Question 1

What is the difference between the compound interest on Rs. 5,000 for 1½ years at 4% per annum when interest is compounded yearly and when it is compounded half-yearly?

Accepted Answer

Introduction: This question compares two different compounding frequencies on the same principal, rate, and time period. The goal is to understand how changing from yearly compounding to half yearly compounding slightly increases the compound interest earned for the same nominal annual rate. Given Data / Assumptions: Principal P = Rs. 5,000. Rate r = 4% per annum. Total time = 1½ years = 1.5 years. Case 1: Interest compounded yearly. Case 2: Interest compounded half-yearly. Concept / Approach: For yearly compounding with a broken period, we apply compound interest for the complete years and then simple interest on the amount for the remaining fraction of the year. For half yearly compounding, we convert the annual rate into a per half year rate and compute the amount over the corresponding number of half years. The required answer is the difference between the two compound interest values. Step-by-Step Solution: Case 1 (Yearly compounding): Amount after 1 year = 5000 * 1.04 = Rs. 5,200. Extra 0.5 year gives simple interest on 5,200. Simple interest = 5200 * 4 * 0.5 / 100 = 5200 * 0.02 = Rs. 104. Total interest (yearly) = 200 + 104 = Rs. 304. Case 2 (Half yearly compounding): Rate per half year = 4 / 2 = 2%. Number of half years in 1.5 years = 3. Amount = 5000 * (1.02)^3 = 5000 * 1.061208 ≈ Rs. 5,306.04. Interest (half yearly) ≈ 5306.04 − 5000 = Rs. 306.04. Difference = 306.04 − 304 = Rs. 2.04. Verification / Alternative check: We can verify by recomputing (1.02)^3 as 1.0404 * 1.02 = 1.061208. The multiplication 5000 * 1.061208 indeed gives approximately 5,306.04, confirming that the difference between the two interest amounts is Rs. 2.04. Why Other Options Are Wrong: Rs. 3.04, Rs. 4.04, and Rs. 5.04 assume much larger gaps between the amounts, which do not occur at such a small rate and period. Rs. 1.04 is smaller than the correct value and results from incorrect handling of the half yearly compounding or the broken period in yearly compounding. Common Pitfalls: Common errors include treating 1½ years as 1 year only or ignoring the fractional year, misusing the quarterly or monthly formula, or forgetting that half yearly compounding changes both the rate and the number of periods. Careful handling of the broken period for yearly compounding is also essential. Final Answer: The difference between the two compound interest amounts is Rs. 2.04.

Question 2

Simple interest on a certain principal at 7% per annum for 4 years is Rs. 2,415. Using this information to find the principal, what will be the compound interest on the same principal at 4% per annum for 2 years, compounded annually?

Accepted Answer

Introduction: This problem links simple interest and compound interest. We first use the given simple interest to determine the unknown principal, and then we use that principal to compute compound interest at a different rate and time. It nicely checks understanding of both formulas in one question. Given Data / Assumptions: Simple interest SI = Rs. 2,415. Simple interest rate = 7% per annum. Simple interest time = 4 years. For compound interest: rate = 4% per annum, time = 2 years, compounded annually. Concept / Approach: First, use the simple interest formula: SI = P * r * t / 100. From this, find the principal P. Then apply the compound interest amount formula for 2 years: A = P * (1 + r/100)^2, and compute compound interest as CI = A − P. Step-by-Step Solution: Step 1: Find the principal using SI. 2415 = P * 7 * 4 / 100. 2415 = P * 28 / 100. P = 2415 * 100 / 28 = 241500 / 28 = Rs. 8,625. Step 2: Apply compound interest for 2 years at 4%. Amount A = 8625 * (1.04)^2. (1.04)^2 = 1.0816. A = 8625 * 1.0816 ≈ Rs. 9,328.80. Step 3: Compound interest. CI = A − P ≈ 9,328.80 − 8,625 = Rs. 703.80. Rounded to the nearest rupee, CI ≈ Rs. 704. Verification / Alternative check: Instead of computing the full amount, we can directly compute: CI = P * [(1.04)^2 − 1] = 8625 * 0.0816. 8625 * 0.0816 ≈ 8625 * 816 / 10000 ≈ Rs. 703.8, which again confirms about Rs. 704. Why Other Options Are Wrong: Rs. 854, Rs. 893, Rs. 914, and Rs. 678 all come from incorrect principal calculations or wrong handling of the compound interest formula. Many of them result from using simple interest instead of compound interest or misusing the percentage value. Common Pitfalls: Learners often forget to first compute the principal from the given simple interest data and instead try to mix both rates directly. Another common mistake is to apply the simple interest formula again for the second part, which will not give the correct compound interest value. Final Answer: The compound interest on the same principal at 4% per annum for 2 years is approximately Rs. 704.

Question 3

Sonika invested an amount of Rs. 5,800 for 2 years at compound interest compounded annually. If she earned a total compound interest of Rs. 594.50 at the end of 2 years, then at what annual rate of compound interest did she invest?

Accepted Answer

Introduction: This is a reverse compound interest problem where the principal, time, and total compound interest are known, and we are asked to find the annual rate of interest. It checks understanding of how the compound interest formula can be rearranged to determine the rate when the amount of interest is given. Given Data / Assumptions: Principal P = Rs. 5,800. Time = 2 years. Total compound interest over 2 years = Rs. 594.50. Interest is compounded annually. We need the annual rate of interest r%. Concept / Approach: Compound interest for 2 years is given by: CI = P * [(1 + r/100)^2 − 1]. We know CI and P, so we can solve for (1 + r/100)^2, then take the square root to find (1 + r/100), and finally obtain r. Recognising perfect squares can make this process easier. Step-by-Step Solution: Step 1: Use the CI formula. 594.50 = 5800 * [(1 + r/100)^2 − 1]. (1 + r/100)^2 − 1 = 594.50 / 5800. 594.50 / 5800 = 0.1025. So (1 + r/100)^2 = 1 + 0.1025 = 1.1025. Step 2: Take square root. We notice 1.1025 = (1.05)^2. Therefore, 1 + r/100 = 1.05. r/100 = 0.05, so r = 5% per annum. Verification / Alternative check: Verify by recomputing CI with r = 5%: Amount A = 5800 * (1.05)^2 = 5800 * 1.1025 = Rs. 6,394.50. Compound interest = 6,394.50 − 5,800 = Rs. 594.50. This exactly matches the given interest, confirming that the rate is correct. Why Other Options Are Wrong: Rates like 6%, 6.5%, 4.5% or 7% would produce different values of (1 + r/100)^2 and would not reproduce Rs. 594.50 as the total compound interest. They come from guessing or incomplete handling of the algebraic steps. Common Pitfalls: Some students confuse simple and compound interest formulas and try to apply P * r * t / 100, which is incorrect here. Others may forget to add 1 to the interest fraction when forming (1 + r/100)^2 or make mistakes when taking the square root. Working carefully with percentages and recognising common decimal squares helps a lot. Final Answer: The required annual rate of compound interest is 5% per annum.

Question 4

At what least number of complete years will a sum of money, invested at 20% per annum compound interest (compounded annually), become more than double its original amount?

Accepted Answer

Introduction: This compound interest question focuses on growth over multiple years and asks after how many complete years an investment will become more than double. It tests your understanding of how to use the compound interest multiplier (1 + r/100)^n and how to compare it with a target multiple like 2 times the principal. Given Data / Assumptions: Rate of compound interest r = 20% per annum. Interest is compounded annually. We want the smallest whole number of years n such that amount > 2 * principal. Concept / Approach: For compound interest with annual compounding, the amount after n years is: A = P * (1 + r/100)^n. We need A > 2P. Since P is common and positive, the condition becomes: (1 + r/100)^n > 2. We substitute r = 20 and test integer values of n until the inequality is satisfied. Step-by-Step Solution: Rate per year = 20%, so multiplier = 1 + 20/100 = 1.20. Compute powers of 1.20: For n = 1: (1.2)^1 = 1.20 (less than 2). For n = 2: (1.2)^2 = 1.44 (still less than 2). For n = 3: (1.2)^3 = 1.728 (still less than 2). For n = 4: (1.2)^4 = 1.728 * 1.2 = 2.0736 (now greater than 2). Therefore, after 4 years the amount becomes more than double its original value. Verification / Alternative check: To check, assume P = 100 for simplicity. Then after 4 years: A = 100 * (1.2)^4 ≈ 100 * 2.0736 = Rs. 207.36, which is clearly more than double 100. After 3 years, A = 100 * 1.728 = Rs. 172.80, which is still less than double, confirming that 4 complete years are needed. Why Other Options Are Wrong: 2 years and 2.5 years do not provide enough growth because the multiplier is far below 2. 3 years gives only about 1.728 times the principal, which is still less than double. 5 years is not the least; it does give more than double but we already crossed 2 times at 4 years. Common Pitfalls: A typical mistake is to use simple interest logic and set 20 * n ≥ 100 to get n ≥ 5, which ignores compounding. Others may stop at 3 years by approximating incorrectly. Always use the compound interest multiplier and test successive integer years when the question asks for the least number of complete years. Final Answer: The least number of complete years required is 4 years.

Question 5

The difference between the compound interest and the simple interest on an amount of Rs. 18,000 for 2 years is Rs. 405. Assuming the same rate of interest per annum for both cases and that compound interest is compounded annually, what is the annual…

Accepted Answer

Introduction: This question uses a special relationship between compound interest and simple interest for a period of 2 years. Instead of computing both interests separately, there is a direct formula for the difference between compound and simple interest over 2 years, which greatly simplifies the process of finding the interest rate. Given Data / Assumptions: Principal P = Rs. 18,000. Time t = 2 years. Difference between CI and SI over 2 years = Rs. 405. Rate r% per annum is the same for both CI and SI, with CI compounded annually. Concept / Approach: For 2 years, the difference between compound interest and simple interest on a principal P at rate r% is: CI − SI = P * (r/100)^2. This result comes from expanding the compound interest formula for 2 years. We use this relation to form an equation in r and then solve for r. Step-by-Step Solution: Formula: CI − SI = P * (r/100)^2. 405 = 18000 * (r/100)^2. (r/100)^2 = 405 / 18000. 18000 / 10000 = 1.8, so 405 / 18000 = 405 / (1.8 * 10000) = 0.0225. Thus (r/100)^2 = 0.0225. r/100 = sqrt(0.0225) = 0.15. Therefore r = 0.15 * 100 = 15% per annum. Verification / Alternative check: Check by computing CI and SI explicitly with r = 15%. For 2 years, SI = 18000 * 15 * 2 / 100 = Rs. 5,400. CI amount = 18000 * (1.15)^2 = 18000 * 1.3225 = Rs. 23,805. CI = 23,805 − 18,000 = Rs. 5,805. Difference = 5,805 − 5,400 = Rs. 405, which matches the given value. Why Other Options Are Wrong: Rates like 12%, 13% or 14% would give much smaller differences between CI and SI, while 10% would give an even smaller difference. None of these satisfy the required difference of Rs. 405 when substituted into the formula. Common Pitfalls: A common mistake is to attempt full calculation of CI and SI separately for each option, which is time consuming and error prone. Others forget the formula for CI − SI for 2 years and try to use an incorrect relation. Remembering the compact formula P * (r/100)^2 is very helpful in such questions. Final Answer: The annual rate of interest is 15% per annum.

Question 6

Rs. 100 doubles to Rs. 200 in 5 years when invested at compound interest, with interest compounded annually. Assuming the same rate of compound interest continues, how many more years will it take for the investment to earn another Rs. 200 in intere…

Accepted Answer

Introduction: This compound interest problem stresses the idea that if an amount doubles in a certain period at a fixed rate, then any amount will also double in the same period at that rate. We are told that Rs. 100 doubles in 5 years, and we must find how much longer it will take for the already doubled amount to double once more and thus gain another Rs. 200 in interest. Given Data / Assumptions: Initial principal P₁ = Rs. 100. After 5 years at some compound rate, amount becomes Rs. 200. So (1 + r/100)^5 = 2. After 5 years, new principal for next phase is P₂ = Rs. 200. We want P₂ to become Rs. 400 at the same rate. Concept / Approach: Under compound interest, growth is multiplicative. If a certain rate doubles any amount in 5 years, it doubles every amount in 5 years. Therefore, the time needed to go from Rs. 200 to Rs. 400 at the same rate is the same as the time needed to go from Rs. 100 to Rs. 200, which is 5 years. Step-by-Step Solution: Step 1: From the first period, we know (1 + r/100)^5 = 2. Step 2: For the second period, the amount starts at 200 and must become 400. This again requires a multiplication by 2. So we need (1 + r/100)^n = 2 for the second stage as well. Step 3: Since (1 + r/100)^5 = 2, the smallest n that satisfies the equation again is 5 years. Thus, it takes another 5 years for Rs. 200 to become Rs. 400 at the same rate. Verification / Alternative check: We can assume any realistic rate that doubles in 5 years, such as r that satisfies the given condition, but the exact numerical value of r is not needed. The key observation is that the doubling time depends only on the rate, not on the starting amount. So the second doubling must also take 5 years. Why Other Options Are Wrong: Options like 3 years, 4 years, 6 years, or 7 years ignore the fact that the growth factor has to reach exactly 2 again, which is tied to the same 5 year period at this rate. These values do not correspond to a full doubling at the same fixed interest rate. Common Pitfalls: Students often think that earning another Rs. 200 on Rs. 200 takes less time than earning the first Rs. 100 on Rs. 100, confusing simple and compound interest. Under compound interest with a fixed rate, percentage growth is constant, so doubling time stays the same regardless of the starting principal. Final Answer: It will take an additional 5 years to earn another Rs. 200 interest and reach Rs. 400.

Question 7

If the compound interest on a certain sum at a rate of 16⅔% per annum (that is, 16 2/3 percent per year) for 3 years is Rs. 1,270, then what will be the simple interest on the same sum at the same rate for the same period of 3 years?

Accepted Answer

Introduction: Here we are given compound interest for 3 years at a fractional rate of 16⅔% per annum and are asked to find the simple interest for the same principal, same rate, and same time. The problem tests skills in handling fractional percentages and in moving between compound interest and simple interest calculations. Given Data / Assumptions: Compound interest CI for 3 years = Rs. 1,270. Rate r = 16⅔% per annum = 16.666...%. Time t = 3 years. The same rate and time apply for computing simple interest. Concept / Approach: First, convert 16⅔% to a fraction: 16⅔% = 50/3 percent, which is 1/6 as a yearly fraction. For compound interest, the amount after 3 years is: A = P * (1 + 1/6)^3 = P * (7/6)^3 = P * 343/216. So CI = A − P = P * (343/216 − 1) = P * (127/216). Given CI, we can solve for P. Then, simple interest for 3 years at rate 1/6 per year equals P * (1/6) * 3 = P/2. Step-by-Step Solution: Step 1: Use CI expression. CI = P * 127/216 = 1,270. P = 1270 * 216 / 127. 1270 / 127 = 10, so P = 10 * 216 = Rs. 2,160. Step 2: Compute simple interest for 3 years. Rate as fraction per year = 1/6. SI = P * (1/6) * 3 = P * 3/6 = P/2. SI = 2160 / 2 = Rs. 1,080. Verification / Alternative check: We can quickly check the CI expression with P = 2,160: CI = 2160 * 127 / 216 = 10 * 127 = Rs. 1,270, which matches the given CI. This confirms that the principal is correct, and therefore the simple interest calculation P/2 is also reliable. Why Other Options Are Wrong: Rs. 1,090, Rs. 1,180, Rs. 1,109, and Rs. 980 all arise from computational errors such as misreading the fractional rate, using an incorrect exponent, or not simplifying P * 3/6 properly. None of them equal P/2 with P = 2,160. Common Pitfalls: Students often misinterpret 16⅔% or treat it as 16.23%. Others may forget that the given Rs. 1,270 is compound interest rather than the amount. Careful fraction handling and stepwise substitution help avoid these mistakes. Final Answer: The simple interest on the same sum for 3 years at the same rate is Rs. 1,080.

Question 8

Albert invested Rs. 8,000 in a fixed deposit scheme for 2 years at a compound interest rate of 5% per annum, compounded annually. What total amount will Albert receive on maturity of this fixed deposit after 2 years?

Accepted Answer

Introduction: This is a direct compound interest amount calculation. Albert puts his money into a fixed deposit with compounding once per year, and we need to calculate the maturity amount after 2 years using the standard compound interest formula. Given Data / Assumptions: Principal P = Rs. 8,000. Rate r = 5% per annum. Time t = 2 years. Compounding is annual. Concept / Approach: For compound interest with annual compounding, the amount after t years is: A = P * (1 + r/100)^t. We substitute P = 8,000, r = 5, and t = 2 to get the maturity amount. The compound interest itself would be A − P, but the question specifically asks for the total amount Albert receives on maturity. Step-by-Step Solution: Step 1: Determine the yearly multiplier. 1 + r/100 = 1 + 5/100 = 1.05. Step 2: Compute the square for 2 years. (1.05)^2 = 1.1025. Step 3: Compute the amount A. A = 8000 * 1.1025. A = 8000 * (1 + 0.1025) = 8000 + 8000 * 0.1025. 8000 * 0.1025 = 820. So A = 8000 + 820 = Rs. 8,820. Verification / Alternative check: We can also compute year by year. After 1 year, amount = 8000 * 1.05 = 8,400. After the second year, interest is 5% of 8,400 which is 420. So final amount = 8,400 + 420 = Rs. 8,820. This matches the earlier result from the formula. Why Other Options Are Wrong: Rs. 8,600 and Rs. 8,500 underestimate the effect of compounding and align more with incorrect simple interest style calculations. Rs. 8,830 brings in a small computational error on the percentage multiplication. Rs. 9,000 is too large and does not correspond to 5% for only 2 years. Common Pitfalls: Some learners may accidentally compute simple interest for 2 years and then add it to the principal, missing the second year interest on the increased amount. Others may square the rate incorrectly or misplace a decimal when calculating 1.05 squared. Double checking every step avoids these errors. Final Answer: On maturity, Albert will receive a total of Rs. 8,820.

Question 9

Find the compound interest on Rs. 15,225 for 9 months at 16% per annum, if the interest is compounded quarterly (that is, every 3 months).

Accepted Answer

Introduction: This problem combines compound interest with quarterly compounding and a time period expressed in months. Such questions are common in banking and finance aptitude sections and require careful conversion of the annual rate and time into quarterly equivalents before applying the compound interest formula. Given Data / Assumptions: Principal P = Rs. 15,225. Annual rate r = 16% per annum. Interest is compounded quarterly. Total time = 9 months. Concept / Approach: Since compounding is quarterly, we need: Rate per quarter = r / 4 = 16 / 4 = 4%. Number of quarters in 9 months = 9 / 3 = 3. Then we apply the quarterly compound interest formula: A = P * (1 + rate per quarter / 100)^(number of quarters). Finally, CI = A − P. Step-by-Step Solution: Step 1: Quarterly rate and periods. Quarterly rate = 4%. Number of quarters n = 3. Step 2: Compute amount. A = 15225 * (1.04)^3. (1.04)^2 = 1.0816. (1.04)^3 = 1.0816 * 1.04 = 1.124864. A ≈ 15225 * 1.124864 ≈ Rs. 17,126.05. Step 3: Compound interest. CI = A − P ≈ 17,126.05 − 15,225 ≈ Rs. 1,901.05. Rounded to the nearest rupee, CI ≈ Rs. 1,901. Verification / Alternative check: We can partially verify by approximating. A 4% increase three times roughly corresponds to an overall increase a little above 12%. Twelve percent of 15,225 is about 1,827. Since compounding makes it slightly higher, a value near 1,900 is reasonable, and the precise calculation yields about 1,901. This confirms our answer lies in the expected range. Why Other Options Are Wrong: Rs. 1,911 and Rs. 1,909 are close but arise from rounding the intermediate steps differently or miscalculating the product 15225 * 1.124864. Rs. 1,907 and Rs. 1,895 are further away and come from cruder approximations or using simple interest instead of compound interest. Common Pitfalls: A very common mistake is to treat 9 months as 9 years or 0.75 years and then apply annual compounding incorrectly. Another error is to divide the rate by 3 instead of by 4 when compounding quarterly. Always align the number of compounding periods with the chosen time units. Final Answer: The compound interest on Rs. 15,225 for 9 months at 16% per annum compounded quarterly is approximately Rs. 1,901.

Question 10

At what annual rate of compound interest, compounded yearly, will a sum of Rs. 50,000 amount to Rs. 73,205 in 2 years?

Accepted Answer

Introduction: This question asks us to determine the unknown compound interest rate when we know the initial principal and the final amount after a specified time. It is a classic reverse compound interest problem that checks algebraic manipulation of the amount formula A = P * (1 + r/100)^n. Given Data / Assumptions: Principal P = Rs. 50,000. Amount after 2 years A = Rs. 73,205. Time t = 2 years. Interest is compounded annually at rate r% per annum. Concept / Approach: The amount formula for compound interest is: A = P * (1 + r/100)^n. We rearrange this to solve for r. For n = 2: (1 + r/100)^2 = A / P. Taking the square root will give 1 + r/100, from which we find r. Recognising familiar decimal squares can speed up the calculation. Step-by-Step Solution: Step 1: Form the ratio A / P. A / P = 73205 / 50000 = 1.4641. So (1 + r/100)^2 = 1.4641. Step 2: Take square root. We know (1.21)^2 = 1.4641 because 121^2 = 14641. Therefore 1 + r/100 = 1.21. Step 3: Solve for r. r/100 = 0.21, so r = 21 percent. Verification / Alternative check: Verify by recomputing the amount using r = 21 percent: A = 50000 * (1.21)^2. (1.21)^2 = 1.4641. A = 50000 * 1.4641 = Rs. 73,205. This matches the given amount exactly, confirming that the rate is correct. Why Other Options Are Wrong: At 19 percent, 17 percent, 15 percent or 23 percent per annum, the squared multipliers would be different from 1.4641 and therefore would not produce an amount of Rs. 73,205 from Rs. 50,000 in 2 years. These options correspond to incorrect guesses or miscalculated square roots. Common Pitfalls: Some learners place A / P directly equal to (1 + r/100) rather than its square, forgetting that the time is 2 years. Others take an approximate square root and do not verify it by squaring again. Always confirm that the square of your candidate multiplier returns the correct ratio A / P. Final Answer: The required annual compound interest rate is 21 percent per annum.

Question 11

The compound interest on a certain sum of money for 2 years at 10% per annum is Rs. 1,155. What will be the simple interest on the same sum at half this rate (that is, 5% per annum) for double the time (that is, 4 years)?

Accepted Answer

Introduction: In this problem we are given a compound interest figure and asked to determine a simple interest figure for a different rate and time, but for the same principal. This links both types of interest and checks whether you can extract the principal from the compound interest data and then apply the simple interest formula correctly. Given Data / Assumptions: Compound interest CI = Rs. 1,155. Rate for CI r₁ = 10% per annum. Time for CI t₁ = 2 years, compounded annually. For SI: rate r₂ = 5% per annum (half of 10%). Time for SI t₂ = 4 years (double of 2 years). Concept / Approach: First, use the compound interest formula for 2 years: A = P * (1 + 10/100)^2 = P * 1.21. Then CI = A − P = 0.21P. Since CI is given, we can solve for P. After getting P, we apply the simple interest formula: SI = P * r₂ * t₂ / 100. Step-by-Step Solution: Step 1: Get principal from CI. CI = 0.21P = 1,155. P = 1155 / 0.21. P = 115500 / 21 = Rs. 5,500. Step 2: Apply SI at 5% for 4 years. SI = P * r₂ * t₂ / 100. SI = 5500 * 5 * 4 / 100. SI = 5500 * 20 / 100 = 5500 / 5 = Rs. 1,100. Verification / Alternative check: We can verify the principal by recomputing the compound interest: A = 5500 * 1.21 = Rs. 6,655. CI = A − P = 6655 − 5500 = Rs. 1,155, which matches the given CI. This confirms that P is correct and thus the simple interest calculation is reliable. Why Other Options Are Wrong: Rs. 5,500 and Rs. 4,120 correspond more to principal or amount figures rather than the interest over 4 years at 5%. Rs. 1,400 and Rs. 950 arise from misapplication of time or rate, such as forgetting that the rate is halved and the time is doubled, or making arithmetic mistakes in multiplying 5500 * 20 / 100. Common Pitfalls: One common error is to treat Rs. 1,155 as simple interest and use SI formula directly to find P, which would give a wrong value. Another mistake is to forget to double the time while halving the rate in the second part. Clear separation of the two stages avoids confusion. Final Answer: The simple interest for 4 years at 5% per annum on the same sum is Rs. 1,100.

Question 12

If you deposit $4,000 into an account paying 6% annual interest compounded quarterly, how much money will be in the account at the end of 5 years (that is, what will be the accumulated amount including interest)?

Accepted Answer

Introduction: This is a standard compound interest problem involving quarterly compounding in a dollars based investment context. It tests the correct use of the compound interest formula with a nominal annual rate converted into a rate per quarter and the number of compounding periods over several years. Given Data / Assumptions: Principal P = $4,000. Nominal annual interest rate r = 6% per annum. Interest is compounded quarterly. Time t = 5 years. Concept / Approach: For quarterly compounding, we use: Rate per quarter = r / 4 = 6 / 4 = 1.5% = 0.015. Number of quarters n = 4 * 5 = 20. The amount formula becomes: A = P * (1 + 0.06 / 4)^(4 * 5) = 4000 * (1.015)^20. We then compute or use known approximations for (1.015)^20 to find A. Step-by-Step Solution: Step 1: Convert annual rate to quarterly. Rate per quarter = 0.06 / 4 = 0.015. Step 2: Determine the number of compounding periods. n = 4 * 5 = 20 quarters. Step 3: Apply the compound interest formula. A = 4000 * (1.015)^20. Using standard compound tables or a precise calculation, (1.015)^20 ≈ 1.346855. A ≈ 4000 * 1.346855 ≈ 5387.42. Thus, the accumulated amount is about $5,387.42. Verification / Alternative check: We can sanity check by approximating. A 6% annual rate for 5 years with annual compounding would roughly give a multiplier near (1.06)^5 ≈ 1.338. Quarterly compounding should be slightly more than this, and 1.346855 is reasonably close and slightly bigger, which matches expectations. Multiplying by 4000 gives a value just above $5,350, so $5,387.42 is plausible. Why Other Options Are Wrong: $3,387.42 is less than the principal and clearly impossible after 5 years of positive interest. $4,387.42 corresponds to a very small overall gain inconsistent with 6% over 5 years. $6,387.42 or $7,487.42 would require much higher effective rates or longer times than those given. Common Pitfalls: Learners sometimes mistakenly use the annual rate directly as 6% for 20 periods, or they use 6% for 5 periods instead of 1.5% for 20 periods. Others forget to convert 6% into 0.06 before dividing. Careful handling of the rate and number of periods is essential for quarterly compounding problems. Final Answer: The amount in the account after 5 years will be approximately $5,387.42.

Question 13

Calculate the final amount on a principal of Rs. 1,875 invested for 2 years at 4% per annum, if the interest is compounded yearly.

Accepted Answer

Introduction: This question asks for the maturity amount on a small principal when compound interest is applied annually for 2 years. It is a straightforward application of the compound interest formula with a modest rate and time period. Given Data / Assumptions: Principal P = Rs. 1,875. Annual rate r = 4%. Time t = 2 years. Compounding is annual. Concept / Approach: The formula for the amount under compound interest is: A = P * (1 + r/100)^t. We will substitute P, r and t, compute the multiplier (1.04)^2, and then multiply by the principal to get the amount. The interest itself could be obtained as A − P, but the question specifically asks for the amount. Step-by-Step Solution: Step 1: Compute the yearly multiplier. 1 + r/100 = 1 + 4/100 = 1.04. Step 2: Square the multiplier for 2 years. (1.04)^2 = 1.0816. Step 3: Compute the final amount. A = 1875 * 1.0816. 1875 * 1 = 1,875. 1875 * 0.0816 ≈ 153. Therefore A ≈ 1,875 + 153 = Rs. 2,028. Verification / Alternative check: We can verify year by year. After 1 year, amount = 1875 * 1.04 = 194,? actually 1875 * 4% = 75, so amount = 1,950. After the second year, interest = 4% of 1,950 which is 78. So final amount = 1,950 + 78 = Rs. 2,028. This matches the answer obtained via the formula. Why Other Options Are Wrong: Rs. 676 and Rs. 776 are far below the principal and therefore impossible as maturity amounts. Rs. 1,778 and Rs. 1,925 are too low and correspond to miscalculations of one year of interest or to ignoring compounding. Only Rs. 2,028 reflects two years of 4% compound interest. Common Pitfalls: A frequent mistake is to compute simple interest only for 2 years and then forget to add interest on the increased amount in the second year. Another is to miscalculate 4% of 1,950. Writing every step neatly helps to avoid arithmetic slips. Final Answer: The final amount after 2 years will be Rs. 2,028.

Question 14

A woman invests Rs. 2,000 at the start of each year at 5% compound interest per annum. Assuming the first Rs. 2,000 is invested for 2 full years and the second Rs. 2,000 is invested for 1 full year, how much will the total value of her investments b…

Accepted Answer

Introduction: This is an example of an annuity due type problem where equal amounts are invested at the start of each year and earn compound interest for different lengths of time. The question tests the ability to handle multiple deposits with varying investment periods under the same interest rate. Given Data / Assumptions: First investment = Rs. 2,000 at the start of year 1. Second investment = Rs. 2,000 at the start of year 2. Rate of compound interest = 5% per annum. Interest is compounded annually. We need the total value at the end of year 2. Concept / Approach: The first deposit earns interest for 2 years, while the second deposit earns interest for only 1 year by the time we reach the end of year 2. We treat each deposit separately using the compound interest amount formula: A = P * (1 + r/100)^n for the appropriate n, and then sum the two resulting amounts. Step-by-Step Solution: Step 1: Compute the value of the first deposit. P₁ = 2000, n₁ = 2 years, r = 5%. A₁ = 2000 * (1.05)^2 = 2000 * 1.1025 = Rs. 2,205. Step 2: Compute the value of the second deposit. P₂ = 2000, n₂ = 1 year. A₂ = 2000 * 1.05 = Rs. 2,100. Step 3: Total value at the end of year 2. Total amount = A₁ + A₂ = 2,205 + 2,100 = Rs. 4,305. Verification / Alternative check: We can reason that the total investment is Rs. 4,000 and that average holding time is between 1 and 2 years, so the total should be somewhat above Rs. 4,200. The computed Rs. 4,305 is consistent with this expectation and matches the precise calculation using the multipliers 1.1025 and 1.05. Why Other Options Are Wrong: Rs. 430 is far too small, clearly missing the principal amount. Rs. 4,355 and Rs. 4,350 represent small arithmetic errors or incorrect exponents. Rs. 4,205 ignores part of the compounding or miscalculates one year of interest. Common Pitfalls: Many students mistakenly treat both amounts as if they were invested for 2 years or both for 1 year. Another mistake is to add Rs. 4,000 and then calculate interest on that total for some arbitrary time. The correct method is to treat each cash flow separately with its appropriate time period. Final Answer: At the end of the 2nd year, her investments will be worth Rs. 4,305 in total.

Question 15

What is the compound interest for 1 year on a sum of Rs. 20,000 at the rate of 40% per annum, if the interest is compounded half yearly (that is, every 6 months)?

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Introduction: In this question we are asked to compute the compound interest earned in 1 year when a high annual rate of 40% is applied, but compounding occurs half yearly. This problem highlights the impact of compounding frequency on the total interest earned. Given Data / Assumptions: Principal P = Rs. 20,000. Nominal annual rate r = 40% per annum. Compounding frequency = half yearly. Total time = 1 year. Concept / Approach: For half yearly compounding: Rate per half year = r / 2 = 40 / 2 = 20%. Number of half years in 1 year = 2. We then use the compound interest formula: A = P * (1 + rate per half year / 100)^2. Compound interest CI is then A − P. Step-by-Step Solution: Step 1: Compute half yearly rate and periods. Rate per half year = 20%. Number of periods n = 2. Step 2: Compute the amount. A = 20000 * (1.20)^2. (1.20)^2 = 1.44. A = 20000 * 1.44 = Rs. 28,800. Step 3: Compute compound interest. CI = A − P = 28,800 − 20,000 = Rs. 8,800. Verification / Alternative check: We can reason that if interest were not compounded half yearly, simple interest at 40% for 1 year would be 0.40 * 20,000 = Rs. 8,000. Because of half yearly compounding, the interest is slightly greater than Rs. 8,000, and Rs. 8,800 aligns with the exact calculation using the 1.44 multiplier. Why Other Options Are Wrong: Rs. 8,000 corresponds to simple interest or annual compounding, not half yearly compounding. Rs. 8,650 and Rs. 8,750 are intermediate guesses that do not match the exact squared multiplier 1.44. Rs. 7,200 is far too small and inconsistent with a 40% nominal rate. Common Pitfalls: A common mistake is to forget that interest is compounded half yearly and to use 40% once. Others divide the rate incorrectly or square 1.4 instead of 1.2. Always remember that half yearly compounding at 40% annual rate means two periods with 20% each, leading to the factor (1.20)^2. Final Answer: The compound interest for 1 year is Rs. 8,800.

Question 16

Shawn invested one half of his total savings in a bond that paid simple interest for 2 years and received Rs. 550 as interest. He invested the remaining half in another bond that paid compound interest, compounded annually for the same 2 years and a…

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Introduction: This is a mixed simple interest and compound interest problem, where the same rate is used in both parts. Shawn splits his savings into two equal halves and earns different interest amounts because of the difference in how interest is calculated. The objective is to find his original total savings by forming and solving equations. Given Data / Assumptions: Total savings = S (unknown). First half invested in simple interest for 2 years gives interest = Rs. 550. Second half invested in compound interest for 2 years gives interest = Rs. 605. Rate of interest r% per annum is the same for both bonds, and CI is compounded annually. Concept / Approach: Let each half be S/2. For the simple interest investment: SI = (S/2) * r * 2 / 100 = S * r / 100 = 550. From this, we can express S in terms of r. For the compound interest half, we use: CI = (S/2) * [(1 + r/100)^2 − 1] = 605. By substituting S from the first relation into the second, r can be found, and then S can be determined. Step-by-Step Solution: Step 1: From the simple interest part. S * r / 100 = 550 ⇒ S * r = 55,000. So S = 55000 / r. Step 2: From the compound interest part. Principal for CI = S/2. CI for 2 years = (S/2) * [(1 + r/100)^2 − 1] = 605. Expand (1 + r/100)^2 − 1 = 2r/100 + (r/100)^2. This equals r(200 + r) / 10000. So 605 = (S/2) * r(200 + r) / 10000. Substitute S = 55000 / r to eliminate S. 605 = (55000 / (2r)) * r(200 + r) / 10000 = 27500 * (200 + r) / 10000. So 605 = 2.75 * (200 + r). 200 + r = 605 / 2.75 = 220 ⇒ r = 20%. Step 3: Find S. S = 55000 / 20 = Rs. 2,750. Verification / Alternative check: Simple interest on S/2 = 2750/2 = 1,375 at 20% for 2 years: SI = 1375 * 20 * 2 / 100 = 1375 * 0.4 = Rs. 550. Compound interest on 1,375 at 20% for 2 years: Amount = 1375 * (1.2)^2 = 1375 * 1.44 = Rs. 1,980. CI = 1980 − 1375 = Rs. 605. Both match the given values, confirming S = 2,750. Why Other Options Are Wrong: All other options produce either wrong simple interest or wrong compound interest when substituted back, because they do not satisfy both conditions simultaneously for a single consistent rate. Common Pitfalls: The most common error is to treat the two halves independently without linking them through the same rate, or to fail to eliminate r properly. Some learners also misexpand (1 + r/100)^2. Keeping track of algebraic expressions step by step is crucial. Final Answer: Shawn's total savings before investing in the two bonds were Rs. 2,750.

Question 17

Mr. Mike borrowed Rs. 8,500 at 4% per annum compound interest, with interest compounded annually. What is the total compound interest for 2 years on this loan?

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Introduction: This is a straightforward compound interest computation where the principal, rate, time, and compounding frequency are all clearly specified. The goal is to find the interest portion only, not the final amount, after 2 years of annual compounding. Given Data / Assumptions: Principal P = Rs. 8,500. Rate r = 4% per annum. Time t = 2 years. Compounding is annual. Concept / Approach: The compound amount after 2 years is: A = P * (1 + r/100)^2. The compound interest CI is A − P. We can compute (1.04)^2 and then multiply by 8,500 to find A and subtract the principal. Step-by-Step Solution: Step 1: Compute the multiplier. 1 + r/100 = 1 + 4/100 = 1.04. (1.04)^2 = 1.0816. Step 2: Compute the amount A. A = 8500 * 1.0816. 8500 * 0.0816 = 8500 * 816 / 10000 ≈ 693.60. So A ≈ 8500 + 693.60 = Rs. 9,193.60. Step 3: Compound interest. CI = A − P = 9,193.60 − 8,500 = Rs. 693.60. Verification / Alternative check: We can do the year by year method. After 1 year, amount = 8500 * 1.04 = 8,840. Interest in the first year = 340. In the second year, interest is 4% of 8,840 which is 353.60. Total interest over 2 years = 340 + 353.60 = Rs. 693.60, which confirms our result. Why Other Options Are Wrong: Rs. 639.60 is close but lower, and it typically comes from mistakenly applying simple interest for 2 years at 4%. Rs. 9,139.60 and Rs. 9,193.60 are amounts, not interest; using these as CI misreads the question. Rs. 600 is a rough approximation that does not match precise calculations. Common Pitfalls: A common mistake is to stop after computing SI for 2 years and assume it equals CI. Others misinterpret the amount given as the interest. Always distinguish carefully between the total amount and the interest portion, and ensure that compounding is applied year by year when required. Final Answer: The total compound interest on the loan for 2 years is Rs. 693.60.

Question 18

If a sum of Rs. 12,500 is invested for 1 year at 12% per annum interest compounded semi-annually (that is, twice a year), then what is the total interest earned in that year?

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Introduction: This question examines the effect of semi annual compounding compared to simple annual interest. The goal is to find the interest earned over 1 year when the nominal annual rate is 12% but compounding occurs twice a year on a principal of Rs. 12,500. Given Data / Assumptions: Principal P = Rs. 12,500. Annual nominal rate r = 12% per annum. Compounding frequency = semi annual (two times per year). Time t = 1 year. Concept / Approach: For semi annual compounding: Rate per half year = r / 2 = 6%. Number of half years = 2. The amount is: A = P * (1 + 0.06)^2. The compound interest CI is A − P. This will be slightly more than simple interest at 12% for 1 year due to compounding. Step-by-Step Solution: Step 1: Compute half yearly rate and periods. Rate per half year = 6%. Number of periods = 2. Step 2: Compute the amount. A = 12500 * (1.06)^2. (1.06)^2 = 1.1236. A = 12500 * 1.1236 = 12500 + 12500 * 0.1236. 12500 * 0.1236 = 1545. So A = 12500 + 1545 = Rs. 14,045. Step 3: Interest earned. CI = A − P = 14,045 − 12,500 = Rs. 1,545. Verification / Alternative check: For simple interest at 12% for 1 year, SI would be 12,500 * 12 / 100 = Rs. 1,500. Since semi annual compounding should yield a bit more than simple interest, an answer slightly above 1,500 such as 1,545 is reasonable and matches the precise calculation. Why Other Options Are Wrong: Rs. 1,505 and Rs. 1,535 are close but do not match the exact product 12500 * 0.1236. Rs. 1,550 is slightly larger and misaligns with the squared multiplier 1.1236. Rs. 1,450 is less than simple interest and therefore impossible for compound interest at the same nominal rate. Common Pitfalls: A common error is to use 12% once for 1 year without adjusting for semi annual compounding. Others may miscalculate (1.06)^2 or forget to subtract the principal to obtain the interest portion only. Always separate the steps of finding the amount and then the interest. Final Answer: The total interest earned in 1 year is Rs. 1,545.

Question 19

What amount will Rs. 1,500 grow to in 3 years if it is invested at 20% per annum compound interest, with interest compounded annually?

Accepted Answer

Introduction: This question is a standard compound interest amount problem. We are asked to find the future value of Rs. 1,500 invested for 3 years at a relatively high rate of 20% per annum, compounded annually. It demonstrates how quickly money can grow under compound interest at higher rates. Given Data / Assumptions: Principal P = Rs. 1,500. Rate r = 20% per annum. Time t = 3 years. Interest is compounded annually. Concept / Approach: The amount under compound interest is given by: A = P * (1 + r/100)^t. Here, (1 + r/100) = 1.20, and we raise this to the power of 3 for 3 years. The amount includes both the original principal and the accumulated interest. Step-by-Step Solution: Step 1: Compute the multiplier. 1 + r/100 = 1 + 20/100 = 1.20. Step 2: Raise to the power 3. (1.20)^2 = 1.44. (1.20)^3 = 1.44 * 1.20 = 1.728. Step 3: Compute the amount. A = 1500 * 1.728. 1500 * 1 = 1,500. 1500 * 0.728 = 1092. So A = 1,500 + 1,092 = Rs. 2,592. Verification / Alternative check: We can verify year by year. End of year 1: 1500 * 1.2 = 1,800. End of year 2: 1,800 * 1.2 = 2,160. End of year 3: 2,160 * 1.2 = 2,592. This stepwise calculation confirms the same result as the direct formula approach. Why Other Options Are Wrong: Rs. 2,492 is slightly low, as if one year of full 20% growth were missed in part. Rs. 2,352 would correspond to a lower effective rate or less than 3 years of compounding. Duplicate Rs. 2,352 and Rs. 2,802 do not match the exact cube of 1.2 multiplied by 1,500. Common Pitfalls: Some students may apply simple interest and calculate 1500 + 1500 * 20 * 3 / 100 = 1500 + 900 = 2,400, which underestimates the amount since it ignores interest on interest. Others may miscalculate powers of 1.2. Breaking the calculation into year by year steps is an easy way to avoid mistakes. Final Answer: The amount after 3 years at 20% compound interest will be Rs. 2,592.

Question 20

Mr. Gupta borrowed a sum of money on compound interest and plans to repay the entire amount at the end of 2 years. Under which of the following sets of statements can the amount to be repaid be uniquely determined? I. The rate of interest is 5% per …

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Introduction / Context: This is a data sufficiency question based on compound interest. Instead of directly asking you to calculate the final amount, it asks which combination of given statements is enough to determine the amount that Mr. Gupta must repay after 2 years at compound interest. Given Data / Assumptions: Mr. Gupta borrows a sum on compound interest for 2 years. I: The rate of interest is 5% per annum. II: Simple interest on the same sum in one year at the same rate is Rs. 600. III: The amount borrowed equals 10 times the simple interest in 2 years at the same rate. We must decide which statements (alone or in combination) are sufficient to determine the amount payable after 2 years. Concept / Approach: We use compound interest and simple interest relations. For principal P and annual rate r%, simple interest for 1 year is P * r / 100 and for 2 years is 2P * r / 100. For compound interest, the amount after 2 years is P * (1 + r/100)^2. A set of statements is sufficient if P and r can be uniquely found, because then the amount is uniquely determined. Step-by-Step Solution: From I alone: we know r = 5% but not P, so the amount cannot be found. Not sufficient. From II alone: P * r / 100 = 600 gives one equation with two unknowns P and r. Not sufficient. From III alone: P = 10 * (2P * r / 100) implies 1 = 20r / 100, so r = 5%. Rate is known but P is still unknown. Not sufficient. I + II together: r = 5%, and P * 5 / 100 = 600 gives P = 12,000. Amount after 2 years can now be found, so this pair is sufficient. II + III together: III gives r = 5%. Using II again gives P = 12,000. This pair is also sufficient. I + III together: both only confirm r = 5% but give nothing new about P. Still insufficient. Verification / Alternative check: Once P = 12,000 and r = 5% are known from II with either I or III, the amount is P * (1.05)^2 = 12,000 * 1.1025 = Rs. 13,230, which is uniquely determined. No other combination besides II with either I or III allows us to determine both P and r. Why Other Options Are Wrong: Only I or only III never give P. Statement II alone leaves both P and r unknown. Even I and III together only fix r. Therefore any option claiming sufficiency without using II in combination is incorrect. Common Pitfalls: Many learners treat data sufficiency as if they must always compute the final numeric answer. Here the focus is only on whether the information is enough. Another mistake is not checking all combinations that involve statement II, which is the key to obtaining P once the rate is known. Final Answer: The correct choice is Statement II together with either statement I or statement III is sufficient.

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