Question 1

Albert invests Rs 8,000 in a fixed deposit scheme that offers 5% per annum interest compounded annually for 2 years. What total amount, in rupees, will Albert receive on maturity of this deposit at the end of 2 years?

Accepted Answer

Introduction / Context: This question involves a standard compound interest calculation where a fixed sum is invested for a known period at a given rate with annual compounding. The twist is that none of the direct numerical options exactly match the correctly computed amount, so the correct answer becomes "None of these". Such questions test not only the ability to apply the compound interest formula but also attention to detail when comparing the result with the given options. Given Data / Assumptions: Principal (P) invested by Albert = Rs 8,000. Rate of interest (R) = 5% per annum. Time (T) = 2 years. Interest is compounded annually. No additional deposits or withdrawals are made during the 2 years. Concept / Approach: For annual compounding, the amount after T years is given by: A = P * (1 + R / 100)^TWe substitute the known values of P, R, and T into this formula. Once we compute A, we compare it precisely with the options. If the exact amount is not present among the numerical options, then we must choose the logical option that indicates this, which is usually labeled "None of these". Step-by-Step Solution: Step 1: Substitute the values in the amount formula: A = 8000 * (1 + 5 / 100)^2.Step 2: Compute 1 + 5 / 100 = 1 + 0.05 = 1.05.Step 3: Square this factor because the time is 2 years: 1.05^2 = 1.1025.Step 4: Multiply this factor by the principal: A = 8000 * 1.1025.Step 5: Perform the multiplication: 8000 * 1.1025 = 8820.Step 6: The maturity amount is Rs 8,820. Verification / Alternative check: We can verify by calculating year wise. At the end of year 1, interest = 8000 * 5 / 100 = 400, so the amount becomes 8400. At the end of year 2, interest is calculated on 8400: interest = 8400 * 5 / 100 = 420. The final amount is 8400 + 420 = 8820. This matches our direct formula calculation and confirms that A = Rs 8,820 is correct. Why Other Options Are Wrong: Rs.8800 is slightly lower than the true amount and would correspond to a miscalculation or rounding error. Rs.8840 is slightly higher and does not arise from correct compounding at 5% per annum. Rs.9000 is much higher than the accurate value and clearly incorrect. Since none of the specific numerical options equal Rs 8,820, the correct choice must be the option labeled "None of these". Common Pitfalls: Many students mistakenly use simple interest, computing interest as 8000 * 5 * 2 / 100 = 800 and then writing the amount as 8800, which matches one of the distractor options. This happens when they forget that the question clearly mentions compound interest. Another frequent mistake is to round the factor 1.05^2 too early or to write it incorrectly as 1.10. Staying careful with the formula and doing the year wise check helps to avoid these errors. Final Answer: The correct maturity amount is Rs 8,820, which does not appear explicitly among the numerical options, so the right choice is None of these.

Question 2

An article currently costs Rs 50. If the general rate of inflation is 300% per year, assuming the price rises at this same rate every year, what will be the cost of this article, in rupees, after 2 years?

Accepted Answer

Introduction / Context: This question is a real life style application of compound growth, where the rate of inflation plays the role of the interest rate. When inflation is very high, prices rise sharply each year, and this can be modeled using the same ideas as compound interest. Here, you are given an initial price, a very large inflation rate of 300% per year, and asked to predict the price after 2 years if this rate continues unchanged. The point is to see that a 300% increase per year means the price multiplies several times each year, not just by a small amount. Given Data / Assumptions: Current price of the article (P) = Rs 50. Inflation rate = 300% per annum. Time period = 2 years. Price increase is assumed to compound yearly at the given rate. No other factors such as discounts or special taxes are considered. Concept / Approach: A 300% inflation rate means that every year the price increases by 300% of its current value. That is, the price becomes original price plus three times that price, which is four times the original. So each year, the price is multiplied by a factor of 4. Mathematically, for compound growth we can use: Future price = Present price * (1 + r)^nHere r is expressed in decimal form. For 300% inflation, r = 3. So the yearly growth factor is (1 + 3) = 4. For 2 years, we apply this factor twice. Step-by-Step Solution: Step 1: Convert the inflation rate into a yearly multiplier. At 300% inflation, r = 300 / 100 = 3.Step 2: Compute the yearly growth factor: 1 + r = 1 + 3 = 4. So price each year becomes four times the previous year.Step 3: After 1 year, price = 50 * 4 = 200 rupees.Step 4: After 2 years, we again multiply by 4: price = 200 * 4 = 800 rupees.Step 5: Therefore, after two full years at 300% inflation per year, the article will cost Rs 800. Verification / Alternative check: Using the compound growth formula, we can write future price after 2 years as: Future price = 50 * (1 + 300 / 100)^2 = 50 * (1 + 3)^2 = 50 * 4^2.Compute 4^2 = 16, so future price = 50 * 16 = 800.This agrees exactly with the stepwise multiplication approach and confirms that the final cost is Rs 800. Why Other Options Are Wrong: Rs.200 would be the price after only 1 year of 300% inflation, not after 2 years. Rs.600 and Rs.400 represent partial or incorrect growth factors and do not match the repeated multiplication by 4 each year. Rs.1000 is larger than the correct price and would require an even higher effective growth rate than 300% per year. Only Rs.800 reflects the correct two year compounding at 300% per annum. Common Pitfalls: A common mistake is to treat 300% as if it means the price becomes 300 rupees after two years, or to add 300% twice (giving 600%) and incorrectly compute 50 * (1 + 6) = 350. Another error is to forget that each year the increase is on the new price, not just on the original price. Recognizing that inflation works like compound interest and using a multiplication factor each year avoids these errors. Final Answer: After 2 years at an annual inflation rate of 300%, the price of the article will be Rs.800.

Question 3

At what least number of complete years will a sum of money invested at 40% annual compound interest become more than three times (trebled) its original value?

Accepted Answer

Introduction / Context: This problem deals with determining how many years it takes for an investment to grow to more than three times its original amount under compound interest. The annual interest rate is quite high at 40%, so the amount grows rapidly each year. The key idea is that the amount after n years of compound interest is given by a power expression in terms of the annual growth factor. We are asked for the smallest whole number of years at which the amount first exceeds three times the principal. Given Data / Assumptions: Principal (P) is some positive amount (exact value is not required). Rate of interest (R) = 40% per annum. Interest is compounded annually. We seek the least integer n such that the amount is greater than 3P. There are no deposits or withdrawals during the investment period. Concept / Approach: Under annual compounding, the amount after n years is: A = P * (1 + R / 100)^nFor R = 40%, the yearly growth factor is 1 + 40 / 100 = 1.4. We want A to be more than 3P, so we need: P * (1.4)^n > 3P(1.4)^n > 3We then test successive integer values of n to find the smallest n that satisfies this inequality. Step-by-Step Solution: Step 1: Compute (1.4)^1 = 1.4, which is less than 3, so 1 year is not enough.Step 2: Compute (1.4)^2 = 1.4 * 1.4 = 1.96, still less than 3, so 2 years are not sufficient.Step 3: Compute (1.4)^3 = 1.96 * 1.4 = 2.744, still less than 3, so 3 years are not sufficient either.Step 4: Compute (1.4)^4 = 2.744 * 1.4 ≈ 3.8416.Step 5: Now (1.4)^4 ≈ 3.84, which is clearly greater than 3.Step 6: Therefore, after 4 years the amount becomes more than three times the original principal.Step 7: Since 3 years is still less than triple and 4 years is more than triple, the least number of complete years required is 4. Verification / Alternative check: We can express the amounts in terms of P for extra clarity. After 1 year, A1 = 1.4P. After 2 years, A2 = 1.4 * 1.4P = 1.96P. After 3 years, A3 = 1.96 * 1.4P ≈ 2.744P. After 4 years, A4 = 2.744 * 1.4P ≈ 3.8416P. Only at year 4 does the value cross 3P. This confirms that the least integer number of years is 4. Why Other Options Are Wrong: Option 3 years gives A ≈ 2.744P, which is less than 3P, so it does not satisfy the condition that the money is more than trebled. Options 5, 6, and 7 years also satisfy the condition but they are not the least number of years, because the amount already exceeds 3P at year 4. Hence they do not answer the specific requirement of the problem. Only 4 years is both sufficient and minimal. Common Pitfalls: Some students may misinterpret "trebled" as simply adding 40% three times, which would give 120% total and only 2.2 times the original, not 3 times. Others may try to solve the inequality using approximate logarithms and then forget to round up to the next whole year. It is often simpler and safer in such questions to test small integer values step by step, especially when the time horizon is likely to be short. Final Answer: The least number of complete years required for the money to become more than three times its original value at 40% compound interest per annum is 4 years.

Question 4

An amount of Rs 8,000 is deposited into an account paying 7% annual interest compounded quarterly. After how many years (correct to one decimal place) will the balance in the account first reach Rs 12,400?

Accepted Answer

Introduction / Context: Here the principal, target amount, interest rate, and compounding frequency are known, and we are asked to determine the time required for the principal to grow to the target amount. This is a classic compound interest problem where we solve for the time variable. Because the interest is compounded quarterly, we must work with a quarterly rate and convert the total number of quarters back into years. This question also reinforces the use of logarithms in financial mathematics. Given Data / Assumptions: Principal (P) = Rs 8,000. Required future amount (A) = Rs 12,400. Nominal annual interest rate (R) = 7% per annum. Interest is compounded quarterly, so m = 4 compounding periods per year. We want the time T in years such that the amount reaches Rs 12,400. Concept / Approach: The general compound interest formula with compounding m times per year is: A = P * (1 + (R / 100) / m)^(m * T)Here, A, P, and R are given. We must solve for T. Rearranging involves dividing both sides by P, then taking logarithms to isolate T. The periodic rate is R / 100 / m and the total number of compounding periods is m * T. Step-by-Step Solution: Step 1: Write the formula with the values: 12400 = 8000 * (1 + 0.07 / 4)^(4 * T).Step 2: Compute the quarterly rate: 0.07 / 4 = 0.0175. So 12400 = 8000 * (1.0175)^(4T).Step 3: Divide both sides by 8000 to get (1.0175)^(4T) = 12400 / 8000.Step 4: Simplify the ratio: 12400 / 8000 = 1.55.Step 5: Thus (1.0175)^(4T) = 1.55.Step 6: Take natural logarithms: ln(1.55) = 4T * ln(1.0175).Step 7: Solve for T: T = ln(1.55) / (4 * ln(1.0175)). Numerically this is approximately equal to 6.3 years. Verification / Alternative check: To check reasonableness, a rough comparison is useful. At about 7% interest with annual compounding, the rule of 72 suggests doubling time is about 72 / 7 ≈ 10.3 years. We are not doubling from 8000 to 16000, but only going to 12400, which is a growth factor of 1.55. That should take noticeably less than 10.3 years. Quarterly compounding is slightly more favorable than annual compounding, giving an effective rate a bit higher than 7%, which further reduces the time. A value around 6 to 7 years is therefore quite reasonable, and 6.3 years matches the precise calculation. Why Other Options Are Wrong: 2.3, 3.3 and 4.3 years are all too short to achieve a 55% increase at only 7% per annum, even with quarterly compounding. An investment cannot grow from 8,000 to 12,400 in such a small time at this modest rate. 5.3 years is closer but still does not satisfy the exact compound interest equation. Only about 6.3 years matches the solution obtained by properly solving the compound growth equation. Common Pitfalls: Common mistakes include using the simple interest formula to estimate time, forgetting to divide the annual rate by 4 for quarterly compounding, or failing to multiply the time by 4 in the exponent. Some students also try to guess the answer rather than using logarithms, which can lead to large errors. Working directly from the compound interest formula and handling the logarithms carefully leads to the correct time period. Final Answer: The deposit of Rs 8,000 will grow to Rs 12,400 at 7% interest compounded quarterly in approximately 6.3 years.

Question 5

A strip bond has a face value of $10,000 and 15 years remaining until maturity. If the prevailing market rate of return is 6.5% per annum, compounded semiannually, what is the fair present value (price) of this strip bond today?

Accepted Answer

Introduction / Context: A strip bond is a fixed income security that pays a single lump sum at maturity and no periodic coupons. Its present value is calculated by discounting the face value back to today using the appropriate market yield and compounding convention. This question requires the application of the compound interest formula in reverse: given the future value, yield rate, compounding frequency, and time to maturity, we must find the bond price today. It reinforces both time value of money concepts and careful handling of semiannual compounding. Given Data / Assumptions: Face value (FV) of the strip bond = $10,000. Time to maturity = 15 years from today. Market rate of return (yield) = 6.5% per annum. Compounding is semiannual, so there are 2 compounding periods per year. The bond pays no coupons; a single payment of $10,000 is received at maturity. Concept / Approach: The present value of a future single sum under compound discounting is: PV = FV / (1 + i)^nHere, i is the periodic interest rate and n is the total number of periods. For a nominal annual yield of 6.5% compounded semiannually, we have: i = 0.065 / 2n = 15 * 2 = 30 periodsWe then substitute FV = 10000, i and n into the formula to obtain the fair price of the bond. Step-by-Step Solution: Step 1: Compute the semiannual rate: i = 6.5% / 2 = 3.25% per half year, or 0.0325 in decimal form.Step 2: Determine the total number of compounding periods: n = 15 years * 2 = 30 semiannual periods.Step 3: Write the present value formula: PV = 10000 / (1 + 0.0325)^30.Step 4: Calculate the denominator (1.0325)^30 using repeated compounding or a calculator. This gives a value larger than 2, reflecting growth over 15 years.Step 5: Divide 10000 by this factor. Numerically, the result is approximately PV = 3830.88.Step 6: Therefore, the fair market value of the strip bond is about $3830.88. Verification / Alternative check: We can check reasonableness. A 6.5% annual yield compounded semiannually over 15 years implies more than doubling, but less than quadrupling, of a sum. If we move from present to future, 3830.88 growing for 30 periods at 3.25% per period should yield roughly 10000. That is, 3830.88 * (1.0325)^30 ≈ 10000. This reverse multiplication confirms that 3830.88 is a suitable discounting result, so the calculation is consistent. Why Other Options Are Wrong: $1710.29 and $2710.29 are too low, suggesting an unrealistically high effective yield. $4710.29 and $5200.00 are too high, corresponding to yields lower than the given 6.5% per annum. Only $3830.88 satisfies the correct present value formula using a 6.5% nominal annual yield with semiannual compounding for 15 years. Common Pitfalls: Common mistakes include using the annual rate of 6.5% directly as i without dividing by 2, or using n = 15 instead of 30, which mixes annual and semiannual compounding incorrectly. Another error is to multiply instead of divide by (1 + i)^n when discounting, confusing present value with future value. Carefully identifying whether we are moving from future to present or from present to future helps avoid such errors. Final Answer: The fair present value of the $10,000 face value strip bond with 15 years to maturity at 6.5% per annum compounded semiannually is approximately $3830.88.

Question 6

A five year promissory note has a face value of $3,500 and bears interest at 11% per annum, compounded semiannually. The note is sold 21 months after its issue date. If the buyer wishes to earn 10% per annum, compounded quarterly, what amount should…

Accepted Answer

Introduction / Context: This problem involves evaluating the fair price of a promissory note that accrues interest at one rate but is sold at a time when the buyer wants a different yield. The note grows at 11% per annum compounded semiannually from issue to maturity, while the buyer requires a yield of 10% per annum compounded quarterly from the purchase date to maturity. We must determine the price that equates the buyer's required yield to the future payoff, which is a standard but advanced time value of money exercise. Given Data / Assumptions: Face value (principal) of the note = $3,500. Maturity time from issue = 5 years. Interest on the note: 11% per annum, compounded semiannually. The note is sold 21 months (1.75 years) after issue. Buyer's required yield = 10% per annum, compounded quarterly. We assume no intermediate coupon payments; the accumulated amount is paid at maturity. Concept / Approach: First, find the maturity value of the promissory note after the full 5 years at 11% compounded semiannually. That is: Maturity value FV = 3500 * (1 + 0.11 / 2)^(2 * 5)Next, from the buyer's perspective, this future value is discounted back from the sale date (which is 21 months after issue) to today at 10% compounded quarterly. The time from sale date to maturity is 5 years - 1.75 years = 3.25 years. With quarterly compounding, this is 3.25 * 4 = 13 quarters. The purchase price PV is then: PV = FV / (1 + 0.10 / 4)^13 Step-by-Step Solution: Step 1: Compute the semiannual rate of the note: i1 = 0.11 / 2 = 0.055 per half year.Step 2: Total number of semiannual periods over 5 years: n1 = 5 * 2 = 10.Step 3: Calculate the maturity value: FV = 3500 * (1.055)^10.Step 4: Numerically, (1.055)^10 is greater than 1.7, giving a future value above $5950. The exact calculation yields a specific FV.Step 5: Determine the time from sale date to maturity: 5 years total minus 21 months (1.75 years) equals 3.25 years.Step 6: Buyer's effective periodic rate: i2 = 0.10 / 4 = 0.025 per quarter, with n2 = 3.25 * 4 = 13 quarters.Step 7: The fair purchase price is PV = FV / (1.025)^13, which evaluates numerically to about $4336.93. Verification / Alternative check: To check the logic, imagine the buyer pays $4336.93 today. If this amount earns 10% per annum compounded quarterly for 13 quarters, its future value should equal the maturity value of the note. That is: 4336.93 * (1.025)^13 ≈ FVIf we have matched the maturity value from the seller's side and the buyer's required yield side, the pricing is internally consistent. This consistency check confirms that $4336.93 is a fair price for the given yield requirements. Why Other Options Are Wrong: $5336, $6336, and $7336 are all higher than the fair price and would result in a yield lower than 10% to the buyer, making the investment unattractive under the stated requirement. $3336.93 is too low and would offer a yield much higher than 10%, which contradicts the idea of a fair market yield at 10%. Only $4336.93 satisfies the time value of money calculations for the specified compounding conventions. Common Pitfalls: Errors often arise from confusing which rate applies over which period. Some candidates incorrectly discount using 11% instead of 10%, or they forget that the sale happens after 21 months, not at issue. Others mis-handle the compounding frequency, using 5 years instead of 10 half years, or 3.25 years instead of 13 quarters. A careful timeline and clear identification of periods and rates help to avoid these mistakes. Final Answer: The amount that the buyer should pay for the note, in order to earn 10% per annum compounded quarterly, is $4336.93.

Question 7

There is a 50% increase in an amount in 5 years when invested at simple interest. Using the same annual rate of interest, what will be the compound interest on Rs 12,000 after 3 years, compounded annually?

Accepted Answer

Introduction / Context: This question connects simple interest and compound interest. First, we are told that a certain principal grows by 50% in 5 years at simple interest, which allows us to deduce the annual rate of interest. Then we must apply this same rate to a new principal of Rs 12,000, but now under compound interest for 3 years. The problem thus tests understanding of both simple interest and compound interest formulas and how to move from one to the other using the underlying rate. Given Data / Assumptions: Under simple interest, an amount increases by 50% in 5 years. This means simple interest over 5 years is 50% of the principal. We infer the annual simple interest rate from this information. For the second part, principal P2 = Rs 12,000. Rate R is the same as the rate found from the simple interest condition. Time T2 = 3 years, and interest is compounded annually. Concept / Approach: For simple interest, the interest is: SI = P * R * T / 100If in 5 years the amount increases by 50%, then SI over 5 years is 50% of P, so: P * R * 5 / 100 = 0.5 * PFrom this we can solve for R. Once R is known, the compound interest on Rs 12,000 for 3 years is found using: A = 12000 * (1 + R / 100)^3CI = A - 12000 Step-by-Step Solution: Step 1: From the simple interest condition, P cancels. We have R * 5 / 100 = 0.5.Step 2: Multiply both sides by 100: 5R = 50, so R = 10% per annum.Step 3: Now apply this rate to Rs 12,000 with compound interest for 3 years. Amount A = 12000 * (1 + 10 / 100)^3.Step 4: Compute 1 + 10 / 100 = 1.10.Step 5: Compute 1.10^3 = 1.10 * 1.10 * 1.10 = 1.331.Step 6: Multiply principal by this factor: A = 12000 * 1.331 = 15972.Step 7: Compound interest CI = A - P = 15972 - 12000 = 3972. Verification / Alternative check: An alternative way to see this is to build the amount year by year. Year 1: amount = 12000 + 10% of 12000 = 12000 + 1200 = 13200. Year 2: amount = 13200 + 10% of 13200 = 13200 + 1320 = 14520. Year 3: amount = 14520 + 10% of 14520 = 14520 + 1452 = 15972. Again, CI = 15972 - 12000 = 3972. This matches the previous calculation and confirms the answer. Why Other Options Are Wrong: Rs.5422, Rs.5489, and Rs.6145 are significantly larger than Rs 3,972 and would require a higher effective rate or longer time. Rs.3600 corresponds to 10% simple interest for 3 years on 12,000 (which would be 12,000 * 10 * 3 / 100 = 3600), but the question specifically asks for compound interest. Therefore, Rs.3600 is a common but incorrect simple interest based answer. Only Rs.3972 correctly reflects compound interest at 10% per annum for 3 years. Common Pitfalls: Students often compute the rate correctly but then revert to simple interest when calculating interest on Rs 12,000, or they mistakenly apply the 50% increase directly to 3 years instead of determining the correct annual rate. Another frequent mistake is to miscalculate 1.10^3, writing it as 1.30 or 1.33 incorrectly. Writing out each multiplication step or using a year by year method is a good way to avoid such errors. Final Answer: The compound interest on Rs 12,000 after 3 years at the same rate that produces a 50% increase in 5 years under simple interest is Rs.3972.

Question 8

The compound interest earned on a certain sum for the second year and the third year are Rs 1,200 and Rs 1,440 respectively. Assuming a constant annual rate with yearly compounding, what is the rate of interest per annum?

Accepted Answer

Introduction / Context: This question involves analyzing compound interest year by year, instead of over the whole period. We are given the interest earned in the second and third years and asked to deduce the annual interest rate. Under compound interest, the interest in each successive year increases because it is calculated on a growing amount. The ratio of interest in consecutive years depends directly on the annual rate. This question tests understanding of that relationship without requiring the explicit principal. Given Data / Assumptions: Compound interest in the second year = Rs 1,200. Compound interest in the third year = Rs 1,440. Interest is compounded annually. The rate of interest per annum is constant. Principal P is unknown and does not need to be found. Concept / Approach: Let the rate of interest be R% per annum and principal be P. Under annual compounding: Amount after year 1 = P * (1 + R / 100).Amount after year 2 = P * (1 + R / 100)^2.Amount after year 3 = P * (1 + R / 100)^3.The interest earned in the second year is the difference between the amount at the end of year 2 and the amount at the end of year 1, and similarly for the third year. However, a simpler fact is that the interest in each successive year is multiplied by a factor of (1 + R / 100). Thus: CI in third year = CI in second year * (1 + R / 100). Step-by-Step Solution: Step 1: Let I2 be the interest in the second year and I3 be the interest in the third year. We are given I2 = 1200 and I3 = 1440.Step 2: Under compound interest, we have I3 = I2 * (1 + R / 100).Step 3: Substitute the known values: 1440 = 1200 * (1 + R / 100).Step 4: Divide both sides by 1200 to get 1440 / 1200 = 1 + R / 100.Step 5: Simplify the fraction: 1440 / 1200 = 1.2.Step 6: Therefore, 1.2 = 1 + R / 100, so R / 100 = 0.2.Step 7: Multiply both sides by 100: R = 20.Step 8: So the rate of interest is 20% per annum. Verification / Alternative check: We can verify using a hypothetical principal P. If the rate is 20%, then the interest in year 2 is based on the amount at the end of year 1. Let amount at end of year 1 be A1. Then year 2 interest is A1 * 20 / 100 = 0.2 * A1. Year 3 interest is A1 * (1.2) * 0.2 = 0.24 * A1. The ratio of year 3 interest to year 2 interest is 0.24 * A1 / (0.2 * A1) = 1.2. Since the given ratio 1440 / 1200 also equals 1.2, the rate 20% fits perfectly. Why Other Options Are Wrong: At 15% per annum, the ratio of interest in successive years would be 1.15, not 1.2. At 18% per annum, the ratio would be 1.18, and at 24%, it would be 1.24. None of these match 1.2. A rate of 12% per annum gives a ratio of 1.12. Thus these options do not fit the observed ratio of third year interest to second year interest. Only 20% per annum produces exactly 1.2 as required by the given data. Common Pitfalls: Some students attempt to find the principal P unnecessarily and get bogged down in lengthy algebra, even though it cancels out. Others mistakenly use the formula for total compound interest instead of focusing on the year-wise interest amounts. A clear understanding that each year's interest under compound interest grows by the factor (1 + R / 100) simplifies the problem significantly. Final Answer: The rate of interest that makes the second year compound interest equal to Rs 1,200 and the third year compound interest equal to Rs 1,440 is 20% per annum.

Question 9

The difference between the simple interest and the compound interest (compounded annually) on a certain sum of money for 2 years at 12% per annum is Rs 900. What is the value of this principal sum, in rupees?

Accepted Answer

Introduction / Context: This question involves a known shortcut relating the difference between compound interest and simple interest for 2 years on the same principal and at the same rate. Instead of computing simple and compound interest separately, we can use a compact formula that directly connects the difference with the principal and the rate. This allows us to work backwards from the given difference to the principal. It is a common type of problem in competitive exams. Given Data / Assumptions: Simple interest (SI) and compound interest (CI) are calculated on the same principal for 2 years. Rate of interest R = 12% per annum. Difference between CI and SI over 2 years = Rs 900. Interest is compounded annually for CI. We must find the principal sum P. Concept / Approach: For 2 years at the same rate R on principal P, with annual compounding, there is a direct relation: Difference (CI - SI) for 2 years = P * (R / 100)^2This result comes from expanding the compound amount and comparing it with simple interest over 2 years. Once we know the difference and the rate, we can solve for P easily by rearranging the formula to: P = Difference / (R / 100)^2 Step-by-Step Solution: Step 1: Write the formula for the difference: CI - SI over 2 years = P * (R / 100)^2.Step 2: Substitute the given data. Here CI - SI = 900 and R = 12.Step 3: Compute (R / 100)^2 = (12 / 100)^2 = (0.12)^2 = 0.0144.Step 4: Set up the equation: 900 = P * 0.0144.Step 5: Solve for P by dividing both sides by 0.0144: P = 900 / 0.0144.Step 6: Convert 0.0144 to a fraction: 0.0144 = 144 / 10000.Step 7: Therefore P = 900 * 10000 / 144. Simplify 900 / 144 = 25 / 4, so P = (25 / 4) * 10000 = 62500.Step 8: Hence the principal sum is Rs 62,500. Verification / Alternative check: We can verify by computing both SI and CI using P = 62,500 and R = 12%. Simple interest for 2 years is SI = 62500 * 12 * 2 / 100 = 62500 * 0.24 = 15000. Compound interest is based on A = 62500 * (1.12)^2 = 62500 * 1.2544 = 78400. CI = 78400 - 62500 = 15900. The difference CI - SI = 15900 - 15000 = 900, which matches the given value. This confirms that the computed principal is correct. Why Other Options Are Wrong: Rs.125000 would give a difference four times as large, Rs.250000 even more, and Rs.187500 somewhere in between, none of which produce exactly Rs 900 as the difference. Rs.100000 also fails to satisfy the difference formula. Only Rs.62500 leads to a difference of Rs 900 between compound and simple interest at 12% per annum over 2 years. Common Pitfalls: Some students attempt to compute SI and CI separately without remembering the compact formula, which leads to more steps and potential arithmetic mistakes. Others misapply the formula, using P * R / 100 instead of P * (R / 100)^2, or forget that it applies specifically to 2 years. Carefully recalling that CI - SI for 2 years equals P * (R / 100)^2 makes the problem straightforward. Final Answer: The value of the principal sum that produces a Rs 900 difference between compound and simple interest in 2 years at 12% per annum is Rs.62500.

Question 10

At what annual rate of compound interest per annum will a sum of Rs 20,000 grow to Rs 23,328 in 2 years, assuming interest is compounded annually?

Accepted Answer

Introduction / Context: In this question, the principal, the amount after a specified time, and the time period are given, and we are asked to find the annual compound interest rate. This is a reverse application of the compound interest formula. Instead of computing the amount from the rate, we must infer the rate from the known principal and amount. This kind of problem is very common in aptitude tests and develops skill in algebraic manipulation of exponential equations. Given Data / Assumptions: Principal (P) = Rs 20,000. Amount after 2 years (A) = Rs 23,328. Time (T) = 2 years. Interest is compounded annually. We must find the annual rate of interest R (in percent). Concept / Approach: For annual compounding, the compound interest amount formula is: A = P * (1 + R / 100)^TWe know A, P, and T, so we can solve for (1 + R / 100) and then for R. For T = 2 years, this becomes: 23328 = 20000 * (1 + R / 100)^2Rearranging gives: (1 + R / 100)^2 = 23328 / 20000Taking the square root yields 1 + R / 100, from which R can be determined. Step-by-Step Solution: Step 1: Write the basic equation: 23328 = 20000 * (1 + R / 100)^2.Step 2: Divide both sides by 20000 to isolate the squared factor: (1 + R / 100)^2 = 23328 / 20000.Step 3: Simplify the ratio 23328 / 20000 = 1.1664.Step 4: So (1 + R / 100)^2 = 1.1664.Step 5: Take the positive square root (since 1 + R / 100 must be positive): 1 + R / 100 = sqrt(1.1664).Step 6: The square root of 1.1664 is 1.08 (since 1.08^2 = 1.1664).Step 7: Therefore, 1 + R / 100 = 1.08, so R / 100 = 0.08.Step 8: Multiply by 100 to find R: R = 8.Step 9: The required annual compound interest rate is 8% per annum. Verification / Alternative check: We can verify by computing the amount using R = 8%. For 2 years, A = 20000 * (1.08)^2. First 1.08^2 = 1.1664. Then A = 20000 * 1.1664 = 23328, which matches the given amount in the problem. This confirms that our calculated rate of 8% per annum is correct. Why Other Options Are Wrong: A rate of 16% would produce (1.16)^2 = 1.3456, giving an amount of 20000 * 1.3456 = 26912, which is too high. A rate of 24% would give (1.24)^2 = 1.5376, and amount 30752, which is much too high. A 12% rate yields (1.12)^2 = 1.2544, giving 25088, again higher than 23328. A 10% rate gives (1.10)^2 = 1.21, amount 24200. None of these match the target amount; only 8% reproduces exactly Rs 23,328. Common Pitfalls: Students sometimes treat the increase from 20000 to 23328 as 16.64% over 2 years and then incorrectly divide by 2 to get about 8.32%, forgetting that compound interest is multiplicative, not additive. Others may take the square root incorrectly or use approximate decimal values without checking. Writing out the steps clearly and verifying by substituting back into the original formula helps avoid these issues. Final Answer: The annual rate of compound interest that turns Rs 20,000 into Rs 23,328 in 2 years is 8 percent per annum.

Question 11

On a certain principal, the simple interest for 2 years is Rs 1,400, while the compound interest for the same 2 years at the same annual rate is Rs 1,449. What is the rate of interest per annum?

Accepted Answer

Introduction / Context: This question links simple interest and compound interest on the same principal and over the same period. We know the simple interest and the compound interest for 2 years, and we need to extract the annual rate of interest. The key idea is that the extra amount in compound interest over simple interest for 2 years is due to interest on interest, and it can be expressed in a compact formula involving the principal and the square of the rate. Combining this with the simple interest information allows us to solve for the rate efficiently. Given Data / Assumptions: Simple interest for 2 years (SI2) = Rs 1,400. Compound interest for 2 years (CI2) = Rs 1,449. Both are calculated on the same principal P and at the same annual rate R. Time period T = 2 years. Interest for CI is compounded annually. Concept / Approach: Let R be the annual rate in percent. First, simple interest for 2 years is: SI2 = P * R * 2 / 100 = 1400So: P * R = 70000The difference between compound interest and simple interest for 2 years on the same principal and at the same rate is: CI2 - SI2 = P * (R / 100)^2We know CI2 - SI2 = 1449 - 1400 = 49. Thus: P * (R / 100)^2 = 49We now have two equations in P and R, which we can solve systematically. Step-by-Step Solution: Step 1: From simple interest, P * R * 2 / 100 = 1400, so P * R = 1400 * 100 / 2 = 70000.Step 2: From the difference, CI2 - SI2 = 49, and CI2 - SI2 = P * (R / 100)^2.Step 3: Substitute P from the first relation into the second. We have P = 70000 / R.Step 4: Then 49 = (70000 / R) * (R / 100)^2.Step 5: Simplify (R / 100)^2 = R^2 / 10000, so 49 = 70000 * (R^2 / 10000) / R.Step 6: This simplifies to 49 = 70000 * R / 10000 = 7 * R.Step 7: Solve 49 = 7 * R, giving R = 7.Step 8: Therefore, the rate of interest is 7% per annum. Verification / Alternative check: We can verify quickly by finding P from P * R = 70000. With R = 7, P = 70000 / 7 = 10000. Simple interest for 2 years at 7% is SI2 = 10000 * 7 * 2 / 100 = 1400, which matches the given SI. For compound interest, the amount after 2 years at 7% is A = 10000 * (1.07)^2 = 10000 * 1.1449 = 11449, so CI2 = 11449 - 10000 = 1449, which matches the given CI. This confirms that R = 7% is correct. Why Other Options Are Wrong: Rates of 3.5%, 5%, 10%, or 14% would not produce the given pair of SI2 and CI2. For example, at 10% per annum, SI2 would be 2000 on a principal of 10000, not 1400. At 14% per annum, the difference CI2 - SI2 would be much larger than 49. These inconsistent outcomes show that such rates cannot satisfy both conditions at the same time. Only 7% per annum fits all the data exactly. Common Pitfalls: Some students try to guess the rate instead of using the relationships systematically and might not test their guesses thoroughly. Others forget or do not know the formula for CI2 - SI2 and therefore attempt longer computations that are more error prone. Remembering that CI2 - SI2 = P * (R / 100)^2 for 2 years greatly simplifies problems of this type. Final Answer: The rate of interest per annum that satisfies the given simple and compound interest values is 7 percent per annum.

Question 12

If the difference between the compound interest and the simple interest on a certain sum at the rate of 5% per annum for 2 years is Rs 20, then what is the value of that sum (principal), in rupees?

Accepted Answer

Introduction / Context: This is another question that uses the relationship between simple interest and compound interest on the same principal over a period of 2 years. The difference between compound interest and simple interest is given, and we must find the principal. The rate is moderate (5% per annum) and the time period is 2 years, which allows use of a direct formula. This type of question is very common in competitive exams because it tests whether a student knows this formula and can manipulate it correctly. Given Data / Assumptions: Rate of interest R = 5% per annum. Time period T = 2 years. Compound interest is computed annually. Difference between compound interest (CI) and simple interest (SI) for 2 years is Rs 20. Principal P is unknown and must be determined. Concept / Approach: For 2 years at the same rate R on a principal P, the difference between CI and SI is given by: CI - SI = P * (R / 100)^2This formula is derived from expanding the 2 year compound amount and comparing it with the simple interest expression for 2 years. Given the difference and the rate, we can solve directly for P. No separate calculation of SI and CI is required. Step-by-Step Solution: Step 1: Write the formula for the difference between CI and SI over 2 years: CI - SI = P * (R / 100)^2.Step 2: Substitute the known values: CI - SI = 20 and R = 5.Step 3: Compute (R / 100)^2 = (5 / 100)^2 = (0.05)^2 = 0.0025.Step 4: So the equation becomes 20 = P * 0.0025.Step 5: Solve for P by dividing both sides by 0.0025: P = 20 / 0.0025.Step 6: Note that 0.0025 = 25 / 10000. So P = 20 * 10000 / 25.Step 7: Simplify 20 / 25 = 4 / 5, hence P = (4 / 5) * 10000 = 8000.Step 8: Therefore, the principal sum is Rs 8,000. Verification / Alternative check: We can verify by directly computing SI and CI on Rs 8,000 at 5% per annum for 2 years. Simple interest SI = 8000 * 5 * 2 / 100 = 800. For compound interest: amount A = 8000 * (1.05)^2. Compute 1.05^2 = 1.1025, so A = 8000 * 1.1025 = 8820. CI = A - P = 8820 - 8000 = 820. The difference CI - SI = 820 - 800 = 20, exactly as given, which confirms that the principal is Rs 8,000. Why Other Options Are Wrong: Rs.2000, Rs.4000, and Rs.6000 would produce smaller differences between CI and SI at 5% over 2 years, because the difference scales directly with P. Rs.10000 would yield a larger difference of 25, not 20. Only Rs.8000 gives a difference of exactly Rs 20 between compound and simple interest at 5% for 2 years. Common Pitfalls: Students sometimes forget the specific formula for CI - SI in 2 years and instead attempt a full SI and CI computation, which is longer but still feasible. Others mistakenly use P * R / 100 instead of P * (R / 100)^2, confusing simple interest for one year with the extra interest in compounding. Correctly recalling that CI - SI = P * (R / 100)^2 for 2 years makes the calculation quick and reliable. Final Answer: The principal sum that produces a difference of Rs 20 between compound interest and simple interest at 5% per annum for 2 years is Rs.8000.

Question 13

Find the compound interest on Rs 7,500 at 4% per annum for 2 years, if the interest is compounded annually. Give your answer in rupees.

Accepted Answer

Introduction / Context: This question is a straightforward computation of compound interest over 2 years with annual compounding. The principal, the rate, and the time are clearly given. We are asked specifically for the compound interest, not the final amount. This type of problem checks basic familiarity with the compound interest formula and the ability to perform simple percentage based calculations accurately. Given Data / Assumptions: Principal (P) = Rs 7,500. Rate of interest (R) = 4% per annum. Time (T) = 2 years. Interest is compounded annually. No intermediate deposits or withdrawals are made. Concept / Approach: With annual compounding, the amount after T years is: A = P * (1 + R / 100)^TOnce we find A, the compound interest CI is simply: CI = A - PFor T = 2 years and R = 4%, the formula becomes: A = 7500 * (1 + 4 / 100)^2 = 7500 * (1.04)^2 Step-by-Step Solution: Step 1: Calculate the growth factor per year: 1 + R / 100 = 1 + 4 / 100 = 1.04.Step 2: Because the time is 2 years, square the factor: (1.04)^2 = 1.04 * 1.04 = 1.0816.Step 3: Multiply the principal by this factor to get the amount: A = 7500 * 1.0816.Step 4: Compute 7500 * 1.0816 = 8112.Step 5: The compound interest is CI = A - P = 8112 - 7500.Step 6: Subtract to find CI: 8112 - 7500 = 612.Step 7: Thus, the compound interest earned is Rs 612. Verification / Alternative check: We can compute year by year for confirmation. In year 1, interest = 7500 * 4 / 100 = 300, so amount = 7500 + 300 = 7800. In year 2, interest is calculated on 7800: interest = 7800 * 4 / 100 = 312, so final amount = 7800 + 312 = 8112. The total interest over 2 years is 8112 - 7500 = 612, which matches the result derived via the formula. Why Other Options Are Wrong: Rs.512, Rs.515, Rs.522, and Rs.600 do not match the total compound interest computed by applying 4% annually for 2 years. For example, Rs.600 would come from approximating 300 each year without updating the principal amount, which is effectively simple interest. The exact compounding adds a little extra interest in the second year, which brings the total to 612. Hence, only Rs.612 is correct. Common Pitfalls: A frequent mistake is to calculate simple interest for 2 years as 7500 * 4 * 2 / 100 = 600 and give this as the answer, ignoring that the question mentions compound interest. Some students also miscalculate 1.04^2 or round intermediate values too early. Using the year by year method as a check is a good way to reinforce correct reasoning and arithmetic. Final Answer: The compound interest on Rs 7,500 at 4% per annum for 2 years, compounded annually, is Rs.612.

Question 14

Simple interest on a certain sum of money for 3 years at 8% per annum is equal to half the compound interest on Rs 4,000 for 2 years at 10% per annum. What is the value of the sum placed on simple interest, in rupees?

Accepted Answer

Introduction / Context: This question combines both simple interest and compound interest in a comparative statement. We are told that simple interest on an unknown sum for a certain time and rate equals half the compound interest on a known sum for a different time and rate. The task is to compute the compound interest on the known principal first and then use the given relationship to find the unknown principal that would produce the given simple interest amount. Such cross linked interest problems are typical in aptitude exams and test a student's ability to coordinate multiple formulas. Given Data / Assumptions: Unknown principal for simple interest = P (to be found). Simple interest rate for P = 8% per annum. Time for simple interest = 3 years. Known principal for compound interest = Rs 4,000. Compound interest rate on 4,000 = 10% per annum. Time for compound interest = 2 years, compounded annually. Simple interest on P for 3 years at 8% equals half the compound interest on Rs 4,000 for 2 years at 10%. Concept / Approach: First compute the compound interest on Rs 4,000 at 10% per annum for 2 years. Then take half of that amount. This half is equal to the simple interest on P for 3 years at 8%. Using the simple interest formula, we can then solve for P. The main formulas are: A = 4000 * (1 + 10 / 100)^2 for the compound amountCI = A - 4000SI = P * R * T / 100 for simple interest Step-by-Step Solution: Step 1: Compute the compound amount on 4,000 for 2 years at 10% per annum. A = 4000 * (1 + 10 / 100)^2 = 4000 * (1.10)^2.Step 2: Calculate (1.10)^2 = 1.21.Step 3: So A = 4000 * 1.21 = 4840.Step 4: Compound interest on 4,000 is CI = A - principal = 4840 - 4000 = 840.Step 5: According to the problem, simple interest on P equals half of this CI, so SI on P = 840 / 2 = 420.Step 6: Use the simple interest formula SI = P * R * T / 100. Here SI = 420, R = 8, and T = 3.Step 7: Substitute the values: 420 = P * 8 * 3 / 100 = P * 24 / 100.Step 8: Rearrange: 420 = 0.24 * P, so P = 420 / 0.24.Step 9: Rewrite 0.24 as 24 / 100. So P = 420 * 100 / 24.Step 10: Simplify 420 / 24 = 17.5. Thus P = 17.5 * 100 = 1750.Step 11: Therefore, the sum placed on simple interest is Rs 1,750. Verification / Alternative check: We can verify by going back to the definitions. Simple interest on 1750 at 8% for 3 years is SI = 1750 * 8 * 3 / 100 = 1750 * 24 / 100 = 1750 * 0.24 = 420. Compound interest on 4000 at 10% for 2 years is 840 as computed earlier. Half of 840 is 420, which matches the simple interest on 1750. This confirms the relationship given in the problem and validates our answer. Why Other Options Are Wrong: At 1550, simple interest would be 1550 * 24 / 100 = 372, which is less than half the compound interest of 840. At 1650, simple interest would be 396, still not equal to 420. At 1850, simple interest becomes 444, greater than 420. At 2000, simple interest would be 480, again too high. None of these match the required 420. Only P = 1750 produces the exact equality specified in the question. Common Pitfalls: Some students misinterpret the statement and mistakenly equate the compound interest directly with simple interest, ignoring the factor of one half. Others compute simple interest and compound interest on the wrong principals or mix the rates and times. Misplacing the factor of 2 in the relationship can lead to completely wrong results. Carefully translating the English sentence into an algebraic equation is crucial in such word problems. Final Answer: The sum placed on simple interest, whose 3 year simple interest at 8% equals half the compound interest on Rs 4,000 at 10% for 2 years, is Rs.1750.

Question 15

In how many years will a principal of Rs 1000 amount to Rs 1331 at 10% per annum, when the interest is compounded annually throughout the period?

Accepted Answer

Introduction / Context: This question tests the core concept of compound interest and how time, rate of interest, and principal interact to produce a given amount. Instead of directly giving the time period, the question provides the initial principal, the final amount, and the annual compound rate, and asks you to work backward to determine the number of years required. Such questions are common in bank exams and aptitude tests because they check conceptual clarity about the compound interest formula and the ability to manipulate powers and ratios. Given Data / Assumptions: Principal P = Rs 1000 Amount after n years A = Rs 1331 Annual rate of interest r = 10% per annum Interest is compounded once every year Concept / Approach: For compound interest with annual compounding, the relationship between amount and principal is given by the standard formula A = P * (1 + r)^n, where r is written in decimal form and n is the number of years. Here, we know P, A, and r, so we rearrange the formula to solve for the time period n. A useful trick is to first compute the ratio A / P and then recognize it as a power of (1 + r). When the numbers are neat, this ratio often corresponds to a small integer power, which allows a quick mental check without heavy calculation. Step-by-Step Solution: Given P = 1000, A = 1331, r = 10% = 0.10 Formula: A = P * (1 + r)^n So, 1331 = 1000 * (1 + 0.10)^n 1331 = 1000 * (1.10)^n Divide both sides by 1000: 1331 / 1000 = (1.10)^n 1331 / 1000 = 1.331 Note that 1.10^2 = 1.21 and 1.10^3 = 1.331 Therefore, (1.10)^3 = 1.331, so n = 3 years Verification / Alternative check: We can verify the answer by directly computing the amount year by year. After 1 year, the amount is 1000 * 1.10 = 1100. After 2 years, it becomes 1100 * 1.10 = 1210. After 3 years, it becomes 1210 * 1.10 = 1331. This matches the required amount exactly. Since after 2 years the amount is still less than 1331 and after 3 years it is exactly 1331, 3 years is the only valid solution. No fractional time is needed because the growth is exact at n = 3. Why Other Options Are Wrong: 2 years would give an amount of 1210, which is less than 1331, so it is too short a period. 4 years would give 1331 * 1.10 = 1464.1, which is greater than the required amount. 5 years would increase the amount even more and therefore does not satisfy the given data. Hence all options other than 3 years are inconsistent with the formula and the given amount. Common Pitfalls: A common mistake is to apply the simple interest idea and assume that Rs 1000 grows by Rs 100 every year and therefore takes 3.31 years, which is incorrect for compound interest. Another error is to approximate the ratio 1331 / 1000 without recognizing it as a neat power of 1.10. Some students also forget to convert the percentage 10% to decimal form 0.10 in the formula, which changes the calculation and leads to a wrong time period. Final Answer: The required time is 3 years for Rs 1000 to grow to Rs 1331 at 10 percent per annum compounded annually.

Question 16

The periodic interest rate on an investment is 1.75% per quarter. What is the equivalent nominal annual rate of interest, quoted per annum, that corresponds to this quarterly rate?

Accepted Answer

Introduction / Context: This question checks your understanding of nominal and periodic interest rates. Financial institutions often quote an annual nominal rate but actually apply interest at shorter intervals such as monthly, quarterly, or half yearly. In some problems, the periodic rate is given and you must work out the corresponding nominal annual rate. Being able to move between these two views of the same rate is an important skill in time value of money calculations and in interpreting bank advertisements and loan offers correctly. Given Data / Assumptions: Periodic rate per quarter i = 1.75% per quarter There are 4 quarters in one year We are asked for the nominal annual rate of interest The nominal rate is quoted without converting it to an effective annual rate Concept / Approach: The nominal annual rate is simply the periodic rate multiplied by the number of compounding periods per year. It is a way of quoting the rate in per year terms but does not reflect compounding within the year. In this question, the compounding period is a quarter, so there are 4 compounding periods in one year. Therefore, the nominal annual rate j is given by j = periodic rate * number of periods per year. We do not yet convert this to an effective annual yield, we only scale the periodic rate linearly by the number of periods. Step-by-Step Solution: Given periodic rate per quarter i = 1.75% Number of quarters per year m = 4 Nominal annual rate j = i * m j = 1.75% * 4 j = 7.00% per annum Verification / Alternative check: You can think of this as simply adding 1.75% four times over the year since nominal rates do not yet incorporate the effect of compounding. If we wanted the effective annual rate, we would calculate (1 + 0.0175)^4 - 1, which gives a slightly higher percentage. However, the question clearly asks for the nominal rate, so the simple multiplication approach is exactly what is needed and the result of 7% per annum is correct. Why Other Options Are Wrong: 8% and 9% per annum would correspond to periodic rates of 2% or 2.25% per quarter, which do not match the given 1.75%. A 10% nominal rate would imply 2.5% per quarter, which is also inconsistent. Therefore 7% per annum is the only option that correctly reflects four quarters at 1.75% each. Common Pitfalls: Learners sometimes confuse the nominal annual rate with the effective annual rate, which includes compounding. Another error is to divide by 4 instead of multiplying, which would be appropriate only if you were given a nominal annual rate and asked for the periodic rate. Reading the phrase per quarter carefully helps avoid this confusion. Final Answer: The nominal annual rate of interest is 7% per annum corresponding to a periodic rate of 1.75 percent per quarter.

Question 17

What is the difference, in rupees, between the compound interest on Rs 1000 for 1 year at 10% per annum when interest is compounded yearly and when interest is compounded half yearly?

Accepted Answer

Introduction / Context: This problem compares two different compounding frequencies applied to the same principal, rate, and total time. Even though the nominal annual rate is the same in both cases, changing the compounding period from yearly to half yearly slightly changes the amount of compound interest earned. Understanding this difference is useful for evaluating investment products and loans, where the compounding frequency can influence the true return or cost. Given Data / Assumptions: Principal P = Rs 1000 Nominal annual rate r = 10% per annum Total time T = 1 year Case 1: Interest compounded yearly (once per year) Case 2: Interest compounded half yearly (twice per year) Concept / Approach: For annual compounding, the amount after 1 year is A1 = P * (1 + r). The compound interest is A1 - P. For half yearly compounding, the yearly rate is divided by 2 to get the half yearly rate, and the number of periods is multiplied by 2. The amount after T years is A2 = P * (1 + r / 2)^(2T). The difference in compound interest is then (A2 - P) minus (A1 - P), which simplifies to A2 - A1. Since P and r are small, we can compute these values exactly without approximation. Step-by-Step Solution: Case 1 (yearly compounding): A1 = P * (1 + r) = 1000 * (1 + 0.10) = 1000 * 1.10 = 1100 CI1 = A1 - P = 1100 - 1000 = Rs 100 Case 2 (half yearly compounding): Half yearly rate = r / 2 = 10% / 2 = 5% = 0.05 Number of half years in 1 year = 2 A2 = P * (1 + 0.05)^2 = 1000 * 1.05^2 = 1000 * 1.1025 = 1102.50 CI2 = A2 - P = 1102.50 - 1000 = Rs 102.50 Difference in compound interest = CI2 - CI1 = 102.50 - 100 = Rs 2.50 Verification / Alternative check: Another way to check is to compute the difference directly from the amounts: A2 - A1 = 1102.50 - 1100 = Rs 2.50. This is the additional interest earned due to more frequent compounding. Both methods are consistent and confirm that the extra interest is Rs 2.50. Why Other Options Are Wrong: A difference of Rs 1.50 or Rs 0.50 would imply much closer amounts for the two compounding schemes and does not match the correctly calculated gap. Rs 3.50 is larger than the true difference and is inconsistent with the computed amounts. Only Rs 2.50 exactly matches the difference between the two correct compound interest values. Common Pitfalls: Students sometimes forget to square the term for half yearly compounding and incorrectly use only one period. Another common error is to compare simple interest versus compound interest instead of comparing two compound interest cases. Some also mistakenly subtract the wrong amounts, mixing up principal and amount. Carefully writing the formulas and performing calculations step by step helps to avoid these mistakes. Final Answer: The difference between the two compound interests is Rs 2.50 in favor of half yearly compounding.

Question 18

For a certain principal invested at compound interest, the amount at the end of the second year is Rs 30250 and the amount at the end of the third year is Rs 33275. What is the annual rate of compound interest?

Accepted Answer

Introduction / Context: This question focuses on using successive compound amounts to determine the annual rate of interest. Instead of giving principal and rate directly, we are told what the investment is worth after two and three years. The key idea is that the ratio of consecutive year end amounts under compound interest is simply 1 plus the annual rate. Such problems test your ability to recognize patterns in the growth of money under compounding and to extract the interest rate from those patterns. Given Data / Assumptions: Amount after 2 years A2 = Rs 30250 Amount after 3 years A3 = Rs 33275 Principal P is unknown but stays constant Interest is compounded once per year at a constant rate r Concept / Approach: Under annual compounding, the amount after each year is multiplied by the same factor (1 + r). Therefore, A2 = P * (1 + r)^2 and A3 = P * (1 + r)^3. Dividing A3 by A2 cancels both P and one power of (1 + r), giving A3 / A2 = 1 + r. This ratio directly equals the growth factor per year, so the annual rate can be found as r = (A3 / A2) - 1. This method avoids having to find the principal first and gives a neat and quick solution. Step-by-Step Solution: A2 = P * (1 + r)^2 A3 = P * (1 + r)^3 Compute A3 / A2: A3 / A2 = [P * (1 + r)^3] / [P * (1 + r)^2] = 1 + r Given A3 = 33275 and A2 = 30250 1 + r = 33275 / 30250 33275 / 30250 = 1.10 So 1 + r = 1.10 and r = 0.10 = 10% per annum Verification / Alternative check: We can verify this by checking whether A2 grows correctly to A3 at 10%. Applying 10% to 30250 gives 30250 * 0.10 = 3025. Adding this interest to 30250 yields 30250 + 3025 = 33275, which matches the given third year amount. This confirms that the annual rate of 10% is consistent with the data, and therefore correct. Why Other Options Are Wrong: A 5% rate would give a factor of 1.05, so A3 would be 30250 * 1.05 which is smaller than 33275. A 15% rate corresponds to factor 1.15, which would produce an amount larger than 33275. A 20% rate would increase the amount even more sharply. Only a 10% rate produces exactly the given progression from 30250 to 33275 in one year. Common Pitfalls: A frequent mistake is to try to calculate the principal first and perform unnecessary algebra. Some learners also incorrectly take the difference A3 - A2 and treat it as simple interest, forgetting the effect of compounding. Remembering that the ratio of successive compound amounts directly equals 1 plus the annual rate makes these problems much easier. Final Answer: The annual rate of compound interest is 10% per annum.

Question 19

A sum of Rs 2000 amounts to Rs 4000 in 2 years at compound interest. In how many years from the start will the same principal amount become Rs 8000 at the same rate of interest?

Accepted Answer

Introduction / Context: This question explores how money grows under compound interest when the amount is doubling and then doubling again. The key idea is that, for a fixed compound interest rate, the growth factor over any equal time interval remains the same. Here, we know that Rs 2000 becomes Rs 4000 in 2 years, and we need to find how long it will take to become Rs 8000 at the same rate. This situation gives a nice opportunity to use ratios and powers rather than complicated calculations. Given Data / Assumptions: Principal P = Rs 2000 Amount after 2 years A2 = Rs 4000 Future target amount A = Rs 8000 Same compound interest rate throughout Interest is compounded annually Concept / Approach: If Rs 2000 grows to Rs 4000 in 2 years, the amount has doubled in that period, so the growth factor over 2 years is 4000 / 2000 = 2. Under compound interest, this factor over the same interval will remain constant. To go from Rs 2000 to Rs 8000, the money must become four times the original principal. Since a factor of 2 is achieved in 2 years, a factor of 4 will be achieved in twice that time, because 4 = 2^2. Therefore, the required time is simply 2 intervals of 2 years each, totaling 4 years. Step-by-Step Solution: Given: 2000 becomes 4000 in 2 years Growth factor for 2 years = 4000 / 2000 = 2 We want to know when 2000 becomes 8000 Required total growth factor = 8000 / 2000 = 4 Notice that 4 = 2^2 Each 2 year period multiplies the amount by 2 Two such periods yield factor 2 * 2 = 4 over 4 years Therefore, required time = 2 years + 2 years = 4 years Verification / Alternative check: We can also express the situation with the compound interest formula. Let r be the annual rate. Then 2000 * (1 + r)^2 = 4000, so (1 + r)^2 = 2. Therefore, 1 + r = square root of 2. For time t, we need 2000 * (1 + r)^t = 8000, so (1 + r)^t = 4. Since (1 + r)^2 = 2, we have (1 + r)^4 = 4, so t = 4. This matches our simple reasoning based on doubling factors. Why Other Options Are Wrong: 2 years would only double the principal once, taking it to Rs 4000, not Rs 8000. 6 years would imply one and a half doubling intervals, which does not fit the neat doubling pattern derived from the given data. 8 years would be four doubling periods, which would produce a growth factor of 2^4 = 16 times the original amount. Only 4 years yields exactly four times the principal. Common Pitfalls: Some learners mistakenly treat this as simple interest and assume a linear growth, which leads to an incorrect time. Others forget that the same factor applies over equal intervals when interest is compounded at a constant rate. Recognizing the doubling pattern and linking it to powers of 2 is the fastest and most reliable method here. Final Answer: The principal will become Rs 8000 in 4 years at the same compound interest rate.

Question 20

In how many years will a principal of Rs 25000 earn Rs 8275 as compound interest at 10% per annum, when interest is compounded annually?

Accepted Answer

Introduction / Context: This question asks you to determine the time period required for a given principal to generate a specified amount of compound interest at a fixed annual rate. It emphasizes understanding the relationship between compound amount, principal, rate, and time, and how to manipulate the standard formula to find time when the interest part, rather than the final amount, is given. This is a typical exam question that checks conceptual clarity and comfort with percentages and powers. Given Data / Assumptions: Principal P = Rs 25000 Compound interest over n years CI = Rs 8275 Annual rate of interest r = 10% per annum Interest is compounded annually We need to find the number of years n Concept / Approach: For annual compounding, the amount A after n years is given by A = P * (1 + r)^n. The compound interest CI is defined as CI = A - P. Therefore, CI = P * [(1 + r)^n - 1]. When P, CI, and r are given, we can first compute the amount A as P + CI, then form the ratio A / P and find n such that (1 + r)^n equals that ratio. In many exam problems, the ratio is chosen so that it matches a simple power, making it easy to identify n without complex calculations. Step-by-Step Solution: P = 25000, CI = 8275, r = 10% = 0.10 First find the amount A: A = P + CI = 25000 + 8275 = 33275 Now use the compound amount formula: A = P * (1 + r)^n 33275 = 25000 * (1.10)^n Divide both sides by 25000: 33275 / 25000 = (1.10)^n 33275 / 25000 = 1.331 We know 1.10^3 = 1.331, so n = 3 years Verification / Alternative check: We can verify by explicitly computing the amount after 3 years at 10%. After 1 year: 25000 * 1.10 = 27500. After 2 years: 27500 * 1.10 = 30250. After 3 years: 30250 * 1.10 = 33275. The compound interest is then 33275 - 25000 = 8275, which matches the given CI. Hence 3 years is correct. Why Other Options Are Wrong: If n = 2 years, the amount is only 30250, giving CI = 5250, which is too low. For 4 years, the amount would be 33275 * 1.10 = 36602.50, giving a much larger CI than 8275. Five years would increase the amount even more. Therefore only 3 years matches the required interest exactly. Common Pitfalls: A common mistake is to treat the given interest as simple interest and apply the simple interest formula P * r * t / 100, which does not work here. Another pitfall is to forget to add the interest to the principal to form the amount A before applying the compound interest formula. Recognizing the ratio 1.331 as 1.10^3 is a useful shortcut that simplifies the calculation. Final Answer: The sum of Rs 25000 will earn Rs 8275 as compound interest in 3 years at 10 percent per annum compounded annually.

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