Question 1

A sum of money amounts to Rs 6690 after 3 years and to Rs 10035 after 6 years at the same compound interest rate. Find the original principal (sum of money) invested.

Accepted Answer

Introduction: This compound interest problem illustrates how knowing the amounts at two different times can help you find both the growth factor and the original principal. You are given the amounts after 3 years and 6 years at the same rate and must work backwards to determine the starting sum. Given Data / Assumptions: Amount after 3 years (A3) = Rs 6690. Amount after 6 years (A6) = Rs 10035. The same compound interest rate is applied for all years. Compounding is assumed to be annual. We must find the principal P. Concept / Approach: Under compound interest, A = P * (1 + r)^n (taking r as a decimal for simplicity). So A6 = P * (1 + r)^6 and A3 = P * (1 + r)^3. If we divide A6 by A3, the principal cancels, leaving (1 + r)^3. From this ratio we find (1 + r) and then use A3 to compute P. Step-by-Step Solution: A3 = P * (1 + r)^3 = 6690A6 = P * (1 + r)^6 = 10035Divide A6 by A3: A6 / A3 = (1 + r)^6 / (1 + r)^3 = (1 + r)^3So (1 + r)^3 = 10035 / 6690 = 1.5Thus 1 + r = (1.5)^(1/3)Now P = A3 / (1 + r)^3 = 6690 / 1.5 = Rs 4460 Verification / Alternative Check: Check the amounts using P = Rs 4460 and (1 + r)^3 = 1.5. After 3 years: 4460 * 1.5 = 6690, which matches A3. After 6 years: 4460 * (1.5)^2 = 4460 * 2.25 = Rs 10035, which matches A6. Hence, P = Rs 4460 is correct. Why Other Options Are Wrong: Rs 4360, Rs 4560, Rs 4660, Rs 4200: When used as principal values, they do not simultaneously produce both 6690 at 3 years and 10035 at 6 years under any single compound rate. They break the ratio relationship between A3 and A6. Common Pitfalls: Students sometimes attempt to compute the rate first using complicated logarithms instead of exploiting the ratio trick. Another error is to treat the difference of amounts as simple interest, which is not valid because compound interest grows multiplicatively, not linearly with time. Final Answer: The original sum of money invested is Rs 4460.

Question 2

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

Accepted Answer

Introduction: This question focuses on the relationship between simple interest and compound interest for a short period of 2 years. There is a well known shortcut formula for the difference between compound interest and simple interest over 2 years, which makes the calculation much faster than computing each interest separately. Given Data / Assumptions: Time period (t) = 2 years. Rate of interest (r) = 5% per annum. Difference between compound interest and simple interest over 2 years = Rs 41. We must find the principal P. Compounding is annual. Concept / Approach: For 2 years at rate r% per annum compounded annually, the difference between compound interest and simple interest is given by the formula: Difference = P * (r/100)^2. This arises from the additional interest earned in the second year on the first year’s interest. Using this compact relation allows us to solve directly for the principal without separately computing CI and SI. Step-by-Step Solution: Difference = P * (r/100)^2Here, r = 5%, so r/100 = 0.05Therefore, Difference = P * (0.05)^2 = P * 0.0025We are given: P * 0.0025 = 41So P = 41 / 0.0025P = 41 * 400 = Rs 16400 Verification / Alternative Check: You can compute SI and CI explicitly. Simple interest for 2 years: SI = P * r * t / 100 = 16400 * 5 * 2 / 100 = Rs 1640. Amount under CI after 2 years: A = 16400 * (1 + 0.05)^2 = 16400 * 1.1025 = Rs 18081. CI = 18081 - 16400 = Rs 1681. Difference = 1681 - 1640 = Rs 41, which confirms the correctness of the principal. Why Other Options Are Wrong: Rs 7200, Rs 9600, Rs 8400, and Rs 8800: Each of these values, when substituted into the formula P * (0.05)^2, gives a difference that is not equal to Rs 41. They therefore cannot satisfy the given condition. Common Pitfalls: An often seen error is to confuse the formula and use r/100 instead of (r/100)^2. Another mistake is to apply the shortcut even when the period is not exactly 2 years or when compounding is not annual, which can lead to incorrect answers in more complex scenarios. Final Answer: The required principal (sum of money) is Rs 16400.

Question 3

Find the compound interest on Rs 10000 in 2 years at 4% per annum, when the interest is compounded half yearly (that is, every six months).

Accepted Answer

Introduction: This problem tests your understanding of compound interest when interest is compounded more frequently than once a year. Here, interest is compounded half yearly, which means the rate per period and the number of periods both change from the standard annual case. Given Data / Assumptions: Principal (P) = Rs 10000. Rate of interest (r) = 4% per annum. Time period = 2 years. Interest is compounded every 6 months (half yearly). We are asked to find the compound interest (CI), not the total amount. Concept / Approach: When interest is compounded half yearly, the annual rate is split into two, and the number of periods doubles. Thus, the rate per half year is r/2 and the number of compounding periods in t years is 2t. We use the amount formula A = P * (1 + r/(2*100))^(2t) and then compute CI as A - P. Step-by-Step Solution: Rate per half year = 4% / 2 = 2% = 0.02Number of half yearly periods in 2 years = 2 * 2 = 4Amount after 4 periods: A = 10000 * (1 + 0.02)^4(1.02)^2 = 1.0404 and (1.02)^4 = (1.0404)^2 ≈ 1.08243216So A ≈ 10000 * 1.08243216 = Rs 10824.32 (rounded)Compound interest CI = A - P = 10824.32 - 10000 = Rs 824.32 Verification / Alternative Check: You may compute step by step: after first half year, amount = 10000 * 1.02 = 10200. After second half year, multiply again by 1.02 and continue for four multiplications. The final amount will again be approximately Rs 10824.32, giving CI ≈ Rs 824.32. Why Other Options Are Wrong: Rs 524.32 and Rs 624.32: These underestimate the interest by ignoring some of the compounding effect.Rs 724.32: Close but still smaller than the correct calculated compound interest.Rs 432.32: This is far too low and might come from using simple interest or treating the time as only 1 year. Common Pitfalls: Students often forget to double the number of periods or divide the interest rate correctly when switching to half yearly compounding. Some also mistakenly apply the simple interest formula, which does not capture the growth on interest itself. Careful handling of the rate per period and the number of periods avoids these errors. Final Answer: The compound interest on Rs 10000 in 2 years at 4% per annum, compounded half yearly, is Rs 824.32.

Question 4

For a certain sum of money, what is the rate of interest per cent per annum? (I) The amount becomes double in 5 years under simple interest. (II) The difference between compound interest and simple interest on a certain amount for 2 years is Rs 400.…

Accepted Answer

Introduction: This is a data sufficiency style question about simple and compound interest. The goal is not to compute the exact rate numerically, but to decide which pieces of information are sufficient to uniquely determine the rate of interest per annum for a given sum. Given Data / Assumptions: Statement I: An amount doubles itself in 5 years on simple interest. Statement II: The difference between compound interest and simple interest on a certain amount in 2 years is Rs 400. Statement III: Simple interest earned per annum is Rs 2000. The rate of interest is constant and is expressed per annum. Concept / Approach: For simple interest, A = P * (1 + r * t / 100). If an amount doubles in t years, then 1 + r * t / 100 = 2. For the difference between compound and simple interest over 2 years, the formula is Difference = P * (r/100)^2. Knowing SI per year also gives P * r / 100. We check which statements alone or in combination give enough equations to solve for r uniquely. Step-by-Step Solution: Using Statement I: 2P = P * (1 + r * 5 / 100) so 1 + 5r/100 = 2Thus 5r/100 = 1, so r = 20%. Hence I alone is sufficient.Using Statement II alone: Difference = P * (r/100)^2 = 400, which contains two unknowns P and r, so II alone is not sufficient.Using Statement III alone: SI per year = P * r / 100 = 2000, again two unknowns P and r, so III alone is not sufficient.Combining II and III: From III, P * r / 100 = 2000.From II, P * (r/100)^2 = 400.Dividing the second equation by the first, we get (P * r^2 / 10000) / (P * r / 100) = 400 / 2000.This simplifies to r / 100 = 1/5, so r = 20%. Hence II and III together are sufficient. Verification / Alternative Check: Both approaches, via Statement I alone and via Statements II and III together, give the same unique rate of 20% per annum. This confirms that I alone is sufficient and also that II and III, when used together, are sufficient. Why Other Options Are Wrong: Option I only: misses the fact that II and III together are also sufficient.Option II and III only: ignores that I alone is already enough.Option All I, II and III: overstates the requirement; you do not need all three.Option None of these: incorrect because sufficient information clearly exists in I and also in II with III. Common Pitfalls: Students may wrongly try to plug numbers into each statement without focusing on sufficiency. Others may treat II or III alone as sufficient, forgetting that both involve two unknowns. It is important to translate each statement into equations, count unknowns and see when you have enough independent equations to solve for the rate. Final Answer: The rate of interest per annum can be determined from Statement I alone or from Statements II and III together, so the correct choice is “I only or II and III only”.

Question 5

A debt of Rs 7620 is to be discharged by equal annual payments made at the end of each year for 3 years, if interest is charged at 16 2/3% per annum compounded annually. What is the amount of each annual payment?

Accepted Answer

Introduction: This is an application of compound interest to loan repayment by equal annual instalments. The idea is that a present debt is equivalent to the present value of a series of future payments, discounted at the given compound interest rate. You must find the fixed instalment that exactly clears the debt in 3 years. Given Data / Assumptions: Present debt (principal) = Rs 7620. Number of annual payments = 3. Interest rate r = 16 2/3% per annum, which is 1/6 in decimal (approximately 16.67%). Interest is compounded annually. Payments are made at the end of each year. Concept / Approach: Let the annual payment be A. The present value of each payment is obtained by discounting it back to today using the factor 1 / (1 + r)^t. The sum of the present values of all payments must equal the current debt of Rs 7620. We then solve for A using the present value of an annuity formula. Step-by-Step Solution: Let r = 1/6 so 1 + r = 7/6.Present value PV of payments: PV = A/(1 + r) + A/(1 + r)^2 + A/(1 + r)^3PV = A / (7/6) + A / (7/6)^2 + A / (7/6)^3PV = A * (6/7 + 36/49 + 216/343)Compute the sum of fractions: 6/7 = 294/343, 36/49 = 252/343, and 216/343 remains as is.Total = (294 + 252 + 216) / 343 = 762 / 343So PV = A * (762/343)Set PV equal to 7620: A * (762/343) = 7620A = 7620 * (343/762) = 10 * 343 = Rs 3430 Verification / Alternative Check: Check by compounding each payment to the end of 3 years. Payment at end of year 1 grows for 2 more years, payment at year 2 grows for 1 year, payment at year 3 does not grow. The sum of these compounded amounts should equal the future value of the debt after 3 years, confirming that the instalment is correct. Why Other Options Are Wrong: Rs 5430, Rs 4430 and Rs 2430: Each of these, when used in the present value equation, does not equal Rs 7620.Rs 2630: This underestimates the payments and would not discharge the full debt with interest. Common Pitfalls: Common errors include forgetting to discount each payment to present value or incorrectly treating the interest as simple interest. Another mistake is mixing future value and present value concepts and equating payments directly to the principal without accounting for the time value of money. Final Answer: The required equal annual payment to discharge the debt is Rs 3430 per year.

Question 6

The difference between the simple interest on a certain sum at the rate of 10% per annum for 2 years and the compound interest on the same sum for the same period, when interest is compounded every 6 months, is Rs 124.05. What is the principal sum?

Accepted Answer

Introduction: This question compares simple interest and compound interest over the same time span and at the same nominal annual rate, but with compounding done half yearly for the compound interest case. You are required to use both formulas and the given difference to determine the principal amount. Given Data / Assumptions: Rate of interest (r) = 10% per annum. Time (t) = 2 years. Simple interest is computed annually. Compound interest is computed half yearly, so every 6 months. Difference (CI − SI) = Rs 124.05. We need to find the principal P. Concept / Approach: Simple interest for 2 years is given by SI = P * r * t / 100. For compound interest with half yearly compounding, the rate per half year is r/2 and the number of periods in t years is 2t. The compound amount is A = P * (1 + r/(2*100))^(2t), and CI = A − P. We set CI − SI equal to the given difference and solve for P. Step-by-Step Solution: Simple interest for 2 years: SI = P * 10 * 2 / 100 = 0.20PHalf yearly rate = 10/2 = 5% = 0.05Number of half year periods in 2 years = 4Compound amount: A = P * (1 + 0.05)^4 = P * (1.05)^4(1.05)^2 = 1.1025, so (1.05)^4 ≈ 1.21550625Thus CI = A − P = P * 1.21550625 − P = 0.21550625PDifference CI − SI = (0.21550625P − 0.20P) = 0.01550625PGiven CI − SI = 124.05, so 0.01550625P = 124.05Therefore P ≈ 124.05 / 0.01550625 ≈ Rs 8000 Verification / Alternative Check: Using P = 8000, SI = 0.20 * 8000 = Rs 1600. Compound amount A = 8000 * (1.05)^4 ≈ 8000 * 1.21550625 ≈ Rs 9724.05. CI = 9724.05 − 8000 ≈ Rs 1724.05. Difference CI − SI ≈ 1724.05 − 1600 = Rs 124.05, which matches the given difference exactly. Why Other Options Are Wrong: Rs 6000 and Rs 12000: Substituting these in the formulas gives differences that are either too small or too large compared to Rs 124.05.Rs 10000: Also fails to produce the exact difference when SI and CI are computed.None of these: Incorrect, because Rs 8000 perfectly satisfies the condition. Common Pitfalls: It is common to forget to adjust the rate and number of periods when changing from annual to half yearly compounding. Some students also mistakenly compare the total amount rather than CI and SI themselves. Careful attention to the formulas and to the meaning of the difference helps avoid these errors. Final Answer: The required principal sum is Rs 8000.

Question 7

If the simple interest on a sum of money at 5% per annum for 3 years is Rs 1200, find the compound interest on the same sum for the same period at the same rate of 5% per annum.

Accepted Answer

Introduction: This question connects simple interest and compound interest for the same principal, rate and time period. Using the given simple interest value, you first find the principal and then use the compound interest formula to compute the compound interest for the same conditions. Given Data / Assumptions: Simple interest (SI) for 3 years = Rs 1200. Rate of interest (r) = 5% per annum. Time (t) = 3 years. Same principal and same rate are used for compound interest. Compounding is annual for the compound interest calculation. Concept / Approach: For simple interest, SI = P * r * t / 100. This allows us to solve for the principal P. Once P is known, we apply the standard compound interest amount formula A = P * (1 + r/100)^t and compute CI = A − P. This systematic approach avoids guesswork and ensures an accurate answer. Step-by-Step Solution: Given SI = 1200, r = 5%, t = 3 years.Using SI formula: SI = P * r * t / 1001200 = P * 5 * 3 / 100 = P * 15 / 100So P = 1200 * 100 / 15 = Rs 8000Now compute compound amount for 3 years at 5%:A = P * (1 + 5/100)^3 = 8000 * (1.05)^3(1.05)^2 = 1.1025 and (1.05)^3 ≈ 1.157625A ≈ 8000 * 1.157625 = Rs 9261Compound interest CI = A − P = 9261 − 8000 = Rs 1261 Verification / Alternative Check: You can also compute compound interest year by year. First year interest: 8000 * 5% = 400 (amount 8400). Second year interest: 8400 * 5% = 420 (amount 8820). Third year interest: 8820 * 5% = 441 (amount 9261). Total CI = 400 + 420 + 441 = 1261, confirming the result. Why Other Options Are Wrong: Rs 1251, Rs 1271 and Rs 1281: These values are close but do not match the exact total interest obtained from correct compounding.Rs 1210: This likely arises from confusion with simple interest or from rounding errors. Common Pitfalls: A frequent mistake is to use the simple interest formula again for compound interest or to apply the simple interest on the original principal for all three years. Another error is rounding intermediate values too early, which can slightly distort the final answer. Using the amount formula directly helps maintain accuracy. Final Answer: The compound interest on the same sum for 3 years at 5% per annum is Rs 1261.

Question 8

A certain sum amounts to Rs 7350 in 2 years and to Rs 8575 in 3 years at the same compound interest rate. Find the original sum of money (principal) and hence determine the principal from the given options.

Accepted Answer

Introduction: This compound interest question gives you two future amounts of the same principal at the same interest rate but for different times. Using these, you can first deduce the annual growth factor and then work backwards to find the original principal. It trains you to use ratios instead of directly solving for the rate first. Given Data / Assumptions: Amount after 2 years (A2) = Rs 7350. Amount after 3 years (A3) = Rs 8575. Same compound interest rate for all years. Interest is compounded annually. We must find the principal P. Concept / Approach: Under compound interest, amounts after 2 and 3 years are A2 = P * (1 + r)^2 and A3 = P * (1 + r)^3, where r is the annual rate in decimal form. Dividing A3 by A2 cancels P, leaving (1 + r). Then you can substitute back into A2 or A3 to solve for P directly. Step-by-Step Solution: A2 = P * (1 + r)^2 = 7350A3 = P * (1 + r)^3 = 8575Divide A3 by A2:A3 / A2 = (1 + r)^3 / (1 + r)^2 = 1 + rSo 1 + r = 8575 / 7350Simplify 8575 / 7350 = 1.166666..., which is 7/6Hence 1 + r = 7/6, so r = 1/6 = 16 2/3% per annumNow find P using A2: 7350 = P * (7/6)^2 = P * 49/36Therefore P = 7350 * 36 / 49P = 7350 * (36/49) = Rs 5400 Verification / Alternative Check: Using P = 5400 and r = 16 2/3%, we can check the amounts. After 2 years: P * (7/6)^2 = 5400 * 49/36 = 7350. After 3 years: P * (7/6)^3 = 5400 * 343/216 = 8575. Both values match the given amounts, confirming that P is correct. Why Other Options Are Wrong: Rs 3400, Rs 4400, Rs 6400 and Rs 4200: None of these values, when combined with a single consistent interest rate, give both 7350 after 2 years and 8575 after 3 years. Each fails at least one of the two conditions. Common Pitfalls: A common mistake is to attempt to calculate the rate directly without using the ratio method, which can lead to messy equations. Another is to misinterpret the amounts as including additional deposits or withdrawals, which is not stated in the question. Remember that under pure compound interest, the ratio of amounts at times differing by 1 year directly gives the growth factor 1 + r. Final Answer: The original principal that grows to Rs 7350 in 2 years and Rs 8575 in 3 years is Rs 5400.

Question 9

Find the compound interest on Rs 16000 at 20% per annum for 9 months, if interest is compounded quarterly (every 3 months).

Accepted Answer

Introduction: This question applies the compound interest formula when the compounding period is quarterly. The time given is 9 months, which corresponds to three quarters of a year, so you must express the time in terms of the number of compounding periods rather than just years. Given Data / Assumptions: Principal (P) = Rs 16000. Rate of interest (r) = 20% per annum. Interest is compounded quarterly (every 3 months). Total time = 9 months. We must find the compound interest (CI) for this period. Concept / Approach: For quarterly compounding, the annual rate is divided by 4 to get the rate per quarter, and the time in months is converted into number of quarters. The amount is given by A = P * (1 + r/(4*100))^n, where n is the number of quarters. Then CI = A − P. Step-by-Step Solution: Quarterly rate = 20% / 4 = 5% = 0.05 per quarterTotal time = 9 months = 9 / 3 = 3 quartersAmount after 3 quarters: A = 16000 * (1 + 0.05)^3(1.05)^2 = 1.1025 and (1.05)^3 ≈ 1.157625So A ≈ 16000 * 1.157625 = Rs 18522Compound interest CI = A − P = 18522 − 16000 = Rs 2522 Verification / Alternative Check: Compute stepwise per quarter: after first quarter, amount = 16000 * 1.05 = 16800. After second quarter: 16800 * 1.05 = 17640. After third quarter: 17640 * 1.05 = 18522. The increase over the original principal is 18522 − 16000 = 2522, matching our previous result. Why Other Options Are Wrong: Rs 2422 and Rs 2322: These underestimate the compounding effect of 20% per annum when applied quarterly.Rs 3522: This overestimates the interest and might come from incorrectly using a full year instead of 9 months.Rs 2022: Much too small and likely based on a simple interest calculation. Common Pitfalls: Typical errors include forgetting to convert 9 months to 3 quarters or using 20% directly as the quarterly rate. Some also treat the period as 1 year, which inflates the interest. Always align the rate per period with the number of compounding periods used in the formula. Final Answer: The compound interest on Rs 16000 at 20% per annum for 9 months, compounded quarterly, is Rs 2522.

Question 10

A sum is lent at 20% per annum compound interest. What is the ratio of the increase in amount (that is, the interest earned) in the 4th year to the increase in amount in the 5th year?

Accepted Answer

Introduction: This question explores how the yearly interest amounts behave under compound interest at a constant rate. It asks for the ratio of the interest earned in the 4th year to the interest earned in the 5th year, without even needing the actual principal value. Given Data / Assumptions: Rate of interest r = 20% per annum. Compounding is annual. We are interested in the increase in amount (interest) during the 4th year and the 5th year. The principal P is not specified, but we will see it cancels out. Concept / Approach: Under compound interest, the amount at the start of year n is P * (1 + r)^(n−1). The interest earned in year n is this amount multiplied by r. Therefore, yearly interests form a geometric sequence. We use this to find the ratio of the interest in the 4th and 5th years. Step-by-Step Solution: Let P be the initial principal and r = 20% = 0.20.Amount at the start of 4th year = P * (1.20)^3Interest in 4th year I4 = P * (1.20)^3 * 0.20Amount at the start of 5th year = P * (1.20)^4Interest in 5th year I5 = P * (1.20)^4 * 0.20Now I4 : I5 = [P * (1.20)^3 * 0.20] : [P * (1.20)^4 * 0.20]Cancel P and 0.20: I4 : I5 = (1.20)^3 : (1.20)^4 = 1 : 1.201 : 1.20 = 5 : 6 Verification / Alternative Check: Assume an easy principal, say P = Rs 100. Compute actual interest values. Interest in 4th year is 100 * (1.2)^3 * 0.2 = 100 * 1.728 * 0.2 = Rs 34.56. Interest in 5th year is 100 * (1.2)^4 * 0.2 = 100 * 2.0736 * 0.2 = Rs 41.472. Ratio 34.56 : 41.472 simplifies to 5 : 6, confirming our earlier result. Why Other Options Are Wrong: 4 : 5 and 5 : 4: These ratios do not match the actual growth pattern under 20% compound interest.Cannot be determined: Incorrect because the principal cancels out, so the ratio can be determined.1 : 1: This would imply equal interest in both years, which is not true under compound interest. Common Pitfalls: Some learners mistakenly calculate simple interest for each year or assume that the interest added each year is constant. Under compound interest, the amount grows, so the interest in later years is always higher than in earlier years at the same rate. Recognizing the geometric pattern makes such ratio questions straightforward. Final Answer: The ratio of the increase in amount in the 4th year to that in the 5th year is 5 : 6.

Question 11

The compound interest on a certain sum of money for 2 years is Rs 832 and the simple interest on the same sum for the same period is Rs 800. What is the difference between the compound interest and simple interest on the same sum for 3 years?

Accepted Answer

Introduction: This problem requires you to work with both simple and compound interest for the same principal and rate, using given information for 2 years to determine the rate and principal, and then extending the calculation to 3 years to find the new difference between compound interest and simple interest. Given Data / Assumptions: Simple interest (SI) for 2 years = Rs 800. Compound interest (CI) for 2 years = Rs 832. Same principal P and rate r for both types of interest. Interest is compounded annually. We need the difference between CI and SI for 3 years. Concept / Approach: First use the simple interest formula SI = P * r * t / 100 to form one equation. Then use the compound interest formula for 2 years to form another equation. Solve the system to find P and r. Then compute SI and CI for 3 years and find the new difference CI − SI. Step-by-Step Solution: For 2 years, SI2 = P * r * 2 / 100 = 800, so P * r / 100 = 400.CI2 = P * (1 + r/100)^2 − P = 832.Let x = r/100. Then CI2 = P * [(1 + x)^2 − 1] = P * (2x + x^2) = 832.But P * x = 400, so 2P * x = 800.Thus 800 + P * x^2 = 832, so P * x^2 = 32.Divide P * x^2 = 32 by P * x = 400 to find x:x^2 / x = 32 / 400, so x = 0.08, hence r = 8%.P * x = 400 means P * 0.08 = 400, so P = 5000.Now for 3 years: SI3 = P * r * 3 / 100 = 5000 * 8 * 3 / 100 = Rs 1200.CI3 = P * (1 + 0.08)^3 − P = 5000 * (1.08)^3 − 5000.(1.08)^3 ≈ 1.259712, so CI3 ≈ 5000 * 1.259712 − 5000 = 6298.56 − 5000 = Rs 1298.56.Difference for 3 years: CI3 − SI3 = 1298.56 − 1200 = Rs 98.56. Verification / Alternative Check: You can also compute year wise interest. After finding P and r, calculate the interest added each year under compound interest and sum over 3 years, then compare with SI over 3 years. This gives the same difference of Rs 98.56. Why Other Options Are Wrong: Rs 48 and Rs 66.56: These are too small and do not match the computed difference for 3 years.Rs 88.56: Close but slightly smaller than the correct value; it may come from rounding errors or miscalculations.None of these: Incorrect because Rs 98.56 fits perfectly. Common Pitfalls: One common error is to incorrectly assume that the difference for 3 years is simply 1.5 times the difference for 2 years, which is not true for compound interest. Others may apply the shortcut formula for difference incorrectly or forget to solve for both P and r before extending the time period. Final Answer: The difference between compound interest and simple interest on the same sum for 3 years is Rs 98.56.

Question 12

In what time will Rs 1000 amount to Rs 1331 at 10% per annum compound interest, if the interest is compounded annually?

Accepted Answer

Introduction: This question asks you to find the time needed for a given principal to grow to a specified amount under compound interest. It is a straightforward application of the compound interest amount formula, solved for the time period when rate and amounts are known. Given Data / Assumptions: Principal (P) = Rs 1000. Amount (A) = Rs 1331. Rate of interest (r) = 10% per annum. Interest is compounded annually. We must find the time t in years. Concept / Approach: For annual compounding, the amount is given by A = P * (1 + r/100)^t. We plug in the known values of A, P and r, and then solve for t. In many exam problems like this, the numbers are chosen so that (1 + r/100)^t becomes a familiar power such as 1.1^3, making the solution simple without logarithms. Step-by-Step Solution: Formula: A = P * (1 + r/100)^tSubstitute: 1331 = 1000 * (1 + 10/100)^t = 1000 * (1.1)^tSo (1.1)^t = 1331 / 1000 = 1.331Recall that 1.1^3 = 1.331 (since 11^3 = 1331).Therefore, (1.1)^t = (1.1)^3, which implies t = 3. Verification / Alternative Check: You can compute the amount year by year. After 1 year: 1000 * 1.1 = 1100. After 2 years: 1100 * 1.1 = 1210. After 3 years: 1210 * 1.1 = 1331. This confirms that it takes exactly 3 years to reach Rs 1331 at 10% compound interest. Why Other Options Are Wrong: 2 years: Gives 1000 * 1.1^2 = Rs 1210, which is less than 1331.4 years: Gives 1000 * 1.1^4 = Rs 1464.10, which is more than 1331.5 years and 6 years: Produce even larger amounts and therefore cannot be correct. Common Pitfalls: Some students try to apply the simple interest formula or linearly scale the interest, which is not appropriate for compound interest. Others may struggle with solving for t algebraically, but recognizing common powers of 1.1 or 1.05 can greatly simplify such questions in competitive exams. Final Answer: The required time for Rs 1000 to amount to Rs 1331 at 10% compound interest is 3 years.

Question 13

There is a 60% increase in an amount in 6 years at simple interest. Using the same rate of interest, what will be the compound interest on Rs 12000 after 3 years, if interest is compounded annually?

Accepted Answer

Introduction: This problem links simple interest and compound interest. First, you must deduce the annual rate of simple interest from the information about a 60% increase over 6 years. Then, using the same rate, you compute the compound interest on a new principal over 3 years. Given Data / Assumptions: Under simple interest, an amount increases by 60% in 6 years. We infer the annual rate from this information. We then consider a principal P = Rs 12000. Time for compound interest calculation = 3 years. Same annual rate is used, with annual compounding. Concept / Approach: If an amount increases by 60% in 6 years under simple interest, then total simple interest over 6 years is 60% of the principal. This gives SI = 0.60P = P * r * 6 / 100, allowing us to find r. Then we use the compound interest formula A = P * (1 + r/100)^3 for P = 12000 and obtain CI = A − P. Step-by-Step Solution: Let the original principal be P (for the 60% increase information).Total SI over 6 years = 0.60P.Using SI formula: SI = P * r * 6 / 100 = 0.60P.Thus P cancels, giving r * 6 / 100 = 0.60.So r = 0.60 * 100 / 6 = 10% per annum.Now consider P = Rs 12000 at 10% compound interest for 3 years.A = 12000 * (1 + 10/100)^3 = 12000 * (1.1)^3.(1.1)^3 = 1.331, so A = 12000 * 1.331 = Rs 15972.Compound interest CI = A − P = 15972 − 12000 = Rs 3972. Verification / Alternative Check: You may also compute CI year by year. First year interest: 12000 * 10% = 1200 (amount 13200). Second year interest: 13200 * 10% = 1320 (amount 14520). Third year interest: 14520 * 10% = 1452 (amount 15972). Total CI = 1200 + 1320 + 1452 = 3972, in agreement with the formula result. Why Other Options Are Wrong: Rs 4972 and Rs 5972: These overestimate the compound interest that would accrue at 10% per annum on Rs 12000 for 3 years.Rs 2972 and Rs 3672: These underestimate the true compound interest, possibly due to mixing up simple and compound interest calculations. Common Pitfalls: One frequent mistake is to treat the 60% increase as directly giving the 10% rate without showing the algebra, which can be error prone if adapted to other numbers. Another is to continue using the simple interest formula when computing the interest on Rs 12000, thereby missing the compounding effect. Final Answer: The compound interest on Rs 12000 after 3 years at the same rate is Rs 3972.

Question 14

On a certain sum of money, the simple interest for 2 years is Rs 660, while the compound interest for the same sum and period is Rs 696.30, the rate of interest being the same in both cases. Find the rate of interest per annum.

Accepted Answer

Introduction: Here you are given both simple interest and compound interest for the same principal and the same time period. This information allows you to determine the annual rate of interest by forming equations and solving for the unknown rate, which is a common type of aptitude question. Given Data / Assumptions: Simple interest SI2 for 2 years = Rs 660. Compound interest CI2 for 2 years = Rs 696.30. Same principal P and rate r for both types of interest. Interest under compound interest is compounded annually. Time t = 2 years. Concept / Approach: Simple interest is SI = P * r * t / 100, which yields one relation between P and r. Compound interest for 2 years is CI2 = P * (1 + r/100)^2 − P. Knowing both SI and CI gives two equations in P and r. Solving them simultaneously yields the rate r. This is an algebraic manipulation exercise combined with interest formulas. Step-by-Step Solution: From simple interest: SI2 = P * r * 2 / 100 = 660So P * r / 100 = 330. Call this equation (1).From compound interest: CI2 = P * (1 + r/100)^2 − P = 696.30Let x = r/100. Then CI2 = P * [(1 + x)^2 − 1] = P * (2x + x^2) = 696.30But from (1), P * x = 330.Thus P * (2x + x^2) = 2 * P * x + P * x^2 = 2 * 330 + P * x^2 = 696.30So 660 + P * x^2 = 696.30, giving P * x^2 = 36.30Now divide P * x^2 by P * x: (P * x^2) / (P * x) = x = 36.30 / 330 = 0.11Hence r/100 = 0.11, so r = 11% per annum. Verification / Alternative Check: Using r = 11%, we can find P from P * r / 100 = 330. So P * 0.11 = 330, which gives P = 3000. Simple interest for 2 years at 11%: 3000 * 11 * 2 / 100 = 660 (checks). Compound amount: 3000 * (1.11)^2 = 3000 * 1.2321 = 3696.30. CI2 = 3696.30 − 3000 = 696.30 (checks again). Why Other Options Are Wrong: 10%, 12% and 13%: Each of these rates, when used with an appropriate principal, does not simultaneously produce SI2 = 660 and CI2 = 696.30.9%: Gives even smaller values of CI and cannot match the given compound interest. Common Pitfalls: Common mistakes include treating the difference CI − SI as directly equal to P * (r/100)^2 without verifying the assumptions or mismanaging the algebra when solving for r. Working step by step with clear substitutions helps keep the work accurate. Final Answer: The rate of interest per annum is 11%.

Question 15

The difference between the simple interest and the compound interest on a certain sum of money at 20% per annum for 2 years is Rs 56. What is the value of the sum (principal)?

Accepted Answer

Introduction: This is a classic shortcut question involving the difference between compound interest and simple interest for 2 years. At a given annual rate, you can use a direct formula to relate this difference to the principal, making the calculation quick and efficient. Given Data / Assumptions: Rate of interest (r) = 20% per annum. Time t = 2 years. Difference between SI and CI for 2 years = Rs 56 (CI − SI or SI − CI in magnitude). Interest is compounded annually for the compound interest case. We must find the principal P. Concept / Approach: For 2 years with annual compounding, the difference between compound interest and simple interest is given by Difference = P * (r/100)^2. This comes from the fact that compound interest earns extra interest in the second year on the first year’s interest, while simple interest does not. Step-by-Step Solution: Use the formula: Difference = P * (r/100)^2Here r = 20%, so r/100 = 0.20Thus Difference = P * (0.20)^2 = P * 0.04We are given: P * 0.04 = 56So P = 56 / 0.04P = 56 * 25 = Rs 1400 Verification / Alternative Check: If P = Rs 1400 and r = 20%, simple interest for 2 years is SI2 = 1400 * 20 * 2 / 100 = 1400 * 0.40 = Rs 560. Compound amount A = 1400 * (1.20)^2 = 1400 * 1.44 = Rs 2016. CI2 = 2016 − 1400 = Rs 616. Difference CI2 − SI2 = 616 − 560 = Rs 56, matching the given difference. Why Other Options Are Wrong: Rs 3680, Rs 2650 and Rs 1170: Each of these values, when used as P, leads to a difference of P * 0.04 that is not equal to Rs 56.Rs 2800: Gives a difference of 2800 * 0.04 = Rs 112, which is double the required value. Common Pitfalls: Some candidates mistakenly use the simple interest formula instead of the dedicated difference formula, or they forget to square the rate fraction. Others misinterpret which difference is given, but since only the magnitude matters here, the formula still applies correctly. Final Answer: The required sum of money (principal) is Rs 1400.

Question 16

The population of a town was 3600 three years ago and is 4800 at present. Assuming that the population has been growing at a constant rate compounded annually, what will be the population of the town three years from now?

Accepted Answer

Introduction: This problem uses the concept of compound growth, commonly applied to population growth, investment growth and similar situations. You are given the population at two times separated by three years and must project the population three years further into the future, assuming the same annual growth rate continues. Given Data / Assumptions: Population 3 years ago (P0) = 3600. Population now (after 3 years) = 4800. Growth rate is constant and compounded annually. We need the population 3 years from now, that is, 6 years after the base time P0. Concept / Approach: Let the annual growth factor be k = (1 + r), where r is the annual growth rate in decimal form. Then population after 3 years is P0 * k^3 and after 6 years is P0 * k^6. Given P0 * k^3 = 4800 and P0 = 3600, we can find k^3, and then k^6. Using k^6, we find the population after 6 years, which is 3 years from now. Step-by-Step Solution: Let P0 = 3600 and let k be the annual growth factor.After 3 years: P0 * k^3 = 4800So 3600 * k^3 = 4800Thus k^3 = 4800 / 3600 = 4/3Now, after 6 years (3 more years from now): population = P0 * k^6But k^6 = (k^3)^2 = (4/3)^2 = 16/9Therefore, population after 6 years = 3600 * 16/93600 / 9 = 400, so population = 400 * 16 = 6400 Verification / Alternative Check: You can think of it another way: from 3 years ago to now, population multiplied by 4/3. Assuming the same growth, over the next 3 years it again multiplies by 4/3. Hence future population = 4800 * 4/3 = 4800 * 1.333... = 6400, which agrees with the previous method. Why Other Options Are Wrong: 6000 and 6500: These values do not correspond to the same compound growth factor that took the population from 3600 to 4800.6600 and 7200: Both are larger than the correctly computed value and are inconsistent with the implied annual growth rate. Common Pitfalls: Some students mistakenly treat the growth as linear and add the same absolute increase (for example, adding 1200 every three years) instead of compounding. Others compute the annual rate and then misapply it over the subsequent 3 years. Using the k^3 and k^6 approach helps keep the computation clean and avoids rate rounding errors. Final Answer: The population of the town three years from now will be 6400.

Question 17

Find the compound interest on Rs 7500 at 4% per annum for 2 years, if the interest is compounded annually.

Accepted Answer

Introduction: This question is a direct application of the compound interest formula with annual compounding. It is a standard type of problem in aptitude tests to check whether you can correctly compute the amount and then derive the compound interest from it. Given Data / Assumptions: Principal (P) = Rs 7500. Rate of interest (r) = 4% per annum. Time (t) = 2 years. Interest is compounded annually. We must find the compound interest (CI) only. Concept / Approach: The standard amount formula for compound interest with annual compounding is A = P * (1 + r/100)^t. The compound interest earned is CI = A − P. We simply substitute the given values of P, r and t into the formula, compute the amount and subtract the principal. Step-by-Step Solution: Given P = 7500, r = 4%, t = 2 years.Amount after 2 years: A = 7500 * (1 + 4/100)^2 = 7500 * (1.04)^2(1.04)^2 = 1.0816So A = 7500 * 1.0816 = Rs 8112Compound interest CI = A − P = 8112 − 7500 = Rs 612 Verification / Alternative Check: You can also compute interest year by year. First year: I1 = 7500 * 4% = 300, amount = 7800. Second year: I2 = 7800 * 4% = 312, amount = 8112. Total interest = 300 + 312 = Rs 612, which confirms the earlier calculation. Why Other Options Are Wrong: Rs 512, Rs 515 and Rs 522: These underestimate the interest, possibly coming from arithmetic mistakes or use of simple interest.Rs 620: Slightly higher than the correct value and may result from rounding errors or miscalculations in squaring 1.04. Common Pitfalls: Students sometimes miscalculate (1.04)^2 or mistakenly use 1.08 instead of 1.04. Others erroneously apply simple interest by multiplying P * r * t / 100, which would give only Rs 600, not the correct compound interest of Rs 612. Final Answer: The compound interest on Rs 7500 at 4% per annum for 2 years is Rs 612.

Question 18

Rs 1301 is divided between A and B so that the amount of A after 7 years is equal to the amount of B after 9 years, if interest is compounded annually at 4% per annum. What is B's share of the money?

Accepted Answer

Introduction: This problem involves dividing a sum between two people so that the future values of their shares at different times become equal under compound interest. It checks your understanding of how amounts grow over time and how to translate equality of future amounts into an equation involving present shares. Given Data / Assumptions: Total sum to be divided between A and B = Rs 1301. Rate of compound interest r = 4% per annum. A's amount is considered after 7 years, B's amount after 9 years. Interest is compounded annually. Amounts of A and B after their respective periods are equal. We must find B's share. Concept / Approach: Let A's present share be x and B's present share be 1301 − x. After 7 years A's amount is x * (1.04)^7. After 9 years B's amount is (1301 − x) * (1.04)^9. Since these future amounts are equal, we equate them and solve for x. From this, we can derive B's share as 1301 − x. Step-by-Step Solution: Let A's present share = x. Then B's present share = 1301 − x.A's amount after 7 years = x * (1.04)^7.B's amount after 9 years = (1301 − x) * (1.04)^9.Given: x * (1.04)^7 = (1301 − x) * (1.04)^9.Divide both sides by (1.04)^7 to simplify:x = (1301 − x) * (1.04)^2.(1.04)^2 = 1.0816, so x = 1.0816 * (1301 − x).x = 1406.0496 − 1.0816x.x + 1.0816x = 1406.0496, so 2.0816x ≈ 1406.05.x ≈ 1406.05 / 2.0816 ≈ 676.Therefore B's share ≈ 1301 − 676 = Rs 625. Verification / Alternative Check: Use approximate shares: A gets Rs 676, B gets Rs 625. A's future amount after 7 years ≈ 676 * (1.04)^7. B's future amount after 9 years ≈ 625 * (1.04)^9. Because (1.04)^9 = (1.04)^7 * (1.04)^2, equality of amounts is consistent with the equation solved above. Using accurate calculator values, the two future amounts match closely, confirming the shares. Why Other Options Are Wrong: Rs 626 and Rs 627: These are very close but do not satisfy the equality of future amounts exactly.Rs 286: Far too small; this would make A's share much larger and disturb the balance of future values.Rs 675: This would leave A with only Rs 626, reversing the roles and not matching the equality with the given rate and times. Common Pitfalls: Students sometimes attempt to divide the sum directly in proportion to the times instead of using compound factors, or they neglect that B's amount accumulates interest for a longer period. It is essential to carefully write the equality of future amounts using the correct compounding factors. Final Answer: B's present share of the money is Rs 625.

Question 19

A certain sum is to be divided between A and B so that after 5 years the amount received by A is equal to the amount received by B after 7 years. The rate of interest is 10% per annum, compounded annually. Find the ratio of the amounts invested by A…

Accepted Answer

Introduction: This compound interest question asks you to divide an investment between two people such that their future amounts, received at different times, are equal. The key is to use the equality of future values and the compound interest formula to derive a ratio of their present investments, without needing the actual total sum. Given Data / Assumptions: A invests an amount that will accumulate for 5 years. B invests an amount that will accumulate for 7 years. Rate of interest r = 10% per annum, compounded annually. The amounts received by A after 5 years and by B after 7 years are equal. We must find the ratio of the present investments of A and B. Concept / Approach: Let A invest an amount a and B invest an amount b. Under compound interest, A's amount after 5 years is a * (1.10)^5 and B's amount after 7 years is b * (1.10)^7. Equating these and simplifying yields a / b = (1.10)^2. Because 1.10 = 11/10, (1.10)^2 = 121/100, so the ratio a : b is 121 : 100. Step-by-Step Solution: Let A's present investment be a and B's present investment be b.A's amount after 5 years: A5 = a * (1 + 10/100)^5 = a * (1.1)^5.B's amount after 7 years: B7 = b * (1.1)^7.Given A5 = B7, so a * (1.1)^5 = b * (1.1)^7.Divide both sides by (1.1)^5: a = b * (1.1)^2.So a / b = (1.1)^2 = (11/10)^2 = 121/100.Therefore, the ratio of their investments is a : b = 121 : 100. Verification / Alternative Check: You may use assumed values such as a = 121x and b = 100x. Then A's amount after 5 years is 121x * (1.1)^5 and B's amount after 7 years is 100x * (1.1)^7. B's amount can be rewritten as 100x * (1.1)^5 * (1.1)^2 = 121x * (1.1)^5, so both amounts are indeed equal, confirming that 121 : 100 is correct. Why Other Options Are Wrong: 125 : 100 and 100 : 125: These ratios would imply a different power relationship, not arising from (1.1)^2.100 : 121: This is simply the inverse of the correct ratio and would make B's present investment higher than A's, which contradicts the condition for equal future values at different times.11 : 10: This corresponds to a single factor of 1.1, whereas we require (1.1)^2 for the 2 year difference in time. Common Pitfalls: Some students mistakenly take the ratio of times 5 : 7 as the ratio of investments or use linear instead of exponential factors for compound interest. Correct handling of powers is crucial: the person whose money is invested for fewer years must start with a larger principal to catch up in future value. Final Answer: The required ratio of amounts invested by A and B is 121 : 100.

Question 20

At compound interest, the principal which amounts to Rs 4913 in 3 years at 6 1/4% per annum compounded annually is to be found. What is this principal (sum invested)?

Accepted Answer

Introduction: This question asks you to reverse the compound interest process. Instead of finding the amount from a known principal, you are given the final amount, rate and time and must determine the original principal. The rate is a mixed fraction, 6 1/4%, which is convenient for expressing as a rational factor. Given Data / Assumptions: Amount after 3 years (A) = Rs 4913. Rate of interest r = 6 1/4% per annum = 6.25%. Time t = 3 years. Interest is compounded annually. We must find the principal P. Concept / Approach: For annual compounding, A = P * (1 + r/100)^t. We know A, r and t. The term (1 + r/100) becomes (1 + 6.25/100) = 1.0625. Notably, 1.0625 can be written as 17/16. Because A and P often come out as perfect cubes in such problems, recognizing (17/16)^3 is helpful. We solve P = A / (1.0625)^3. Step-by-Step Solution: Convert the rate: r = 6.25% = 6 1/4%.1 + r/100 = 1 + 6.25/100 = 1 + 0.0625 = 1.0625.Note that 1.0625 = 17/16.So the amount relation is A = P * (17/16)^3.Given A = 4913, so 4913 = P * (17/16)^3.Therefore P = 4913 / (17/16)^3 = 4913 * (16/17)^3.Notice that 4913 = 17^3, so:P = 17^3 * (16^3 / 17^3) = 16^3 = 4096. Verification / Alternative Check: Verify by forward calculation. Starting with P = 4096 at 6.25% per annum: factor per year = 1.0625 = 17/16. After 3 years, amount = 4096 * (17/16)^3. Since 4096 = 16^3, we get 16^3 * 17^3 / 16^3 = 17^3 = 4913, which matches the given amount. Why Other Options Are Wrong: Rs 3096, Rs 4076 and Rs 4085: None of these values, when grown at 6.25% compound interest for 3 years, will produce exactly Rs 4913.Rs 3500: Also does not match the given amount when the compound interest formula is applied. Common Pitfalls: Students often mishandle the mixed fraction rate or neglect to simplify 6 1/4% to 6.25%. Others attempt to compute (1.0625)^3 in decimal form and round too aggressively, leading to small discrepancies. Recognizing the rational form 17/16 and the cube structure helps achieve an exact and elegant solution. Final Answer: The required principal that amounts to Rs 4913 in 3 years at 6 1/4% compound interest is Rs 4096.

Compound Interest Questions