Question 1

A loan shark lends $100 to a borrower who agrees to repay $120 at the end of one month; assuming simple interest, what is the equivalent annual rate of interest in percent per annum for this short term loan?

Accepted Answer

Introduction / Context: This question illustrates an extremely high interest rate charged over a very short period, which is then converted to an annual rate using simple interest. The loan shark lends 100 dollars and demands 120 dollars after one month, representing a large one month interest. The problem shows how apparently small short term loans can correspond to huge annual percentage rates when scaled up. Given Data / Assumptions: - Principal P = 100 dollars. - Amount repaid after one month A = 120 dollars. - Time T for the loan = 1 month. - Interest is calculated using simple interest, not compounding. - Required quantity is the equivalent annual rate of interest R percent per annum. Concept / Approach: First find the one month simple interest as A − P. Then compute the monthly simple interest rate as (interest for one month) divided by principal, expressed in percent. Under simple interest, the annual rate is obtained by multiplying the monthly rate by 12, because interest grows linearly with time. This scaling from monthly to yearly rate is valid since no compounding is assumed. Step-by-Step Solution: Step 1: Compute one month interest: SI month = A − P = 120 − 100 = 20 dollars. Step 2: Monthly rate r month (in percent) is (SI month * 100) / P. Step 3: Substitute values: r month = (20 * 100) / 100 = 20 percent per month. Step 4: Under simple interest, annual rate R = monthly rate * 12. Step 5: Compute R = 20 * 12 = 240 percent per annum. Step 6: Therefore, the equivalent annual rate is 240 percent per year. Verification / Alternative check: We can also treat the annual rate R as unknown in the simple interest formula. For one month, which is 1 / 12 of a year, SI should be 20 dollars. Using SI = (P * R * T) / 100 with P = 100, T = 1 / 12, we get 20 = (100 * R * (1 / 12)) / 100. This simplifies to 20 = R / 12. Multiplying both sides by 12 gives R = 240 percent, confirming the earlier calculation. Why Other Options Are Wrong: A rate of 210 percent would give a monthly equivalent of 210 / 12 = 17.5 percent and only 17.50 dollars of interest, not 20 dollars. A rate of 220 percent corresponds to about 18.33 percent per month, again too low. A rate of 230 percent gives roughly 19.17 percent per month, still not enough to produce 20 dollars interest on 100 dollars in one month. Only 240 percent per annum corresponds exactly to a 20 percent monthly simple interest rate. Common Pitfalls: Learners might assume a compounding interpretation and incorrectly raise (1 + monthly rate) to a power, which is not required here. Others may forget to convert months to a fraction of a year when using the simple interest formula. Remember that for simple interest, the annual rate scales linearly with time, so multiplying the monthly rate by 12 is valid. Final Answer: The equivalent annual simple interest rate charged by the loan shark is 240% per annum.

Question 2

What is the simple interest on Rs 7,500 at a rate of 10% per annum for a period of 5 years, assuming standard simple interest calculations?

Accepted Answer

Introduction / Context: This is a direct and basic application of the simple interest formula. The principal, rate, and time are all given, and the question asks only for the interest earned over the specified period. Such problems are often used early in the study of interest calculations to build familiarity with the formula and the meaning of percentage rates per annum. Given Data / Assumptions: - Principal P = Rs 7,500. - Annual simple interest rate R = 10 percent per annum. - Time T = 5 years. - Interest is calculated using simple interest. - Required quantity is the simple interest SI over 5 years. Concept / Approach: The formula for simple interest is SI = (P * R * T) / 100. We substitute the values of principal, rate, and time directly into this formula. There is no need to find the total amount unless explicitly asked, but we can always compute it as A = P + SI as an additional step. Here we focus on the interest only, which is straightforward. Step-by-Step Solution: Step 1: Identify P = Rs 7,500, R = 10 percent, T = 5 years. Step 2: Use SI = (P * R * T) / 100. Step 3: Substitute: SI = (7,500 * 10 * 5) / 100. Step 4: Compute 7,500 * 10 = 75,000. Step 5: Multiply 75,000 * 5 = 3,75,000. Step 6: Divide by 100: SI = 3,75,000 / 100 = Rs 3,750. Step 7: Hence, the simple interest for 5 years is Rs 3,750. Verification / Alternative check: We can also think in terms of yearly interest. At 10 percent per annum on Rs 7,500, yearly interest is (7,500 * 10 * 1) / 100 = Rs 750. Over 5 years, total interest is 750 * 5 = Rs 3,750. This matches the result obtained from the formula, confirming the calculation. Why Other Options Are Wrong: Rs 2,750 would correspond to a lower effective rate or shorter time. Rs 1,750 and Rs 750 are even smaller and do not represent 10 percent per year for 5 full years. Only Rs 3,750 equals 10 percent of the principal each year, multiplied by 5 years. Common Pitfalls: Students may forget to divide by 100 after multiplying P, R, and T, which would give 3,75,000 instead of 3,750. Another error is to multiply the principal by the rate only once and forget to include time. Writing the full formula explicitly and following it step by step avoids such mistakes. Final Answer: The simple interest for 5 years is Rs 3,750.

Question 3

At the same rate of simple interest and on the same principal, what will be the ratio of the simple interest earned in 4 years to the simple interest earned in 6 years?

Accepted Answer

Introduction / Context: This question focuses on the proportional nature of simple interest with respect to time. When the principal and rate remain the same, simple interest grows linearly with time. Therefore, the interest earned for different time periods is directly proportional to those time periods. Here we compare interest over 4 years and 6 years and find the ratio of the two amounts of interest. Given Data / Assumptions: - Principal P is the same in both cases. - Annual simple interest rate R is the same in both cases. - Time periods considered are T1 = 4 years and T2 = 6 years. - Interest is calculated using simple interest only. - Required ratio is SI for 4 years : SI for 6 years. Concept / Approach: Simple interest is given by SI = (P * R * T) / 100. For a fixed principal and rate, SI is directly proportional to time T. This means SI1 / SI2 = T1 / T2. Hence, the ratio of simple interest for 4 years and 6 years is simply the ratio of 4 to 6. Simplifying this ratio gives the final answer without needing to know the actual principal or rate values. Step-by-Step Solution: Step 1: Write SI1 = (P * R * T1) / 100 for T1 = 4 years. Step 2: Write SI2 = (P * R * T2) / 100 for T2 = 6 years. Step 3: Compute SI1 / SI2 = [(P * R * 4) / 100] / [(P * R * 6) / 100]. Step 4: Cancel common factors P, R, and 100 to obtain SI1 / SI2 = 4 / 6. Step 5: Simplify the fraction 4 / 6 by dividing numerator and denominator by 2 to get 2 / 3. Step 6: Therefore, the ratio of simple interest for 4 years to that for 6 years is 2 : 3. Verification / Alternative check: Choose a convenient principal and rate to verify. For example, let P = Rs 100 and R = 10 percent per annum. For 4 years, SI1 = (100 * 10 * 4) / 100 = Rs 40. For 6 years, SI2 = (100 * 10 * 6) / 100 = Rs 60. The ratio SI1 : SI2 = 40 : 60, which simplifies to 2 : 3, confirming the conceptual reasoning. Why Other Options Are Wrong: A ratio of 1 : 2 would mean the interest over 6 years is exactly double that over 4 years, which is not correct because 6 is not double 4. A ratio of 2 : 1 would incorrectly imply that interest is larger for 4 years than for 6 years. A ratio of 3 : 2 is the inverse of the correct ratio. Only 2 : 3 matches the proportional relationship between 4 years and 6 years. Common Pitfalls: Learners may mistakenly think of compound interest behaviour and assume a non linear relation with time. Others might try to assign arbitrary values to principal and rate without realising that these cancel out. Remembering that simple interest is directly proportional to time for fixed principal and rate simplifies many ratio questions. Final Answer: The ratio of simple interest for 4 years to that for 6 years is 2 : 3.

Question 4

If Rs 2 produces Rs 11 as simple interest over a period of 30 years, what is the annual rate of simple interest in percent per annum for this very long term investment?

Accepted Answer

Introduction / Context: This question presents a very small principal and interest but over a very long period of 30 years. It asks for the annual simple interest rate that would produce a total interest of Rs 11 on a principal of Rs 2. The main skill tested is correct substitution into and rearrangement of the simple interest formula to solve for the annual rate. Given Data / Assumptions: - Principal P = Rs 2. - Simple interest earned SI = Rs 11. - Time period T = 30 years. - Interest is calculated using simple interest. - Required quantity is the annual rate R in percent per annum. Concept / Approach: The simple interest formula SI = (P * R * T) / 100 can be rearranged to find the rate as R = (SI * 100) / (P * T). We substitute the known values for SI, P, and T and then simplify. Because the principal is small and the time is large, the rate obtained is relatively high. However, mathematically, the process is the same as for any other simple interest problem. Step-by-Step Solution: Step 1: Start with SI = (P * R * T) / 100. Step 2: Rearrange for R: R = (SI * 100) / (P * T). Step 3: Substitute SI = 11, P = 2, T = 30. Step 4: Compute denominator P * T = 2 * 30 = 60. Step 5: Compute numerator SI * 100 = 11 * 100 = 1,100. Step 6: Divide numerator by denominator: R = 1,100 / 60. Step 7: Compute 1,100 / 60 = 18.3333 recurring. Step 8: Round to two decimal places to get R ≈ 18.33 percent per annum. Verification / Alternative check: Verify by recalculating SI with R = 18.33 percent. SI ≈ (2 * 18.33 * 30) / 100. First 2 * 18.33 = 36.66. Then 36.66 * 30 ≈ 1,099.8. Dividing by 100 gives SI ≈ 10.998, which is very close to 11, with the tiny difference due to rounding of the rate. If we keep more decimal places for R, the interest matches 11 exactly. This confirms that 18.33 percent is the correct approximate rate. Why Other Options Are Wrong: At 17.25 percent, the interest would be (2 * 17.25 * 30) / 100 = 10.35, which is less than 11. At 18.14 percent, interest is about 10.884, still smaller than 11. At 17.35 percent, the interest is also below 11. Among the options, 18.33 percent is the only rate that produces an interest essentially equal to Rs 11 over 30 years on Rs 2. Common Pitfalls: Learners may mistakenly treat SI as the final amount, or may forget to multiply by 100 when solving for R. Some may be surprised by the large rate and suspect a calculation error. However, with such a tiny principal over a long time, a high rate is necessary to earn Rs 11. Following the formula carefully shows that the result is consistent. Final Answer: The annual rate of simple interest is approximately 18.33% per annum.

Question 5

In a simple interest calculation in quantitative aptitude, a certain principal sum is invested at 18% per annum for 2 years instead of at 12% per annum for the same time period, and therefore the simple interest received is greater by Rs. 840. What …

Accepted Answer

Introduction / Context: This aptitude question tests the basic relationship between principal, rate, time, and simple interest, and how a change in rate of interest affects the total interest earned. By comparing the interest at two different rates for the same principal and time period, we can back-calculate the value of the original principal sum from the given difference in interest, here Rs. 840. Given Data / Assumptions: • The same principal sum P is invested for 2 years. • First case: rate of simple interest R1 = 18% per annum. • Second case: rate of simple interest R2 = 12% per annum. • Time T is the same in both cases and equals 2 years. • Extra interest earned at the higher rate is Rs. 840. • We assume ordinary simple interest: SI = (P * R * T) / 100. Concept / Approach: The key idea is that the extra interest arises only from the difference in rates, since principal and time remain the same. Therefore, we can write the difference of interests directly in terms of the difference in rates and equate it to Rs. 840 in order to find the principal P. Step-by-Step Solution: Simple interest formula: SI = (P * R * T) / 100. Interest at 18%: SI1 = (P * 18 * 2) / 100. Interest at 12%: SI2 = (P * 12 * 2) / 100. Difference in interest: SI1 - SI2 = Rs. 840. Compute SI1 - SI2 using rates: SI1 - SI2 = P * (18 - 12) * 2 / 100 = P * 6 * 2 / 100. So, P * 12 / 100 = 840. Therefore, P = 840 * 100 / 12. P = 840 * (100 / 12) = 840 * (25 / 3) = 7000. Hence, the original principal sum is Rs. 7000. Verification / Alternative check: We can quickly verify: At 12% for 2 years, interest = 7000 * 12 * 2 / 100 = 1680. At 18% for 2 years, interest = 7000 * 18 * 2 / 100 = 2520. Difference = 2520 - 1680 = 840, which matches the given excess interest. Therefore, Rs. 7000 is correct. Why Other Options Are Wrong: Rs. 5600, Rs. 6300, Rs. 8400, and Rs. 9000 all produce a difference of interest at 18% and 12% over 2 years that is not exactly Rs. 840. Each of these values either gives a smaller or larger difference than 840, so they cannot satisfy the equation based on the simple interest formula. Common Pitfalls: A common mistake is to compute each full interest amount separately without focusing on the difference in rates, which leads to unnecessary calculations. Another error is to forget the factor of time T = 2 years when forming the difference. Some students also divide by the wrong percentage or misplace decimal points when working with percentages, which changes the final value of P. Final Answer: The original principal sum invested is Rs. 7000.

Question 6

An automobile financier claims to lend money at simple interest, but in practice he adds the interest calculated every six months to the principal and then charges interest again, effectively using half-yearly compounding. If his stated (nominal) ra…

Accepted Answer

Introduction / Context: This problem focuses on the difference between simple interest and compound interest, and on how a financier can effectively charge a higher rate than the one declared. Instead of truly using simple interest, he compounds the interest every six months. The question asks us to convert the nominal rate of 8% per annum with semi-annual compounding into an equivalent effective annual rate. Given Data / Assumptions: • Stated (nominal) annual rate R = 8% per annum. • Interest is added every six months, so there are 2 compounding periods in a year. • Each half-yearly rate r_half = 8% / 2 = 4% per half-year. • We want the effective annual rate, meaning the single rate that produces the same increase over one full year. Concept / Approach: For compound interest with n compounding periods per year, the effective annual rate is calculated using the formula: Effective rate = (1 + r_half / 100)^n - 1, expressed as a percentage. Here, n = 2 because interest is compounded twice a year. We compare this effective rate with the nominal 8% to see how much extra the borrower is actually paying. Step-by-Step Solution: Half-yearly rate r_half = 8 / 2 = 4%. Number of compounding periods in a year n = 2. Effective annual multiplier = (1 + r_half / 100)^n. Substitute values: (1 + 4 / 100)^2 = (1 + 0.04)^2. Compute: 1.04 * 1.04 = 1.0816. Effective annual rate = (1.0816 - 1) * 100%. Effective annual rate = 0.0816 * 100% = 8.16% per annum. Verification / Alternative check: Consider Rs. 100 as principal. In one year, at 8% simple interest, interest is Rs. 8, so the amount is Rs. 108. Under the financier’s method with semi-annual compounding at 4% per half-year, amount after one year is 100 * 1.0816 = Rs. 108.16. The interest is Rs. 8.16, which corresponds to 8.16% on Rs. 100, confirming our effective annual rate. Why Other Options Are Wrong: 8% ignores the compounding effect and represents only the nominal rate. 8.5%, 9%, and 10.25% all overstate the growth; using them would give amounts significantly above Rs. 108.16 on Rs. 100 in one year, which does not match the computed compound amount at 4% per half-year. Common Pitfalls: Students often forget that semi-annual compounding requires using half the annual rate and squaring the growth factor. Another mistake is to simply add 8% and half of 8% or to treat the situation as if it were simple interest. Ignoring the compounding periods leads to incorrect effective rates. Final Answer: The effective annual rate of interest charged by the financier is 8.16% per annum.

Question 7

A sum of money is invested at simple interest and amounts to Rs. 415 in 2 years and to Rs. 514 in 4 years. Using the change in amount over time, what is the value of the original principal sum deposited?

Accepted Answer

Introduction / Context: This simple interest problem uses the fact that the amount increases linearly with time when the interest is simple. We are given the total amounts at the end of 2 years and 4 years and are asked to determine the original principal. The difference between these two amounts directly reveals the interest earned in an additional 2-year period, from which we can find the annual interest and then the principal. Given Data / Assumptions: • Amount after 2 years, A2 = Rs. 415. • Amount after 4 years, A4 = Rs. 514. • The same principal P and rate R are used throughout. • Simple interest is applied, so interest per year is constant. • Relation: Amount A = P + SI, where SI = (P * R * T) / 100. Concept / Approach: Under simple interest, the extra amount between year 2 and year 4 is just the interest earned for 2 more years. That difference allows us to calculate the yearly interest, and then we subtract the total 2-year interest from the 2-year amount to recover the principal P. Step-by-Step Solution: Difference between amounts: A4 - A2 = 514 - 415 = Rs. 99. This extra Rs. 99 is the simple interest for 2 additional years. So yearly interest = 99 / 2 = Rs. 49.50 per year. Interest for 2 years = 2 * 49.50 = Rs. 99. Amount after 2 years A2 = P + interest for 2 years. So P = A2 - 2-year interest = 415 - 99. P = Rs. 316. Therefore, the original principal sum is Rs. 316. Verification / Alternative check: If P = 316 and interest per year = 49.50, then after 2 years, amount = 316 + 2 * 49.50 = 316 + 99 = 415, matching A2. After 4 years, amount = 316 + 4 * 49.50 = 316 + 198 = 514, matching A4. This confirms that our principal calculation is correct. Why Other Options Are Wrong: Rs. 251, Rs. 294, Rs. 350, and Rs. 154 do not produce consistent amounts of Rs. 415 and Rs. 514 when a constant yearly interest is applied. Substituting these values for P leads to either different amounts or non-integer yearly interest values that do not match the given data for both 2 years and 4 years simultaneously. Common Pitfalls: Learners sometimes try to find the rate first, which introduces extra steps and possible errors. Another frequent mistake is to divide the difference 99 by 4, thinking it spans 4 years, instead of recognizing it is the interest for only the extra 2 years. Forgetting that simple interest grows linearly with time also leads to confusion. Final Answer: The original principal sum is Rs. 316.

Question 8

Avinash lends Rs. 5400 to Rajeev at a simple interest rate of 8% per annum for 4 years. Rajeev, in turn, lends the same Rs. 5400 to Chanukya at a simple interest rate of 6% per annum for 4 years. At the end of 4 years, how much extra amount, in rupe…

Accepted Answer

Introduction / Context: This problem is a chain-lending simple interest question that compares the interest Rajeev has to pay to Avinash with the interest Rajeev receives from Chanukya. Because Rajeev borrows and lends the same amount for the same time period but at different rates of interest, he will either gain or lose money based on the spread between the two rates. Given Data / Assumptions: • Principal amount in both transactions P = Rs. 5400. • Avinash lends to Rajeev at R1 = 8% per annum simple interest for T = 4 years. • Rajeev lends to Chanukya at R2 = 6% per annum simple interest for the same T = 4 years. • Simple interest formula: SI = (P * R * T) / 100. • We want the extra amount Rajeev must pay from his own pocket at the end of 4 years. Concept / Approach: Since both loans have the same principal and duration, the difference in total interest is purely due to the difference in rates. Rajeev pays interest at 8% but earns at 6%. The shortfall in interest is exactly what he must add from his own money in order to pay Avinash the full amount due. Step-by-Step Solution: Interest Rajeev must pay Avinash: SI1 = (5400 * 8 * 4) / 100. Compute SI1: 5400 * 8 = 43200; 43200 * 4 = 172800; SI1 = 172800 / 100 = Rs. 1728. Interest Rajeev receives from Chanukya: SI2 = (5400 * 6 * 4) / 100. Compute SI2: 5400 * 6 = 32400; 32400 * 4 = 129600; SI2 = 129600 / 100 = Rs. 1296. Difference in interest: SI1 - SI2 = 1728 - 1296 = Rs. 432. Thus, Rajeev is short by Rs. 432 and must pay this amount himself. Verification / Alternative check: Total amount Avinash expects after 4 years = 5400 + 1728 = Rs. 7128. Total amount Chanukya pays to Rajeev = 5400 + 1296 = Rs. 6696. The gap is 7128 - 6696 = Rs. 432, exactly the extra amount Rajeev must contribute, confirming the result. Why Other Options Are Wrong: Rs. 364, Rs. 410, Rs. 498, and Rs. 540 do not equal the difference between the two simple interest amounts. Using any of these values would imply an incorrect spread between 8% and 6% over 4 years on Rs. 5400 and would not balance the total required and received amounts. Common Pitfalls: A frequent error is to calculate only one side of the transaction or to subtract the rates 8% and 6% but forget to multiply by both principal and time. Some students mistakenly compare only yearly interests and forget to scale by 4 years, while others confuse simple and compound interest formulas. Final Answer: Rajeev must pay an extra Rs. 432 from his own pocket.

Question 9

The simple interest on Rs. 10 for a period of 4 months is calculated at a rate of 3 paise per rupee per month. When this monthly rate is applied, the interest is how many rupees in total?

Accepted Answer

Introduction / Context: This is a straightforward simple interest calculation involving a small principal and a rate given in paise per rupee per month. The question checks whether the student can correctly convert paise to rupees, interpret the rate per month, and apply it for the given number of months to find the total interest in rupees. Given Data / Assumptions: • Principal P = Rs. 10. • Time T = 4 months. • Rate is 3 paise per rupee per month. • 100 paise = 1 rupee. • We use simple interest, so interest each month is constant on the principal. Concept / Approach: The rate of 3 paise per rupee per month means that for every rupee of principal, the interest is Rs. 0.03 each month. Therefore, the monthly rate in percentage is 3%. Since the principal is Rs. 10 and the time is 4 months, we can treat the period directly in months and multiply the monthly interest accordingly. Step-by-Step Solution: Convert 3 paise to rupees: 3 paise = 3 / 100 = Rs. 0.03. Therefore, monthly rate per rupee is Rs. 0.03, which is 3% of 1 rupee. Interest per month on Rs. 10 = 10 * 3% = 10 * 0.03 = Rs. 0.30. Time is 4 months, so total interest = 4 * 0.30. Total simple interest = Rs. 1.20. Verification / Alternative check: We can also apply the simple interest formula in month units: SI = (P * R * T) / 100 with P = 10, R = 3% per month, T = 4 months. So SI = (10 * 3 * 4) / 100 = 120 / 100 = Rs. 1.20. This confirms that the interest is Rs. 1.20. Why Other Options Are Wrong: Rs. 0.03 is only the interest on Rs. 1 for one month, not on Rs. 10 for 4 months. Rs. 0.12 and Rs. 0.30 correspond to partial calculations, such as using only one month or incorrect multiplication. Rs. 3.00 greatly overestimates the interest and would require either a much longer time or a much higher rate. Common Pitfalls: A common mistake is to forget the conversion between paise and rupees, leading to using 3 rupees instead of Rs. 0.03. Others may apply the rate for only one month or misinterpret it as an annual rate instead of a monthly rate. Careful interpretation of the units avoids these errors. Final Answer: The total simple interest earned is Rs. 1.20.

Question 10

A certain sum of money becomes double of its original value in 15 years at a fixed rate of simple interest. At the same rate of simple interest, in how many years will this sum become five times its original value?

Accepted Answer

Introduction / Context: This question explores how the time required to reach different multiples of the principal changes under simple interest. Since simple interest grows linearly with time, once we know the time taken for the sum to double, we can determine the rate and then find the time required for the sum to become any other multiple, such as five times the original principal. Given Data / Assumptions: • The principal is some sum P (unknown but constant). • The sum becomes 2P (double) in 15 years. • Interest is simple interest at rate R% per annum. • We are asked for the time required for the amount to become 5P. Concept / Approach: Under simple interest, amount A after T years is A = P + SI = P + (P * R * T) / 100 = P * (1 + R * T / 100). We first use the doubling condition (A = 2P in 15 years) to find R. Then we apply the same rate R with A = 5P to calculate the new time T needed for the amount to become five times the principal. Step-by-Step Solution: For doubling: 2P = P * (1 + R * 15 / 100). Divide both sides by P: 2 = 1 + (R * 15 / 100). So, R * 15 / 100 = 1, which gives R = 100 / 15 = 6.666...% per annum. Now we want A = 5P: 5P = P * (1 + R * T / 100). Divide both sides by P: 5 = 1 + R * T / 100. So R * T / 100 = 4, therefore T = 4 * 100 / R. Substitute R = 100 / 15: T = 4 * 100 / (100 / 15) = 4 * 15 = 60 years. Verification / Alternative check: Once we know that 15 years of interest equals the principal P (because the sum doubled), each block of 15 years adds another P. To go from P to 5P, we need an increase of 4P in interest. Each P takes 15 years, so time required is 4 * 15 = 60 years. This short reasoning confirms the detailed calculation. Why Other Options Are Wrong: 37.5, 45, 75, and 30 years arise from miscounting how many times the principal needs to be added or from incorrectly scaling the time period. For example, 45 years corresponds to tripling, not quintupling, the principal, and 75 years would imply 5P plus an extra half P, which is not required here. Common Pitfalls: Many students assume that if doubling takes 15 years, then five times might be obtained by simply multiplying 15 by 2 or 3 without carefully thinking about how much interest is needed. Others mistakenly apply compound interest reasoning. Remember that under simple interest, the increase is linear, so each additional P of interest always takes the same amount of time. Final Answer: The sum will become five times its original value in 60 years.

Question 11

A certain sum of money becomes three times its original value in 5 years at simple interest. At the same constant rate of simple interest, in how many years will this sum become five times its original value?

Accepted Answer

Introduction / Context: This simple interest problem considers a sum that grows from its original value to three times in 5 years, and then asks for the time needed to reach five times at the same rate. Because simple interest produces linear growth over time, we can use the triple condition to find the rate and then extend the time to reach five times the principal. Given Data / Assumptions: • Principal P becomes 3P in 5 years at simple interest. • Rate of simple interest is R% per annum (constant). • We want the time T needed so that the amount becomes 5P. • Simple interest formula: A = P * (1 + R * T / 100). Concept / Approach: First, we find R using the condition that the amount becomes three times the principal in 5 years. Then we plug the same R into the amount formula with A = 5P and solve for T. Linear growth under simple interest makes this two-step approach straightforward and intuitive. Step-by-Step Solution: For tripling: 3P = P * (1 + R * 5 / 100). Divide both sides by P: 3 = 1 + (R * 5 / 100). So R * 5 / 100 = 2, which means R = 2 * 100 / 5 = 40% per annum. Now we want A = 5P: 5P = P * (1 + R * T / 100). Divide both sides by P: 5 = 1 + (40 * T / 100). So 40 * T / 100 = 4, giving T = 4 * 100 / 40 = 10 years. Therefore, it will take 10 years to become five times the principal. Verification / Alternative check: In 5 years the amount becomes 3P, so the total interest gained is 2P. This means an interest of 2P in 5 years, or 0.4P per year, consistent with a 40% rate. To reach 5P from P, we need interest of 4P. Since 1 year gives 0.4P, time required is 4P / 0.4P per year = 10 years, confirming our result. Why Other Options Are Wrong: 5 years would only triple the sum, not make it five times. 8, 12, and 15 years correspond to different multiples of P under a 40% simple interest rate and do not yield exactly 5P. Using those times in the formula would lead to amounts that are either less than or greater than five times the principal. Common Pitfalls: Some learners confuse three times the principal with three times the interest. Others mistakenly think the time to go from 3P to 5P is the same as the first 5 years. It is important to calculate the actual rate first, then apply it consistently to reach the required multiple. Mixing simple interest and compound interest ideas is another frequent error. Final Answer: The sum will become five times its original value in 10 years.

Question 12

The simple interest on Rs. 2000 for 2 years is calculated at a rate described as Rs. 75 per thousand rupees per annum. When this rate is applied, what is the total simple interest earned in rupees over the 2-year period?

Accepted Answer

Introduction / Context: This question is a basic test of interpreting a slightly unusual way of expressing the interest rate. Instead of giving a direct percentage, the lender states the rate as Rs. 75 per thousand rupees per annum. We must convert this description into a percentage rate and then apply the simple interest formula to find the total interest for 2 years on Rs. 2000. Given Data / Assumptions: • Principal P = Rs. 2000. • Time T = 2 years. • Rate is Rs. 75 interest per Rs. 1000 per year. • Simple interest formula: SI = (P * R * T) / 100. Concept / Approach: If Rs. 75 is charged as interest on Rs. 1000 for 1 year, then the percentage rate per annum is (75 / 1000) * 100% = 7.5% per year. Once we have the percentage rate, we can plug it into the simple interest formula along with the principal and time to get the total interest. Step-by-Step Solution: Convert the rate: R = (75 / 1000) * 100 = 7.5% per annum. Principal P = 2000, time T = 2 years. Use SI formula: SI = (P * R * T) / 100. SI = (2000 * 7.5 * 2) / 100. First compute 2000 * 7.5 = 15000. Then SI = (15000 * 2) / 100 = 30000 / 100 = Rs. 300. Verification / Alternative check: Another way: If Rs. 1000 earns Rs. 75 per year, then Rs. 2000 (twice as much) earns Rs. 150 per year. Over 2 years, total interest = 2 * 150 = Rs. 300, matching our earlier calculation. Why Other Options Are Wrong: Rs. 150 corresponds to only one year of interest on Rs. 2000 at this rate, not 2 years. Rs. 400, Rs. 450, and Rs. 600 are larger than the correct value and arise from mistakenly treating the rate as 10%, 12%, or double-counting the time or principal. Common Pitfalls: A typical error is to misread “Rs. 75 per thousand per annum” as a flat amount regardless of principal size, or to forget to convert it into a percentage. Others forget to multiply by the full 2-year period or incorrectly divide by 100 more than once. Keeping track of units and conversion steps avoids these issues. Final Answer: The total simple interest earned in 2 years is Rs. 300.

Question 13

A total sum of Rs. 10,000 is lent out partly at 6% per annum and the remaining amount at 10% per annum, both on simple interest. If the average rate of simple interest earned on the whole sum is 9.2% per annum, how much money, in rupees, is lent at …

Accepted Answer

Introduction / Context: This is a classic mixture or allocation question involving different simple interest rates. The total principal is divided into two parts lent at 6% and 10% per annum, and we are told the effective average rate on the entire Rs. 10,000 is 9.2%. Using this average, we determine how much was invested at the higher rate of 10% per annum. Given Data / Assumptions: • Total principal P_total = Rs. 10,000. • Let x be the amount lent at 10% per annum. • Then (10000 - x) is lent at 6% per annum. • Average simple interest rate on entire sum = 9.2% per annum. • Same time period is assumed for both parts, so time cancels out in average rate calculations. Concept / Approach: Because the time is the same for both parts, the overall average rate depends only on the weighted average of the two rates, using principal amounts as weights. We equate the total interest per year from both parts to 9.2% of the full Rs. 10,000 and solve for x, the amount at 10%. Step-by-Step Solution: Interest per year from the amount at 10%: I1 = x * 10 / 100 = 0.10x. Interest per year from the amount at 6%: I2 = (10000 - x) * 6 / 100 = 0.06 * (10000 - x). Total yearly interest = I1 + I2. Average yearly interest at 9.2% on Rs. 10000 = 10000 * 9.2 / 100 = Rs. 920. Form equation: 0.10x + 0.06 * (10000 - x) = 920. Simplify: 0.10x + 600 - 0.06x = 920. Combine x terms: 0.04x + 600 = 920. So 0.04x = 320, giving x = 320 / 0.04 = 8000. Verification / Alternative check: Check interest: At 10%, Rs. 8000 earns 800 per year. At 6%, Rs. 2000 earns 120 per year. Total interest per year = 800 + 120 = 920. On Rs. 10000, 920 corresponds to an average rate of 9.2%, confirming that Rs. 8000 is correct. Why Other Options Are Wrong: Rs. 5000, Rs. 6500, Rs. 8500, and Rs. 9000 all yield total interest per year that is not equal to Rs. 920. Each incorrect allocation either gives a lower or higher average rate than 9.2% when the total interest is divided by 10000, so they do not satisfy the condition. Common Pitfalls: Some students try to apply a simple average of 6% and 10% without considering the different principal portions, which would give 8%, not 9.2%. Others set up the equation incorrectly by mixing up which amount is at which rate. Remember that average rate must be based on weighted contributions of each part. Final Answer: The amount lent at 10% per annum is Rs. 8000.

Question 14

A sum invested at simple interest amounts to Rs. 1392 in 2 years and to Rs. 1488 in 3 years. Using these two successive amounts, what is the rate of interest per annum (in percentage) being applied?

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Introduction / Context: This question uses the property of simple interest that the amount increases by the same fixed interest amount each year. We are given amounts after 2 and 3 years and must determine the annual rate. The difference between the two amounts directly tells us the interest for one year, from which we can find the rate as a percentage of the principal. Given Data / Assumptions: • Amount after 2 years A2 = Rs. 1392. • Amount after 3 years A3 = Rs. 1488. • Principal P is the same in both cases. • Simple interest is used, so annual interest is constant. • Relation: A = P + SI = P * (1 + R * T / 100). Concept / Approach: The extra amount from year 2 to year 3 is just one year’s simple interest. Once we know the yearly interest, we subtract 2 years’ interest from A2 to find P, and then compute the rate R using SI = (P * R * T) / 100 with T = 1 year. Step-by-Step Solution: Difference between A3 and A2: 1488 - 1392 = Rs. 96. This Rs. 96 is the interest for 1 additional year. So yearly interest I_year = Rs. 96. Interest for 2 years = 2 * 96 = Rs. 192. Principal P = A2 - interest for 2 years = 1392 - 192 = Rs. 1200. Now, for 1 year: SI = 96 = (P * R * 1) / 100 = 1200 * R / 100. So 96 = 12R, giving R = 96 / 12 = 8% per annum. Verification / Alternative check: Check using P = 1200 and R = 8%. Amount after 2 years: A2 = 1200 * (1 + 8 * 2 / 100) = 1200 * (1 + 0.16) = 1200 * 1.16 = 1392. Amount after 3 years: A3 = 1200 * (1 + 8 * 3 / 100) = 1200 * (1.24) = 1488. Both match the given values, confirming R = 8%. Why Other Options Are Wrong: 8.5%, 9%, 10%, and 12% produce different yearly interest amounts and thus lead to different A2 and A3 values. When substituted, they do not maintain the difference of exactly Rs. 96 between the second and third year while also matching the given amounts. Common Pitfalls: Students sometimes divide the total 3-year amount difference by 3 or misidentify which difference corresponds to one year. Another common issue is trying to find P and R simultaneously with unnecessary algebra instead of first extracting the yearly interest from the consecutive amounts. Final Answer: The rate of interest per annum is 8%.

Question 15

On a certain principal, the simple interest for 2 years is Rs. 4800, whereas the compound interest on the same principal for the same 2 years is Rs. 5088. Based on this difference between simple and compound interest over 2 years, what is the annual…

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Introduction / Context: This question compares simple interest and compound interest on the same principal over the same period of 2 years. Because compound interest adds interest on interest, it is slightly higher than simple interest for the same rate and time. The difference between the amounts of simple and compound interest allows us to determine the underlying annual rate of interest. Given Data / Assumptions: • Simple interest for 2 years on principal P is Rs. 4800. • Compound interest for 2 years on the same P is Rs. 5088. • Time T = 2 years in both cases. • Rate R is the same in both cases and is to be found. • Simple interest formula: SI = (P * R * T) / 100. • Compound amount for 2 years: A = P * (1 + R / 100)^2, and CI = A - P. Concept / Approach: From the simple interest, we can relate P and R. From the compound interest, we relate P and R in another way. Solving these two relations together yields both P and R. However, we do not need to explicitly know P to find R; we can use the equations strategically to solve for R directly. Step-by-Step Solution: From simple interest: SI = 4800 = P * R * 2 / 100. So P * R = 4800 * 100 / 2 = 240000. From compound interest: CI = 5088 = P * [(1 + R / 100)^2 - 1]. Let k = R / 100. Then (1 + k)^2 - 1 = 2k + k^2. Thus, 5088 = P * (2k + k^2). We also know P * k = P * R / 100 = (P * R) / 100 = 240000 / 100 = 2400. So express 5088 as P * (2k + k^2) = P * 2k + P * k^2 = 2 * 2400 + P * k^2. This gives 5088 = 4800 + P * k^2, so P * k^2 = 288. But P * k^2 = (P * k) * k = 2400 * k, so 2400 * k = 288. Therefore k = 288 / 2400 = 0.12, so R = 0.12 * 100 = 12 percent per annum. Verification / Alternative check: Using R = 12% and P from P * R = 240000, we get P = 240000 / 12 = 20000. Simple interest for 2 years: SI = 20000 * 12 * 2 / 100 = 4800, as given. Compound amount for 2 years: A = 20000 * (1.12)^2 = 20000 * 1.2544 = 25088, so CI = 25088 - 20000 = 5088, matching the question. This confirms R = 12%. Why Other Options Are Wrong: Rates of 6 percent, 10 percent, 18 percent, or 24 percent produce different relationships between SI and CI and do not lead to a simple interest of 4800 and compound interest of 5088 on the same principal in 2 years. When substituted, the differences between SI and CI are either too large or too small. Common Pitfalls: One common mistake is to assume that the difference between CI and SI over 2 years is simply P * (R / 100)^2, but misapplying this can cause algebra errors. Others try random substitution of rates without using the given SI value to form equations. Systematically using both interest expressions keeps the work clear and accurate. Final Answer: The annual rate of interest is 12 percent per annum.

Question 16

A sum of Rs. 2000 becomes Rs. 3645 in 2 years at a certain rate of compound interest. At the same annual rate of compound interest, what will be the amount of this sum after 4 years in total?

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Introduction / Context: Here we have a compound interest question where the amount after 2 years is known and the principal is Rs. 2000. We first determine the annual compound interest rate that transforms Rs. 2000 into Rs. 3645 in 2 years. Once we know this rate, we apply it for a total of 4 years to find the new amount of the same sum at compound interest. Given Data / Assumptions: • Principal P = Rs. 2000. • Amount after 2 years A2 = Rs. 3645. • Time for first stage T1 = 2 years. • Same annual rate R is used for compound interest. • We want the amount A4 after total time T2 = 4 years. Concept / Approach: For compound interest with annual compounding, amount A after T years is given by A = P * (1 + R / 100)^T. We use A2 = 2000 * (1 + R / 100)^2 = 3645 to find the growth factor (1 + R / 100). Once we have this factor, we raise it to the power 4 to get A4 = 2000 * (1 + R / 100)^4. Step-by-Step Solution: Let (1 + R / 100) = k. Then A2 = P * k^2, so 3645 = 2000 * k^2. Thus, k^2 = 3645 / 2000 = 1.8225. Taking the positive square root: k = sqrt(1.8225) = 1.35. So 1 + R / 100 = 1.35, giving R / 100 = 0.35 and R = 35% per annum. Now for 4 years, amount A4 = 2000 * k^4 = 2000 * (1.35)^4. Compute (1.35)^2 = 1.8225 and then (1.35)^4 = (1.8225)^2. (1.8225)^2 = 3.32150625 approximately. So A4 ≈ 2000 * 3.32150625 = Rs. 6643.0125. Verification / Alternative check: We can also compute year by year: After 1 year, amount = 2000 * 1.35 = 2700. After 2 years, 2700 * 1.35 = 3645, which matches the given amount. After 3 years, 3645 * 1.35 = 4920.75. After 4 years, 4920.75 * 1.35 = 6643.0125, confirming our calculation and matching the given option. Why Other Options Are Wrong: The other amounts (5942.0125, 7000.0000, 7243.0125, and 7498.1250) correspond to either different rates or different time periods. None of them matches the exact growth obtained by compounding at 35% for 4 years on Rs. 2000 after passing through Rs. 3645 at year 2. Common Pitfalls: A common error is to treat the situation as simple interest and incorrectly scale the 2-year increase linearly to 4 years. Another mistake is using an approximate rate without solving for the precise growth factor k, which leads to rounding errors that do not match any option exactly. Keeping the compound interest formula clear prevents confusion. Final Answer: The amount after 4 years will be approximately Rs. 6643.0125.

Question 17

If, in 3 years at simple interest, a principal increases by 18% of its own value, what will be the compound interest earned on Rs. 25,000 in 3 years at the same annual rate of interest?

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Introduction / Context: This problem connects a simple interest statement with a compound interest calculation. We are told that in 3 years at simple interest, a principal grows by 18% of itself. From this we deduce the annual rate. Then, using that rate, we calculate the compound interest on a different principal (Rs. 25,000) over the same 3-year period with annual compounding. Given Data / Assumptions: • Under simple interest, in 3 years, the increase is 18% of the principal. • Therefore, for simple interest: total interest over 3 years = 18% of P. • Hence, annual simple interest rate R% must satisfy 3R = 18. • New principal for compound interest calculation is P_new = Rs. 25,000. • Time for compound interest T = 3 years. Concept / Approach: First, we find the annual rate R by using the simple interest relation. Once R is known, we apply the compound interest formula A = P * (1 + R / 100)^T for P = 25000 and T = 3 years. The compound interest is then CI = A - P. Careful calculation is needed to keep decimal places correct. Step-by-Step Solution: From simple interest information: in 3 years increase is 18% of principal. So 3R = 18, which gives R = 18 / 3 = 6% per annum. Now we use R = 6% to compute compound interest on Rs. 25,000 for 3 years. Amount A = 25000 * (1 + 6 / 100)^3 = 25000 * (1.06)^3. Compute (1.06)^2 = 1.1236. Then (1.06)^3 = 1.1236 * 1.06 = 1.191016. So A = 25000 * 1.191016 = Rs. 29775.40. Compound interest CI = A - P = 29775.40 - 25000 = Rs. 4775.40. Verification / Alternative check: We can also compute year by year: After 1 year, amount = 25000 * 1.06 = 26500. After 2 years, amount = 26500 * 1.06 = 28090. After 3 years, amount = 28090 * 1.06 = 29775.40. Subtracting the original 25000 yields 4775.40, confirming the compound interest. Why Other Options Are Wrong: 4557.40 and 5000.00 assume incorrect rate values or mix simple and compound interest ideas. 5575.40 and 5774.40 correspond to higher effective rates than 6% or to extra years. Only Rs. 4775.40 matches the compound interest obtained from a 6% rate over 3 years on Rs. 25,000. Common Pitfalls: A common mistake is to treat the 18% as a yearly rate instead of a 3-year cumulative rate, which would incorrectly set R = 18%. Another error is to stop after computing simple interest instead of compound interest, or to miscalculate (1.06)^3. Careful handling of exponents and interpretation of the given statement avoids these issues. Final Answer: The compound interest on Rs. 25,000 in 3 years at this rate is Rs. 4775.40.

Question 18

Dalajit lends Rs. 10,800 to Jaabir for 3 years and Rs. 7,500 to Kabir for 2 years at the same rate of simple interest, and receives a total of Rs. 1422 as interest from both loans together. What is the common rate of interest per annum (in percent)?

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Introduction / Context: In this question, one lender, Dalajit, gives two loans to two different people for different time periods but at the same unknown simple interest rate. The total interest he receives from both loans is known. We must use this combined information to find the common annual rate of simple interest. Given Data / Assumptions: • Loan 1: Principal P1 = Rs. 10,800 lent to Jaabir for T1 = 3 years. • Loan 2: Principal P2 = Rs. 7,500 lent to Kabir for T2 = 2 years. • Common rate R% per annum for both loans. • Total simple interest from both loans: SI_total = Rs. 1422. • Simple interest formula: SI = (P * R * T) / 100. Concept / Approach: We write separate simple interest expressions for each loan using the same rate R, then add them and set the sum equal to the given total interest of Rs. 1422. This gives a single equation in one unknown R, which we can solve straightforwardly to obtain the rate per annum. Step-by-Step Solution: Interest from first loan: SI1 = (P1 * R * T1) / 100 = (10800 * R * 3) / 100. Simplify SI1 = 10800 * 3 * R / 100 = 32400R / 100. Interest from second loan: SI2 = (P2 * R * T2) / 100 = (7500 * R * 2) / 100. Simplify SI2 = 7500 * 2 * R / 100 = 15000R / 100. Total interest: SI_total = SI1 + SI2 = (32400R + 15000R) / 100 = 47400R / 100. We are given SI_total = 1422, so 47400R / 100 = 1422. Thus, 474R = 1422, giving R = 1422 / 474 = 3% per annum. Verification / Alternative check: Compute SI1 at 3%: SI1 = 10800 * 3 * 3 / 100 = 10800 * 0.09 = 972. Compute SI2 at 3%: SI2 = 7500 * 3 * 2 / 100 = 7500 * 0.06 = 450. Total = 972 + 450 = 1422, which matches the given total interest, confirming that R = 3% per annum is correct. Why Other Options Are Wrong: Rates of 3.5 percent, 4 percent, 4.5 percent, or 2.5 percent lead to interest totals that differ from Rs. 1422 when substituted into the formula. For example, a higher rate like 4 percent would produce a larger total interest, while 2.5 percent would produce less. Common Pitfalls: A common error is to try to find separate rates for each loan, even though the question clearly states the rate is the same. Another pitfall is arithmetic mistakes when combining the two simple interest expressions or when solving for R. Keeping all steps aligned with SI = (P * R * T) / 100 ensures accuracy. Final Answer: The common rate of interest per annum is 3 percent.

Question 19

The nominal rate of interest is 15% per annum, payable half-yearly (that is, interest is compounded twice a year). What is the corresponding effective annual rate of interest, expressed as a percentage per year?

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Introduction / Context: This problem asks us to convert a nominal annual interest rate with semi-annual compounding into an effective annual rate. The nominal rate is 15% per annum, but because interest is actually applied twice a year, the true annual growth is slightly higher than 15%. The task is to compute that effective annual rate. Given Data / Assumptions: • Nominal annual rate R_nominal = 15% per annum. • Compounding is half-yearly, so there are 2 compounding periods per year. • Each half-yearly rate r_half = 15 / 2 = 7.5%. • We seek the effective annual rate R_effective that produces the same growth over one full year. Concept / Approach: If the interest is compounded twice a year at 7.5% each half-year, then after one year, the amount multiplies by (1 + 7.5 / 100)^2. The effective annual rate is the net percentage increase of this multiplier compared to the original principal. Mathematically, R_effective = [(1 + r_half / 100)^2 - 1] * 100%. Step-by-Step Solution: Compute half-yearly rate: r_half = 15 / 2 = 7.5%. Growth factor per half-year = 1 + 7.5 / 100 = 1.075. Effective annual growth factor = (1.075)^2. Calculate (1.075)^2 = 1.075 * 1.075 = 1.155625. Effective annual rate R_effective = (1.155625 - 1) * 100%. R_effective = 0.155625 * 100% = 15.5625%. Rounded suitably, this is approximately 15.56 percent per annum. Verification / Alternative check: Consider a principal of Rs. 100. At 15% simple interest, after one year it becomes Rs. 115. At 7.5% compounded half-yearly, after the first half-year it becomes 100 * 1.075 = 107.50. After the second half-year it grows to 107.50 * 1.075 = 115.5625. The net increase is 15.5625%, which matches the earlier calculation of about 15.56%. Why Other Options Are Wrong: 15 percent ignores the extra growth due to compounding. 15.75 percent and 31.13 percent significantly overstate the effect of semi-annual compounding. 30 percent would correspond to simply doubling the nominal rate, which is incorrect for this context. Only 15.56 percent correctly reflects the true annual growth factor from two periods at 7.5% each. Common Pitfalls: Students often either forget to divide the nominal rate by the number of compounding periods or fail to square the growth factor for two periods. Others may incorrectly add 15% and half of 15% instead of compounding. Always remember that compounding uses multiplication of growth factors, not simple addition of percentages. Final Answer: The effective annual rate of interest is approximately 15.56 percent per annum.

Question 20

A certain sum becomes Rs. 2286 in 3 years and Rs. 2448 in 4 years at simple interest. Using these two consecutive amounts under simple interest, what is the annual rate of interest in percentage?

Accepted Answer

Introduction / Context: In this simple interest question, we know the amount after 3 years and after 4 years. Because simple interest adds the same interest every year, the difference between the amounts for years 3 and 4 tells us the interest for one year. We can then use that yearly interest to find the principal and compute the annual interest rate as a percentage. Given Data / Assumptions: • Amount after 3 years A3 = Rs. 2286. • Amount after 4 years A4 = Rs. 2448. • Principal P remains constant. • Simple interest applies, so yearly interest is constant. • Simple interest formula: A = P + (P * R * T) / 100. Concept / Approach: The difference A4 - A3 represents the interest for one extra year. Once we know the yearly interest, we compute the total 3-year interest and subtract it from A3 to get the principal P. Finally, we use SI for one year and the simple interest formula to solve for the annual rate R. Step-by-Step Solution: Difference between A4 and A3: 2448 - 2286 = Rs. 162. This Rs. 162 is the simple interest for 1 year. Thus, yearly interest I_year = Rs. 162. Total interest for 3 years = 3 * 162 = Rs. 486. Principal P = A3 - 3-year interest = 2286 - 486 = Rs. 1800. Now, for 1 year: SI = 162 = (P * R * 1) / 100 = 1800 * R / 100. So 162 = 18R, giving R = 162 / 18 = 9% per annum. Verification / Alternative check: Check with P = 1800 and R = 9%. Amount after 3 years: A3 = 1800 * (1 + 9 * 3 / 100) = 1800 * (1 + 0.27) = 1800 * 1.27 = 2286, matching the given A3. Amount after 4 years: A4 = 1800 * (1 + 9 * 4 / 100) = 1800 * 1.36 = 2448, which matches A4. This confirms that R = 9%. Why Other Options Are Wrong: Rates of 8%, 10%, 11%, or 12% produce yearly interest values different from Rs. 162 on the principal that fits A3 and A4. Substituting those rates into the formula would lead to inconsistent amounts that do not match both given values for 3 and 4 years simultaneously. Common Pitfalls: Some students mistakenly divide the total difference between 4-year and 3-year amounts by the entire 4 years, rather than recognizing that it represents just one year of interest. Others try to find R directly without isolating the principal. Using the stepwise method of finding annual interest, then principal, then rate, helps avoid errors. Final Answer: The annual rate of interest is 9% per annum.

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