Adding these again results in 3,402. Since both methods give the same answer, the calculations are internally consistent. Why Other Options Are Wrong: Rs 850 and Rs 1,701 are much too small compared to a 15% half-yearly rate on two deposits of 7,200 each. Rs 6,803 is too large and would require substantially higher rates or more periods of compounding. Rs 2,552 is also smaller than the correctly computed combined interest. Only Rs 3,402 matches the detailed calculation from the bank's compounding rule. Common Pitfalls: A common mistake is to treat both deposits as if they were invested for the full year, which overstates the interest on the second deposit. Another error is to convert 15% per half-year into 30% and then apply it as a single annual simple interest rate, ignoring compounding. Always track how many compounding periods each deposit experiences and apply the compound interest formula to each cash flow separately. Final Answer: The total interest earned by the end of the year is Rs 3,402.

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