Question 1

The difference between the compound interest and the simple interest on a certain sum at 12% per annum for 2 years is Rs. 90. What will be the total amount (principal plus interest) at the end of 3 years at the same rate of compound interest?

Accepted Answer

Introduction / Context: This question combines both simple interest and compound interest ideas and asks you to use the known difference between them to deduce the principal. After finding the principal, you must compute the compound amount for a longer period. It is a classic multi-step aptitude question that checks your conceptual understanding of interest formulas. Given Data / Assumptions: Rate of interest r = 12% per annum Time T1 = 2 years Difference between compound interest and simple interest for 2 years = Rs. 90 Same principal P in both cases We need the amount at compound interest after T2 = 3 years Concept / Approach: For the same principal P, rate r and time 2 years, the difference between compound interest and simple interest is: Difference = P * (r / 100)^2This comes from expanding the compound interest expression for 2 years. Once P is found, the amount after 3 years at compound interest is: A3 = P * (1 + r / 100)^3 Step-by-Step Solution: Step 1: Use the difference formula for 2 years.Difference = P * (r / 100)^290 = P * (12 / 100)^2 = P * (144 / 10000)P = 90 * 10000 / 144 = 6250Step 2: Use compound interest formula for 3 years.A3 = 6250 * (1 + 12 / 100)^3A3 = 6250 * (1.12)^3(1.12)^3 = 1.404928A3 = 6250 * 1.404928 = Rs. 8780.80 Verification / Alternative check: You can double-check the principal by computing simple and compound interest for 2 years. Simple interest for 2 years: SI2 = P * r * 2 / 100 = 6250 * 12 * 2 / 100 = Rs. 1500Compound amount after 2 years: A2 = 6250 * (1.12)^2 = 6250 * 1.2544 = 7840CI2 = A2 - P = 7840 - 6250 = Rs. 1590Difference = CI2 - SI2 = 1590 - 1500 = Rs. 90This matches the given difference. Why Other Options Are Wrong: Rs. 8560, Rs. 8673, Rs. 8746, Rs. 8620: These values correspond to using incorrect principal, rate factor, or computing only 2 years instead of 3 years of compounding. Common Pitfalls: One common error is to treat Rs. 90 as the interest for a third year, which is incorrect. Another mistake is forgetting that the difference formula applies specifically for 2 years and involves the square of the rate fraction. Some students also wrongly use simple interest formulas for 3 years instead of compound interest. Keeping track of formulas and time periods avoids these errors. Final Answer: The value of the amount at the end of 3 years at 12% compound interest is Rs. 8780.80.

Question 2

If you deposit $5000 into an account paying 6% annual interest compounded monthly, approximately how many years will it take until the balance in the account reaches $8000?

Accepted Answer

Introduction / Context: This problem tests your understanding of compound interest with monthly compounding and the use of exponents to solve for time. Such questions model real savings or investment accounts where interest is added many times per year, and they require comfort with logarithms or careful estimation. Given Data / Assumptions: Initial deposit (principal) P = $5000 Final amount A = $8000 Nominal annual interest rate r = 6% per year Compounding frequency: monthly (12 times per year) We must find the time T in years Concept / Approach: For nominal annual rate r, compounded monthly, the periodic rate i is: i = r / 12The compound amount formula with monthly compounding is: A = P * (1 + i)^(12 * T)We are given A, P, r, and need to solve for T. This involves taking logarithms or using approximate calculation and then comparing with the options given. Step-by-Step Solution: Step 1: Compute the monthly rate.i = 6% / 12 = 0.06 / 12 = 0.005Step 2: Write the compound interest equation.8000 = 5000 * (1.005)^(12 * T)Step 3: Divide both sides by 5000.8000 / 5000 = (1.005)^(12 * T)1.6 = (1.005)^(12 * T)Step 4: Take natural logarithms on both sides.ln(1.6) = (12 * T) * ln(1.005)T = ln(1.6) / (12 * ln(1.005))Step 5: Approximating, ln(1.6) is about 0.470, ln(1.005) is about 0.004987.T ≈ 0.470 / (12 * 0.004987) ≈ 0.470 / 0.0598 ≈ 7.85 yearsThis is closest to 7.9 years. Verification / Alternative check: You can verify by plugging T = 7.9 years into the formula. Compute the exponent 12 * 7.9 = 94.8 months. Then estimate (1.005)^(94.8) which is approximately 1.6. Multiplying 5000 by 1.6 yields about 8000, confirming that 7.9 years is a very good approximation consistent with the multiple-choice options. Why Other Options Are Wrong: 6.9 years: Too small; the account would not yet grow from 5000 to 8000.8.9 and 9.9 years: These would result in an amount significantly higher than $8000.5.9 years: Much too short a time to increase the investment by 60% at only 6% nominal rate. Common Pitfalls: Some learners mistakenly use annual compounding instead of monthly, which leads to incorrect time estimates. Others try to avoid logarithms but then guess poorly. Remember that for such growth problems with time in the exponent, logarithms are the correct mathematical tool. Also ensure that the rate is converted to a decimal and divided by the correct compounding frequency. Final Answer: It will take approximately 7.9 years for $5000 to grow to $8000 at 6% interest compounded monthly.

Question 3

Calculate the future value of an investment of $3000 at 7% annual interest compounded yearly for 5 years.

Accepted Answer

Introduction / Context: This question is a direct application of the compound interest formula to find the future value of a single lump-sum investment. It reflects typical situations in savings accounts or fixed deposits where interest is compounded annually at a fixed rate over a specified number of years. Given Data / Assumptions: Principal P = $3000 Annual interest rate r = 7% per annum Time T = 5 years Compounding is annual We must find the future value (amount) A after 5 years Concept / Approach: The standard compound amount formula for annual compounding is: A = P * (1 + r / 100)^THere, the deposit is made once at the beginning, and interest is added each year on the growing balance. No additional deposits or withdrawals occur during the 5 years. Step-by-Step Solution: Step 1: Substitute the given values in the formula.A = 3000 * (1 + 7 / 100)^5Step 2: Simplify the rate term.1 + 7 / 100 = 1.07Step 3: Compute (1.07)^5.(1.07)^2 ≈ 1.1449(1.07)^3 ≈ 1.2250(1.07)^4 ≈ 1.3108(1.07)^5 ≈ 1.40255 (more precisely 1.4025517)Step 4: Multiply by the principal.A ≈ 3000 * 1.4025517 ≈ $4207.66 Verification / Alternative check: You can verify using a financial calculator or spreadsheet function such as FV. Setting rate = 0.07, nper = 5, pmt = 0, pv = -3000 will return a future value close to 4207.66, confirming the calculation. Any small difference comes only from rounding intermediate steps. Why Other Options Are Wrong: $5207 and $5687: These assume either a much higher interest rate or a longer time period.$4376: Slightly higher than the correct value, likely from miscalculating (1.07)^5.$4000: Too low; it corresponds roughly to a lower effective interest rate. Common Pitfalls: Students sometimes apply simple interest instead of compound interest, or confuse the number of years. Another common error is to round early when computing powers, leading to noticeable errors in the final amount. Keep more decimal places in intermediate calculations and round only at the end. Final Answer: The future value of $3000 invested at 7% per annum compounded annually for 5 years is approximately $4207.66.

Question 4

Money can be invested at 8% annual interest compounded quarterly. Which is financially larger: receiving $2500 now, or receiving $3800 in 5 years? Compute the present value of $3800 to justify your answer.

Accepted Answer

Introduction / Context: This question compares a lump sum available now with a larger amount to be received in the future, when money can earn compound interest. The key idea is the time value of money: a rupee or dollar today is worth more than the same amount in the future because it can earn interest. We use present value to compare the two options on an equal basis today. Given Data / Assumptions: Option 1: $2500 received now (today) Option 2: $3800 received after 5 years Nominal annual interest rate r = 8% Compounding quarterly, so 4 periods per year We must find the present value of $3800 and decide which amount is larger today Concept / Approach: The present value PV of a future amount A under nominal rate r compounded quarterly is: PV = A / (1 + r / 4)^(4 * T)Here T is time in years. If the present value of $3800 is greater than $2500, the future payment is more attractive at this interest rate. Otherwise, taking $2500 now would be better. Step-by-Step Solution: Step 1: Determine periodic rate and periods.r = 8% per annum, so rate per quarter i = 0.08 / 4 = 0.02Number of quarters in 5 years = 4 * 5 = 20Step 2: Compute present value of $3800.PV = 3800 / (1.02)^20(1.02)^20 ≈ 1.4859PV ≈ 3800 / 1.4859 ≈ $2557.29Step 3: Compare with $2500.Since $2557.29 > $2500, the $3800 in 5 years has a higher present value. Verification / Alternative check: We can also compute the future value of $2500 after 5 years at the same rate: FV = 2500 * (1.02)^20 ≈ 2500 * 1.4859 ≈ $3714.87, which is less than $3800. This again shows that the $3800 received in 5 years is financially better than $2500 now under 8% quarterly compounding. Why Other Options Are Wrong: $1557.29: Far too low, as if a much higher discount rate were used.$2567.00 and $2457.00: Close but not equal to the correctly computed present value, indicating rounding or formula errors.$2500.00: This is simply the immediate amount and not the present value of $3800. Common Pitfalls: Students often confuse present value and future value or forget that quarterly compounding requires dividing the rate and multiplying the time by 4. Another error is comparing 2500 directly with 3800 without adjusting for time value. Using the correct present value formula is essential to make a fair comparison. Final Answer: The present value of $3800 in 5 years at 8% compounded quarterly is approximately $2557.29, which is larger than $2500 now, so the $3800 in 5 years is financially more valuable.

Question 5

Two payments of $10,000 each are due one year and four years from now. If money earns 9% annual interest compounded monthly, what single payment made two years from now would be economically equivalent to these two scheduled payments?

Accepted Answer

Introduction / Context: This problem involves the concept of equivalent cash flows under compound interest. Instead of two separate payments at different future times, we want to replace them with a single payment at an intermediate date that has the same economic value when interest is 9% compounded monthly. This type of calculation is common in loan restructuring and financial planning. Given Data / Assumptions: Payment 1: $10,000 at t = 1 year Payment 2: $10,000 at t = 4 years Interest rate: 9% nominal per annum, compounded monthly We must find a single equivalent payment X at t = 2 years All comparisons are made using the same interest rate and compounding convention Concept / Approach: To find an equivalent payment at a different time, we move each original payment to the target time using compound interest. The value at t = 2 of each payment is: Value at t = 2 of amount A at time t1: A * (1 + i)^n or A / (1 + i)^nwhere i is monthly rate and n is number of months moved forward or backward. Then we sum these equivalent values and set them equal to the single payment X at t = 2. Step-by-Step Solution: Step 1: Monthly interest rate.i = 9% / 12 = 0.09 / 12 = 0.0075 per monthStep 2: Move first payment from t = 1 year to t = 2 years (12 months forward).Value1 = 10000 * (1.0075)^(12)Step 3: Move second payment from t = 4 years to t = 2 years (24 months backward).Value2 = 10000 / (1.0075)^(24)Step 4: Equivalent single payment at t = 2 is:X = Value1 + Value2Using accurate calculation, X ≈ 19296.38, rounded to $19,296. Verification / Alternative check: If you discount the single payment X back to t = 0 and compare it with the present value of the original two payments, you will find they match. This confirms that the replacement stream is economically equivalent under 9% interest compounded monthly. Why Other Options Are Wrong: $19,396, $19,496, $19,596: These correspond to slightly different interest assumptions or compounding periods and do not result from the correct formula.$18,296: Too low; it undervalues the combined effect of the two $10,000 payments. Common Pitfalls: Students sometimes use annual compounding instead of monthly, or simply add the two payments without time adjustment. Another mistake is to move both payments in the wrong direction (both forward or both backward) or to use the wrong number of months. Careful attention to the time diagram and compounding frequency avoids these errors. Final Answer: The equivalent single payment two years from now is approximately $19,296.

Question 6

A chartered bank offers a five-year Escalator Guaranteed Investment Certificate (GIC) that pays annual interest rates of 4%, 4.5%, 5%, 5.5%, and 6% in successive years, compounded at the end of each year. It also offers a regular five-year GIC payin…

Accepted Answer

Introduction / Context: This question compares two investment products: an Escalator GIC with changing annual interest rates and a regular GIC with a constant rate. To compare them, it is useful to know the exact maturity value of the regular five-year 5% GIC on a $1000 investment. This highlights how average rates and compounding interact in real financial products. Given Data / Assumptions: Principal P = $1000 Regular GIC rate r = 5% per annum Time T = 5 years Compounding annually Escalator GIC rates (for comparison) are 4%, 4.5%, 5%, 5.5%, and 6% per year, but the numeric answer sought is the maturity value of the regular GIC Concept / Approach: For the regular GIC, the same 5% rate applies every year. The amount after 5 years with annual compounding is: A = P * (1 + r / 100)^TAfter we find this amount, we can qualitatively compare it with the escalator GIC, whose maturity value is very close but slightly lower than that of the regular GIC. Step-by-Step Solution: Step 1: Use the compound amount formula.A = 1000 * (1 + 5 / 100)^5Step 2: Simplify the factor.1 + 5 / 100 = 1.05Step 3: Compute (1.05)^5.(1.05)^2 = 1.1025(1.05)^3 ≈ 1.157625(1.05)^4 ≈ 1.21550625(1.05)^5 ≈ 1.27628156Step 4: Multiply by the principal.A ≈ 1000 * 1.27628156 ≈ $1276.28 Verification / Alternative check: If we also compute the maturity of the Escalator GIC: A ≈ 1000 * 1.04 * 1.045 * 1.05 * 1.055 * 1.06 ≈ $1276.14. This is very close to, but slightly less than, $1276.28. This confirms that a constant 5% rate over 5 years yields almost the same result as the average of the given escalating rates. Why Other Options Are Wrong: $1234 and $1256: These underestimate the growth at 5% over five years.$1278 and $1260: These are approximations that do not match the precisely calculated compound amount for 5 years at 5%. Common Pitfalls: Students sometimes average the rates and multiply principal by (1 + 0.05 * 5) as if interest were simple, forgetting about compounding. Others may miscalculate (1.05)^5 by rounding too early. Always apply the compound interest formula correctly and keep adequate precision until the final step. Final Answer: The maturity value of $1000 invested in the regular five-year GIC paying 5% compounded annually is $1276.28.

Question 7

If the rate of inflation for the next 20 years is expected to be 2.5% per year, what annual income will be needed 20 years from now to have the same purchasing power as a Rs. 30,000 annual income today?

Accepted Answer

Introduction / Context: This question illustrates the effect of inflation on purchasing power. Even if your nominal income stays the same, rising prices mean that you can buy less in the future with the same rupee amount. To maintain today's standard of living, you need to adjust your future income upward using the inflation rate. Given Data / Assumptions: Current annual income with desired purchasing power = Rs. 30,000 Annual inflation rate = 2.5% per year Time horizon T = 20 years Prices grow at a steady rate of 2.5% every year We want the future nominal income that has the same purchasing power as Rs. 30,000 today Concept / Approach: The future cost of the same basket of goods is increased by inflation. If prices rise at rate i per year, then after T years, the price level multiplies by (1 + i)^T. Therefore, the required future income F to match a present income P in real terms is: F = P * (1 + i)^THere P = 30000 and i = 2.5% = 0.025. Step-by-Step Solution: Step 1: Identify the formula.F = 30000 * (1.025)^20Step 2: Compute (1.025)^20 approximately.(1.025)^10 ≈ 1.28 (approximate)Then ((1.025)^10)^2 ≈ 1.28^2 ≈ 1.6384More accurately, (1.025)^20 ≈ 1.6386Step 3: Multiply by the present income.F ≈ 30000 * 1.6386 ≈ 49,158 Verification / Alternative check: If we instead used a financial calculator or spreadsheet with P = -30000, rate = 0.025, nper = 20, we would again obtain a future value around 49158. This confirms that about Rs. 49,158 is required in 20 years to buy what Rs. 30,000 buys today, assuming 2.5% inflation each year. Why Other Options Are Wrong: Rs. 39,158: This would correspond to a lower effective inflation rate than 2.5%.Rs. 59,158 and Rs. 69,158: These correspond to higher inflation assumptions or a longer time horizon.Rs. 45,000: A rough guess that does not match the compound inflation calculation. Common Pitfalls: A frequent mistake is to treat inflation linearly and multiply 2.5% by 20 to get 50% total, then calculate 30000 * 1.5 = 45000. This ignores compounding of inflation. In reality, inflation itself compounds, so the total increase is more than 50%. Using (1 + i)^T is essential for accurate long-term financial planning. Final Answer: An annual income of approximately Rs. 49,158 will be needed 20 years from now to match the purchasing power of Rs. 30,000 today at 2.5% annual inflation.

Question 8

Originally, payments of $2000 and $1000 were scheduled to be made one year and five years from today, respectively. They are to be replaced by a $1500 payment due four years from today and another payment due two years from today. If money earns 7% …

Accepted Answer

Introduction / Context: This is a cash flow equivalence problem under compound interest with semiannual compounding. Two original payments at different future dates are to be replaced by two new payments at other dates. To preserve economic equivalence, the present value of both streams must be the same when discounted at the given interest rate. Given Data / Assumptions: Original payments: $2000 at t = 1 year, $1000 at t = 5 years Replacement payments: $1500 at t = 4 years, unknown amount X at t = 2 years Interest rate: 7% per annum, compounded semiannually Semiannual rate i = 7% / 2 We compare present values at t = 0 Concept / Approach: For nominal rate r compounded semiannually, the period rate is: i = r / 2and each year has 2 periods. The present value of a future amount A at time t years is: PV = A / (1 + i)^(2 * t)Economic equivalence requires that: PV(original stream) = PV(replacement stream)We solve this equation for X. Step-by-Step Solution: Step 1: Compute semiannual rate.i = 0.07 / 2 = 0.035Step 2: Present value of original payments.PV1 = 2000 / (1.035)^(2 * 1) + 1000 / (1.035)^(2 * 5)PV1 ≈ 2000 / (1.035)^2 + 1000 / (1.035)^10PV1 ≈ 2575.94 (approximate)Step 3: Present value of replacement payments.PV2 = 1500 / (1.035)^(2 * 4) + X / (1.035)^(2 * 2)PV2 = 1500 / (1.035)^8 + X / (1.035)^4Step 4: Set PV1 = PV2 and solve for X.2575.94 = 1500 / (1.035)^8 + X / (1.035)^4Rearranging and multiplying by (1.035)^4 gives X ≈ 1648.79 ≈ $1648 Verification / Alternative check: Substitute X = 1648 back into PV2 and recompute with more precise calculator values. PV2 will match PV1 up to rounding error, confirming the correctness of the unknown payment. Why Other Options Are Wrong: $1548 and $1448: These are smaller than required and would reduce the present value of the replacement stream below the original.$1748: Too large; it would make the replacement stream more valuable than the original.$1348: Much too low to be equivalent at the given interest rate. Common Pitfalls: Some learners mix up the number of compounding periods (using years instead of half-years), or compare future values instead of present values. Others incorrectly move all payments in the wrong direction in time or forget that both streams must be evaluated at the same reference date. Drawing a time line and carefully applying the present value formula for each payment help avoid mistakes. Final Answer: The unknown replacement payment due two years from today must be approximately $1648 to be economically equivalent to the original payment stream.

Question 9

The compound interest on a certain sum for 2 years at 10% per annum is Rs. 1155. What is the simple interest on the same sum for double the time (4 years) at half the rate (5% per annum)?

Accepted Answer

Introduction / Context: This question links compound interest information to a simple interest calculation on the same principal. You must first deduce the principal using the compound interest data, and then use that principal with different time and rate in the simple interest formula. It is a good example of multi-step interest reasoning often seen in aptitude tests. Given Data / Assumptions: Compound interest CI for 2 years at 10% per annum = Rs. 1155 Rate for CI r1 = 10% per annum Time for CI T1 = 2 years Same principal P in both cases Required: simple interest SI for 4 years at 5% per annum (double time, half rate) Concept / Approach: For compound interest with annual compounding: A = P * (1 + r1 / 100)^2CI = A - PFrom CI we can find P. Once P is known, we use simple interest formula for the second part: SI = P * r2 * T2 / 100where r2 = 5% and T2 = 4 years. Step-by-Step Solution: Step 1: Express CI in terms of P.A = P * (1.10)^2 = P * 1.21CI = A - P = P * 1.21 - P = P * 0.21Step 2: Use given CI to find P.P * 0.21 = 1155P = 1155 / 0.21 = 5500Step 3: Use simple interest formula for double time and half rate.r2 = 5%, T2 = 4 yearsSI = P * r2 * T2 / 100SI = 5500 * 5 * 4 / 100 = 5500 * 20 / 100 = 5500 * 0.2 = Rs. 1100 Verification / Alternative check: We may quickly re-check the CI step. If principal is Rs. 5500, amount after 2 years at 10% CI is 5500 * 1.21 = 6655, so compound interest is 6655 - 5500 = 1155, matching the given value. This confirms that P = 5500 is correct and thus our simple interest calculation is reliable. Why Other Options Are Wrong: Rs. 5500: This is the principal, not the simple interest.Rs. 1400 and Rs. 4120: These arise from misusing the rate, time, or compound interest formula.Rs. 1200: A rough guess that is close to but not equal to the correct simple interest. Common Pitfalls: Students sometimes treat the given CI as if it were simple interest and directly apply proportion, which fails here. Another common mistake is to miscompute 10% compound interest for 2 years and obtain the wrong principal. Keeping formulas clear and solving stepwise avoids these errors. Final Answer: The simple interest on the same sum for 4 years at 5% per annum is Rs. 1100.

Question 10

At 6% per annum interest compounded half-yearly, in what time will Rs. 3300 amount to Rs. 3399?

Accepted Answer

Introduction / Context: This question tests understanding of compound interest when the interest is compounded half-yearly. The increase in amount over the principal is relatively small, so the time period is likely to be a fraction of a year. Identifying the correct compounding period and rate is the key step. Given Data / Assumptions: Principal P = Rs. 3300 Final amount A = Rs. 3399 Nominal annual rate r = 6% per annum Compounding half-yearly, that is twice per year We need to find the time T in years Concept / Approach: For nominal rate r compounded half-yearly, the period rate i is: i = r / 2 = 6% / 2 = 3% per half-yearThe compound amount after n half-yearly periods is: A = P * (1 + i / 100)^nWe are given both A and P and must find n, then convert n to years by noting that 2 periods correspond to 1 year. Step-by-Step Solution: Step 1: Use the compound amount formula with half-yearly compounding.3399 = 3300 * (1.03)^nStep 2: Divide both sides by 3300.3399 / 3300 = (1.03)^n3399 / 3300 = 1.03Therefore (1.03)^n = 1.03Step 3: Solve for n.n = 1 half-yearly periodStep 4: Convert to years.Since 1 period corresponds to half a year, T = 0.5 year = half year Verification / Alternative check: Directly check one half-year of compounding: Amount after one half-year: A = 3300 * (1 + 3 / 100) = 3300 * 1.03 = 3399This exactly matches the required amount, confirming that the time is one half-year. Why Other Options Are Wrong: Quarter year or 1.5 years: These would correspond to partial or multiple periods and would not produce exactly Rs. 3399.1 year or 2 years: These correspond to 2 or 4 half-yearly periods, which would give significantly larger amounts than Rs. 3399. Common Pitfalls: Students may forget that the quoted 6% is per annum and that for half-yearly compounding, they must use 3% per half-year. Others confuse the number of periods with the number of years. Always relate the number of compounding periods to the given compounding frequency to interpret the time correctly. Final Answer: The required time for Rs. 3300 to amount to Rs. 3399 at 6% per annum compounded half-yearly is half year.

Question 11

What is the compound interest on Rs. 5000 for 3 years if the rate of interest is 4% for the first year, 3% for the second year and 2% for the third year, with interest compounded annually at these varying rates?

Accepted Answer

Introduction / Context: Unlike standard compound interest questions with a constant rate, this problem uses different interest rates for each of the three years. This reflects real-life situations where interest rates change over time. The approach is still based on compounding, but we apply each year's rate sequentially to the growing amount. Given Data / Assumptions: Principal P = Rs. 5000 Rate for year 1 = 4% Rate for year 2 = 3% Rate for year 3 = 2% Compounding takes place annually We need the total compound interest over 3 years Concept / Approach: When rates vary each year, we multiply the principal by each year's growth factor in sequence. The total amount after 3 years is: A = P * (1 + 0.04) * (1 + 0.03) * (1 + 0.02)Then compound interest CI is: CI = A - P Step-by-Step Solution: Step 1: Compute the amount after year 1.A1 = 5000 * 1.04 = Rs. 5200Step 2: Compute the amount after year 2 with 3% on A1.A2 = 5200 * 1.03 = Rs. 5356Step 3: Compute the amount after year 3 with 2% on A2.A3 = 5356 * 1.02 = Rs. 5463.12Step 4: Find compound interest.CI = A3 - P = 5463.12 - 5000 = Rs. 463.12 Verification / Alternative check: We can also calculate using a single combined multiplier: overall factor = 1.04 * 1.03 * 1.02 ≈ 1.092624. Then: A = 5000 * 1.092624 = 5463.12The difference between amount and principal again gives CI = Rs. 463.12. This matches our stepwise result. Why Other Options Are Wrong: Rs. 435.21, Rs. 453.12, Rs. 436.12: These come from misapplying one of the yearly rates or from incorrect multiplication.Rs. 420.00: Too low; it may be an approximation using simple interest ideas instead of compounding. Common Pitfalls: Common errors include averaging the rates and using a single rate for all three years, or applying each rate on the original principal instead of the accumulated amount. Always remember that compound interest grows on the updated balance each year, especially when rates change. Final Answer: The compound interest on Rs. 5000 for 3 years at rates 4%, 3%, and 2% respectively is Rs. 463.12.

Question 12

A bank offers compound interest at the rate of 15% per half year. A customer deposits Rs. 2400 on 1st January and another Rs. 2400 on 1st July of the same year. What total interest will the customer earn by the end of that year?

Accepted Answer

Introduction / Context: This question demonstrates how compound interest works when multiple deposits are made at different times during the year and interest is compounded at fixed intervals. The key is to treat each deposit separately, considering how long it earns interest before the final date, and then sum the interest from both deposits. Given Data / Assumptions: Interest rate = 15% per half year First deposit = Rs. 2400 on 1st January Second deposit = Rs. 2400 on 1st July We are interested in the amount at the end of the year (31st December) Compounding occurs every half year Concept / Approach: Since interest is 15% per half year, we treat each half of the year as one compounding period. The deposit on 1st January earns interest for two half-year periods (January–June and July–December). The deposit on 1st July earns interest for one half-year period (July–December). We compute the future amount of each deposit and then deduct the total principal to get the total interest. Step-by-Step Solution: Step 1: Amount from first deposit after two half-years.Rate per period i = 15% = 0.15A1 = 2400 * (1 + 0.15)^2A1 = 2400 * (1.15)^2 = 2400 * 1.3225 = Rs. 3174Step 2: Amount from second deposit after one half-year.A2 = 2400 * (1 + 0.15) = 2400 * 1.15 = Rs. 2760Step 3: Total amount at year end.Total amount = A1 + A2 = 3174 + 2760 = Rs. 5934Step 4: Total interest.Total principal = 2400 + 2400 = Rs. 4800Interest = 5934 - 4800 = Rs. 1134 Verification / Alternative check: Another way is to compute interest separately. For the first deposit: interest = 2400 * 1.3225 - 2400 = 774. For the second deposit: interest = 2400 * 1.15 - 2400 = 360. Sum = 774 + 360 = 1134, which matches the previous result. Why Other Options Are Wrong: Rs. 2268: This is exactly double the correct interest and would correspond to mistakenly thinking both deposits earn for two periods.Rs. 567 and Rs. 283: These are partial amounts and represent miscalculation or omitting one of the deposits or periods.Rs. 900: A rounded guess that ignores the exact compound growth factors. Common Pitfalls: Students may incorrectly treat the whole year as one compounding period or assume that both deposits earn for the same length of time. It is crucial to carefully track when each deposit is made and how many compounding periods it experiences before the final date. Final Answer: The customer will earn a total interest of Rs. 1134 by the end of the year.

Question 13

If the amount received at 10% per annum compound interest after 3 years is Rs. 19,965, what was the principal amount invested (in rupees)?

Accepted Answer

Introduction / Context: This is a reverse compound interest problem where the final amount, rate, and time are known, and we need to find the original principal. Such questions are typical when checking whether a given investment amount matches a known maturity value after a certain time at a fixed interest rate. Given Data / Assumptions: Amount after 3 years A = Rs. 19,965 Rate of interest r = 10% per annum Time T = 3 years Compounding is annual We must find the principal P Concept / Approach: The compound amount formula for annual compounding is: A = P * (1 + r / 100)^TGiven A, r, and T, we solve for P: P = A / (1 + r / 100)^TBecause 10% is a common rate, we can quickly compute (1.1)^3. Step-by-Step Solution: Step 1: Compute the growth factor.1 + r / 100 = 1 + 10 / 100 = 1.1(1.1)^3 = 1.331Step 2: Use the formula for principal.P = 19965 / 1.331Step 3: Perform the division.P ≈ Rs. 15,000Indeed, 15000 * 1.331 = 19965 Verification / Alternative check: We can directly compute the amount produced by a principal of Rs. 15,000 at 10% compound interest for 3 years and verify that it equals Rs. 19,965: A = 15000 * (1.1)^3 = 15000 * 1.331 = 19965This exactly matches the given amount, confirming that the principal is Rs. 15,000. Why Other Options Are Wrong: Rs. 16,000, Rs. 17,000, Rs. 18,000: These would yield amounts higher than Rs. 19,965 when multiplied by 1.331.Rs. 14,000: This would yield an amount less than Rs. 19,965, so it cannot be the correct principal. Common Pitfalls: Some learners mistakenly subtract interest or use simple interest formulas. Others forget to divide by the full growth factor (1.1)^3 and divide only by (1.1). Always remember that compound growth over multiple years is exponential, not linear, and that the principal is found by dividing the final amount by the total growth factor. Final Answer: The original principal amount invested was Rs. 15,000.

Question 14

The compound interest earned in 2 years at 15% per annum on a certain sum is Rs. 20,640. What is the principal sum invested (in rupees)?

Accepted Answer

Introduction / Context: Here, the total compound interest over a two-year period is given, rather than the final amount. You are asked to work backward to obtain the original principal. This requires understanding the relationship between amount, principal, and compound interest at a known rate and time period. Given Data / Assumptions: Compound interest CI for 2 years = Rs. 20,640 Rate r = 15% per annum Time T = 2 years Compounding is annual Principal P is unknown Concept / Approach: For compound interest at rate r for 2 years, the amount A is: A = P * (1 + r / 100)^2and the compound interest is: CI = A - PWe can write: CI = P * [(1 + r / 100)^2 - 1]Given CI, we solve this equation for P. Step-by-Step Solution: Step 1: Compute the 2-year factor at 15%.1 + r / 100 = 1 + 15 / 100 = 1.15(1.15)^2 = 1.3225Step 2: Express CI in terms of P.CI = P * (1.3225 - 1) = P * 0.3225Step 3: Use the given CI to solve for P.20640 = P * 0.3225P = 20640 / 0.3225P = Rs. 64,000 Verification / Alternative check: Verify by directly computing amount and interest from P = 64,000. Amount after 2 years: A = 64000 * (1.15)^2 = 64000 * 1.3225 = Rs. 84,640CI = A - P = 84640 - 64000 = Rs. 20,640This matches the given compound interest, confirming the calculation. Why Other Options Are Wrong: Rs. 60,000, Rs. 56,000, Rs. 52,000, Rs. 48,000: Substituting any of these values into the formula A = P * 1.3225 produces compound interest different from 20,640. They therefore cannot be the correct principal. Common Pitfalls: Some students mistakenly treat 20,640 as the amount or use simple interest formulas. Others fail to compute the squared factor correctly or forget to subtract 1 when forming the CI expression P * [(1 + r / 100)^2 - 1]. Careful algebra and attention to the formula structure prevent such errors. Final Answer: The principal sum invested was Rs. 64,000.

Question 15

On a certain principal, the compound interest received in the second year is Rs. 1200 at a rate of 10% per annum. What will be the compound interest for the fourth year on the same principal at the same rate?

Accepted Answer

Introduction / Context: This question focuses on the concept of interest earned in specific years under compound interest. Instead of total interest over many years, we are given the interest in the second year and asked to find the interest in the fourth year. This requires recognizing the pattern of annual interest amounts under compound growth. Given Data / Assumptions: Compound interest in 2nd year I2 = Rs. 1200 Rate of interest r = 10% per annum We need compound interest in 4th year I4 Principal P is the same in all years Compounding is annual Concept / Approach: Under compound interest, interest in a particular year grows because it is calculated on the amount at the beginning of that year. The interest in year n is: In = P * (1 + r / 100)^(n - 1) * (r / 100)Thus: I2 = P * (1 + r / 100)^(1) * (r / 100)I4 = P * (1 + r / 100)^(3) * (r / 100)Therefore: I4 / I2 = (1 + r / 100)^2We can find I4 without explicitly computing P. Step-by-Step Solution: Step 1: Compute the growth factor.1 + r / 100 = 1 + 10 / 100 = 1.1Step 2: Use the ratio of interests.I4 / I2 = (1.1)^2(1.1)^2 = 1.21Step 3: Compute I4.I4 = I2 * 1.21I4 = 1200 * 1.21 = Rs. 1452 Verification / Alternative check: If we wanted, we could find principal P. Let A1 = P * 1.1, interest in 2nd year is 10% of A1: I2 = 0.1 * P * 1.1 = 0.11P = 1200P = 1200 / 0.11 = 10909.09 (approx)Amount at beginning of 4th year is P * (1.1)^3, so interest in 4th year is 10% of that amount: I4 = 0.1 * P * (1.1)^3 ≈ 0.1 * 10909.09 * 1.331 ≈ 1452This confirms the earlier ratio method. Why Other Options Are Wrong: Rs. 1320 and Rs. 1420: These values are too low and do not reflect the correct growth factor of (1.1)^2.Rs. 1552 and Rs. 1600: These overestimate the increase from the second to the fourth year. Common Pitfalls: Some learners think that the interest per year stays the same or increases linearly, but under compound interest, it increases multiplicatively. Others try to calculate total amounts year by year and risk arithmetic errors. Using the ratio method based on (1 + r / 100)^2 is both faster and less error-prone. Final Answer: The compound interest for the fourth year on the same principal at 10% per annum is Rs. 1452.

Question 16

In a compound interest calculation, what will be the compound interest on a principal of Rs. 25,000 invested for 2 years at 5% per annum, if the interest is compounded annually?

Accepted Answer

Introduction: In this question we apply the basic compound interest formula for a fixed principal, a known annual rate, and a small number of years. The aim is to compute only the compound interest earned, not the final amount, on Rs. 25,000 invested at 5% per annum for 2 years with annual compounding. Given Data / Assumptions: Principal (P) = Rs. 25,000 Rate of interest (r) = 5% per annum Time (t) = 2 years Interest is compounded once every year (annually) Concept / Approach: The compound amount after t years at rate r with principal P is given by: A = P * (1 + r/100)^tOnce we find A, the compound interest (CI) is simply: CI = A - PThis approach ensures that we first compute the total amount and then subtract the original principal to obtain the interest portion only. Step-by-Step Solution: Step 1: Convert the percentage rate to a decimal factor: r/100 = 5/100 = 0.05.Step 2: Use the formula A = P * (1 + r/100)^t.Step 3: Compute the growth factor (1 + r/100) = 1 + 0.05 = 1.05.Step 4: Raise this factor to the power of 2 years: (1.05)^2 = 1.05 * 1.05 = 1.1025.Step 5: Multiply by the principal to get the amount: A = 25,000 * 1.1025 = 27,562.5.Step 6: Subtract the principal to obtain the compound interest: CI = A - P = 27,562.5 - 25,000 = 2,562.5.Thus the compound interest earned over 2 years is Rs. 2,562.5. Verification / Alternative check: We can cross check by computing the interest year by year. For the first year, interest is 5% of 25,000 which is Rs. 1,250. The new amount becomes 26,250. For the second year, interest is 5% of 26,250 which equals Rs. 1,312.5. Adding the two years interests gives 1,250 + 1,312.5 = 2,562.5, which matches the previous result. Why Other Options Are Wrong: Option 2500 ignores the effect of compounding and treats the situation as simple interest only. Option 2425.25 corresponds to an incorrect rate or incorrect number of years. Option 2600 and option 2700 are rounded or guessed values that do not result from the correct compound interest calculation. Common Pitfalls: Students often confuse simple interest and compound interest and may simply multiply principal * rate * time without considering compounding. Another common mistake is to forget to square the factor (1.05) for 2 years, which leads to incorrect results. Careful attention to the formula and exponent is essential. Final Answer: The correct compound interest earned on Rs. 25,000 at 5% per annum compounded annually for 2 years is Rs. 2,562.5.

Question 17

The compound interest on Rs. 24,000 invested at 10% per annum for 1.5 years, when interest is compounded semi-annually, is equal to how much amount (in rupees)?

Accepted Answer

Introduction: This problem tests understanding of compound interest with a compounding period that is not yearly. The investment is for one and a half years, and the interest is compounded semi-annually, meaning twice per year. You must correctly convert the annual rate into a half yearly rate and use the right number of compounding periods. Given Data / Assumptions: Principal (P) = Rs. 24,000 Annual nominal rate (r) = 10% per annum Compounding frequency = semi-annual (twice a year) Total time = 1.5 years Concept / Approach: If interest is compounded semi-annually, we divide the annual rate by 2 and multiply the number of years by 2 to find the number of periods. The amount after n periods is given by: A = P * (1 + i)^nwhere i is the rate per period and n is the number of periods. The compound interest CI is then: CI = A - P Step-by-Step Solution: Step 1: Compute the half yearly rate: i = r/2 = 10% / 2 = 5% = 0.05 per half year.Step 2: Compute the number of half yearly periods: n = 1.5 years * 2 = 3 periods.Step 3: Use the formula A = P * (1 + i)^n.Step 4: Substitute values: A = 24,000 * (1 + 0.05)^3.Step 5: Calculate the growth factor: (1.05)^3 = 1.05 * 1.05 * 1.05 = 1.157625.Step 6: Multiply by the principal: A = 24,000 * 1.157625 = 27,783.Step 7: Find compound interest: CI = A - P = 27,783 - 24,000 = 3,783.Thus, the compound interest is Rs. 3,783. Verification / Alternative check: You can approximately check by noting that for 1 year at 10%, simple interest would be Rs. 2,400. For half a year, simple interest would be around Rs. 1,200. Adding a small extra due to compounding suggests a figure slightly above Rs. 3,600. The value Rs. 3,783 fits this expectation well. Why Other Options Are Wrong: Rs. 3,600 and Rs. 3,720 are closer to simple interest approximations and ignore the exact compounding over three half yearly periods. Rs. 3,840 and Rs. 3,900 are overestimates resulting from using an incorrect rate or wrong number of compounding periods. Common Pitfalls: A frequent mistake is to treat 1.5 years as 1 year or 2 years, or to apply 10% directly for 1.5 years without adjusting for semi annual compounding. Another common error is incorrectly computing (1.05)^3. Always ensure that both the rate per period and number of periods are consistent with the compounding frequency. Final Answer: The compound interest on Rs. 24,000 at 10% per annum, compounded semi annually for 1.5 years, is Rs. 3,783.

Question 18

The compound interest earned in 2 years at a rate of 12% per annum is Rs. 10,176. What was the original principal sum (in rupees) that was invested, assuming annual compounding?

Accepted Answer

Introduction: This question reverses the usual compound interest calculation. Instead of finding the interest from a known principal, we are given the compound interest and asked to find the original principal that generated this interest at a known rate over 2 years. Given Data / Assumptions: Compound interest (CI) for 2 years = Rs. 10,176 Rate of interest (r) = 12% per annum Time (t) = 2 years Interest is compounded annually Concept / Approach: For annual compounding, the amount A after t years is: A = P * (1 + r/100)^tThe compound interest is: CI = A - PWe know CI and r and t, but not P. We first express CI in terms of P and then solve for P using algebra. Step-by-Step Solution: Step 1: Write the formula for the amount over 2 years: A = P * (1 + 12/100)^2 = P * (1.12)^2.Step 2: Compute (1.12)^2 = 1.2544.Step 3: So A = 1.2544 * P.Step 4: The compound interest CI is A - P, so CI = 1.2544 * P - P = (1.2544 - 1) * P = 0.2544 * P.Step 5: We are given CI = 10,176, so 0.2544 * P = 10,176.Step 6: Solve for P by dividing both sides: P = 10,176 / 0.2544.Step 7: Perform the division: P = 40,000.So the original principal sum invested is Rs. 40,000. Verification / Alternative check: To verify, compute the amount from P = 40,000 at 12% for 2 years. First year amount: 40,000 * 1.12 = 44,800. Second year amount: 44,800 * 1.12 = 50,176. The total compound interest is 50,176 - 40,000 = 10,176, which matches the given CI. Hence the calculation is consistent. Why Other Options Are Wrong: Rs. 50,000, Rs. 60,000, and Rs. 80,000 all give much larger compound interest at 12% over 2 years. Rs. 30,000 would give significantly less than Rs. 10,176 as compound interest. Only Rs. 40,000 produces exactly the stated interest figure. Common Pitfalls: Some learners plug numbers directly into the simple interest formula instead of using the compound interest relation. Others forget that CI depends on both the principal and accumulated interest, making linear methods incorrect. Always express CI in terms of P using the correct compound interest factor, then solve algebraically for P. Final Answer: The principal that generated Rs. 10,176 as compound interest at 12% per annum for 2 years is Rs. 40,000.

Question 19

If the simple interest on a sum of money at 5% per annum for 3 years is Rs. 1,200, then what is the compound interest on the same principal for the same period at the same rate, assuming annual compounding?

Accepted Answer

Introduction: This question connects simple interest and compound interest for the same principal, rate, and time. We are given the simple interest earned over 3 years and asked to find the corresponding compound interest at the same rate and period, which requires us to first determine the principal and then apply the compound interest formula. Given Data / Assumptions: Simple interest (SI) for 3 years = Rs. 1,200 Rate of interest (r) = 5% per annum Time (t) = 3 years Same principal, same rate, and interest compounded annually for CI Concept / Approach: For simple interest, the formula is: SI = (P * r * t) / 100We can rearrange this to find P. Once we have P, we apply the compound interest amount formula: A = P * (1 + r/100)^tThen the compound interest is: CI = A - P Step-by-Step Solution: Step 1: Use SI = (P * r * t) / 100.Step 2: Substitute SI = 1,200, r = 5, and t = 3.Step 3: 1,200 = (P * 5 * 3) / 100 = (15P) / 100.Step 4: Rearrange to find P: P = 1,200 * 100 / 15 = 8,000.Step 5: Apply the compound amount formula: A = 8,000 * (1 + 5/100)^3 = 8,000 * (1.05)^3.Step 6: Compute (1.05)^3 = 1.157625.Step 7: So A = 8,000 * 1.157625 = 9,261.Step 8: Compound interest CI = A - P = 9,261 - 8,000 = 1,261.Thus, the compound interest is Rs. 1,261. Verification / Alternative check: We can verify by computing interest year by year: Year 1 interest = 5% of 8,000 = 400, new amount = 8,400. Year 2 interest = 5% of 8,400 = 420, new amount = 8,820. Year 3 interest = 5% of 8,820 = 441, new amount = 9,261. Total compound interest = 400 + 420 + 441 = 1,261, agreeing with the formula result. Why Other Options Are Wrong: Rs. 1,251 and Rs. 1,271 are close but correspond to slight miscalculations in (1.05)^3 or rounding errors. Rs. 1,281 is higher than the correct total and comes from adding an extra incorrect interest component. Rs. 1,240 is too low and does not reflect the compounding effect accurately. Common Pitfalls: One common error is to treat compound interest as simple interest, leading to SI = Rs. 1,200 being used as if it were CI. Another is miscomputing (1.05)^3 or rounding prematurely. Always compute the principal accurately using the simple interest formula and then use the full compound interest formula with correct exponents. Final Answer: The compound interest on the same sum for 3 years at 5% per annum is Rs. 1,261.

Question 20

The compound interest on Rs. 30,000 at 7% per annum amounts to Rs. 4,347. For how many years was the money invested at this rate of compound interest, assuming annual compounding?

Accepted Answer

Introduction: Here we are given a principal, a rate of compound interest, and the total compound interest earned. Instead of finding the interest, we are required to determine the time period for which the money must have been invested at the given rate to generate the specified compound interest. Given Data / Assumptions: Principal (P) = Rs. 30,000 Rate (r) = 7% per annum Compound interest (CI) = Rs. 4,347 Interest is compounded annually Concept / Approach: The compound amount after t years is: A = P * (1 + r/100)^tThe compound interest is: CI = A - PSo we can write: CI = P * (1 + r/100)^t - P = P * [(1 + r/100)^t - 1]We know CI, P, and r, so we solve for t by isolating the term (1 + r/100)^t. Step-by-Step Solution: Step 1: Substitute the known values: 4,347 = 30,000 * [(1 + 7/100)^t - 1].Step 2: Convert the rate: 1 + 7/100 = 1.07.Step 3: Divide both sides by 30,000: 4,347 / 30,000 = (1.07)^t - 1.Step 4: Compute the fraction: 4,347 / 30,000 = 0.1449.Step 5: So (1.07)^t - 1 = 0.1449, therefore (1.07)^t = 1.1449.Step 6: Compare with powers of 1.07. We know (1.07)^2 = 1.07 * 1.07 = 1.1449.Step 7: Therefore t = 2 years.The money was invested for exactly 2 years. Verification / Alternative check: Verify by computing the amount for 2 years directly: A = 30,000 * (1.07)^2 = 30,000 * 1.1449 = 34,347. Compound interest = 34,347 - 30,000 = 4,347, which matches the given CI, confirming that the period is 2 years. Why Other Options Are Wrong: For t = 1 year, CI would be only 7% of 30,000, that is Rs. 2,100, which is too small. For t = 3, 4, or 5 years, the interest accumulated would be much larger than Rs. 4,347 due to repeated compounding, so these options cannot be correct. The option of 1 year is also inconsistent with the given CI value. Common Pitfalls: Students may try to treat the problem using simple interest or by guessing the time period without checking powers of (1 + r/100). Others may make errors when dividing CI by the principal or neglect the minus one in the expression (1.07)^t - 1. Carefully manipulating the formula and checking plausible integer values of t is a reliable approach. Final Answer: The money must have been invested for 2 years at 7% per annum compound interest to earn Rs. 4,347 as compound interest.

Compound Interest Questions