Question 1

In what time will Rs. 1,860 amount to Rs. 2,641.20 at 12% per annum simple interest?

Accepted Answer

Introduction / Context: This question focuses on finding the time taken for a sum of money to grow to a given amount at a specified simple interest rate. Instead of directly giving the interest, the exam provides the final amount, so the first step is to find the interest and then use the simple interest formula to solve for time. Given Data / Assumptions: Principal (P) = Rs. 1,860 Final amount (A) = Rs. 2,641.20 Rate (R) = 12% per annum Simple interest is applied Concept / Approach: First compute simple interest as the difference between amount and principal: SI = A - PThen apply the formula for simple interest and rearrange to find time: SI = (P * R * T) / 100T = (SI * 100) / (P * R)Careful arithmetic with decimals is important to reach the exact time value. Step-by-Step Solution: SI = A - P = 2641.20 - 1860 = Rs. 781.20Now use SI formula: 781.20 = (1860 * 12 * T) / 100Compute denominator: (1860 * 12) / 100 = 1860 * 0.12 = 223.20So 781.20 = 223.20 * TT = 781.20 / 223.20T = 3.5 years Verification / Alternative check: Check by forward calculation. Interest per year at 12% is: Yearly interest = (1860 * 12) / 100 = Rs. 223.20In 3.5 years, total interest = 223.20 * 3.5 = 781.20Final amount = principal + interest = 1860 + 781.20 = 2641.20This matches the given amount, confirming that 3.5 years is correct. Why Other Options Are Wrong: For 2.9 years, interest would be significantly less than 781.20, giving a lower final amount. For 4.2 or 4.7 years, the interest would exceed 781.20 and the final amount would be higher than Rs. 2,641.20. Only 3.5 years gives an exact match. Common Pitfalls: Errors often arise from incorrectly subtracting the principal from the amount or mishandling the decimal arithmetic when dividing 781.20 by 223.20. Also, some students forget to divide by 100 in the simple interest formula. Writing each step clearly helps avoid these issues. Final Answer: The required time is 3.5 years.

Question 2

A man borrows a total of Rs. 24,000 from two moneylenders. On one part of the loan he pays 15% per annum simple interest and on the remaining part he pays 18% per annum simple interest. At the end of 1 year he pays Rs. 4,050 as interest in all. How …

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Introduction / Context: This is a typical mixture of rates problem in simple interest. A total principal is split into two parts, each invested at a different rate. We know the total interest and must find how much was invested at one of the rates. It is a direct application of linear equations with simple interest. Given Data / Assumptions: Total principal = Rs. 24,000 Part of the amount is at 15% per annum Remaining amount is at 18% per annum Time = 1 year Total simple interest paid = Rs. 4,050 We must find the amount at 18% per annum Concept / Approach: Let the amount borrowed at 15% be x rupees. Then the amount at 18% is 24000 - x. We compute the interest for each part using the simple interest formula for 1 year and equate the sum to 4050. Solving the resulting linear equation will give x and then the amount at 18%. Step-by-Step Solution: Let amount at 15% = xThen amount at 18% = 24000 - xInterest on x at 15% for 1 year = (x * 15 * 1) / 100 = 0.15xInterest on (24000 - x) at 18% for 1 year = (24000 - x) * 18 / 100 = 0.18(24000 - x)Total interest = 0.15x + 0.18(24000 - x)Given total interest = 4050So 0.15x + 0.18(24000 - x) = 40500.15x + 4320 - 0.18x = 40504320 - 0.03x = 40500.03x = 4320 - 4050 = 270x = 270 / 0.03 = 9000Amount at 15% = Rs. 9,000Amount at 18% = 24000 - 9000 = Rs. 15,000 Verification / Alternative check: Compute interest using the found values. Interest on Rs. 9,000 at 15% is 9000 * 15 / 100 = Rs. 1,350. Interest on Rs. 15,000 at 18% is 15000 * 18 / 100 = Rs. 2,700. Total interest = 1,350 + 2,700 = Rs. 4,050, which matches the given total interest. This confirms that the amounts are correct. Why Other Options Are Wrong: If the amount at 18% were 16,000, interest at 18% on that part plus interest on 8,000 at 15% would not sum to 4,050. Similar checks show that 12,000 or 13,000 at 18% also do not satisfy the total interest condition. Only Rs. 15,000 gives the correct total interest. Common Pitfalls: Students may mistakenly assume that the 4,050 is divided equally between the two rates or that the principal is split in the same ratio as the rates. The correct method is always to form and solve the linear equation based on the simple interest formula. Final Answer: The man borrowed Rs. 15,000 at 18% per annum.

Question 3

The simple interest on Rs. 25 for 4 months at the rate of 3 paise per rupee per month is equal to how many rupees?

Accepted Answer

Introduction / Context: This question uses an interest rate expressed in paise per rupee per month rather than a standard annual percentage. Many bank style problems adopt such wording, and it is important to convert the given rate into a usable percentage or directly work with it in fractional form. The time period is in months, and the principal is small, so the result will also be a small rupee amount. Given Data / Assumptions: Principal (P) = Rs. 25 Rate = 3 paise per rupee per month Time = 4 months Concept / Approach: Three paise per rupee per month means that for every rupee of principal, the interest per month is 0.03 rupee. In percentage terms, 3 paise per rupee is 3%, since 1 rupee has 100 paise. Because the period is 4 months, and the problem is clearly monthly, we can work directly at the monthly rate. The simple interest formula in this context is: SI = P * Monthly rate * Number of months Step-by-Step Solution: Monthly rate = 3 paise per rupee = 3% per monthTime = 4 monthsSimple interest = P * rate * time / 100SI = (25 * 3 * 4) / 100SI = (25 * 12) / 100 = 300 / 100SI = Rs. 3 Verification / Alternative check: Another way is to think in rupees directly. For each rupee, interest per month is 0.03 rupee. For 25 rupees, interest per month is 25 * 0.03 = 0.75 rupee. For 4 months, total interest = 0.75 * 4 = 3 rupees. This matches the earlier calculation and thus confirms the answer. Why Other Options Are Wrong: 240 paise is Rs. 2.40, which would be the interest if the rate or time were lower. Rs. 1.40 and Rs. 2.20 are even smaller and do not correspond to the given rate and time. Only Rs. 3 is consistent with both methods of calculation. Common Pitfalls: Candidates may mistakenly treat 3 paise per rupee per month as 0.03% instead of 3%. Others may forget that the rate is monthly and try to convert it to an annual rate incorrectly. Working directly in terms of rupees per month and then multiplying by the number of months is usually the safest method here. Final Answer: The simple interest is Rs. 3.

Question 4

Manju borrows Rs. 5,000 for 2 years at 4% per annum simple interest from Sudha. She immediately lends the same amount to another person at 6.25% per annum simple interest for 2 years. What is her gain in this transaction per year?

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Introduction / Context: This problem is similar in structure to earlier simple interest gain questions. Manju acts as an intermediary, borrowing money at a lower rate and lending it out at a higher rate. The difference in interests over the same principal and time period becomes her profit. We are required to find her profit per year. Given Data / Assumptions: Principal borrowed and lent = Rs. 5,000 Time = 2 years Rate paid by Manju = 4% per annum simple interest Rate charged by Manju = 6.25% per annum simple interest We must find Manju's gain per year Concept / Approach: The formula for simple interest is: SI = (P * R * T) / 100We calculate the interest Manju pays to Sudha at 4%, the interest she receives at 6.25%, and then take the difference, which is her total gain over 2 years. Dividing this by 2 gives the gain per year. Step-by-Step Solution: Interest paid to Sudha at 4% for 2 years:SI1 = (5000 * 4 * 2) / 100 = 5000 * 0.08 = Rs. 400Interest received from the other person at 6.25% for 2 years:SI2 = (5000 * 6.25 * 2) / 100 = 5000 * 0.125 = Rs. 625Total gain over 2 years = SI2 - SI1 = 625 - 400 = Rs. 225Gain per year = 225 / 2 = Rs. 112.50 Verification / Alternative check: We can interpret the profit as arising from the net extra rate of interest. Net rate advantage = 6.25% - 4% = 2.25% per annum. Gain per year is therefore: (5000 * 2.25) / 100 = 5000 * 0.0225 = Rs. 112.50This matches our earlier calculation, confirming consistency. Why Other Options Are Wrong: Values like Rs. 101, Rs. 98.5 or Rs. 124.3 do not correspond to the correct difference between the two simple interest amounts. They either understate or overstate the net gain that would arise from the given rate difference on Rs. 5,000 over 2 years. Common Pitfalls: Students may forget to divide the total gain for 2 years by 2, thereby giving the gain over the whole period instead of per year. Another mistake is to average the two rates instead of subtracting them to find the net advantage. Always remember that profit here is driven by the difference in interest rates applied to the same principal and time. Final Answer: Manju's gain in the transaction per year is Rs. 112.50.

Question 5

A person lends Rs. 1,540 for 5 years and Rs. 1,800 for 4 years, both at the same rate of simple interest. If the total interest received from the two loans together is Rs. 1,788, what is the annual rate of interest (in percent per annum)?

Accepted Answer

Introduction / Context: This question combines two simple interest loans with the same rate but different principals and time periods. The total interest from both loans is known, and we are asked to find the common annual rate of interest. It tests the ability to form and solve a single equation involving multiple simple interest terms. Given Data / Assumptions: First loan: Principal P1 = Rs. 1,540, time T1 = 5 years Second loan: Principal P2 = Rs. 1,800, time T2 = 4 years Rate of interest for both loans = R% per annum Total simple interest from both loans = Rs. 1,788 Concept / Approach: Simple interest for each loan is given by: SI = (P * R * T) / 100We compute the simple interest expressions for both loans in terms of R, sum them, and set the result equal to 1,788. Solving this equation gives us the rate R. Step-by-Step Solution: Interest from first loan: SI1 = (1540 * R * 5) / 100Interest from second loan: SI2 = (1800 * R * 4) / 100Total interest = SI1 + SI2 = 1788SI1 = (1540 * 5 * R) / 100 = (7700R) / 100SI2 = (1800 * 4 * R) / 100 = (7200R) / 100Total interest = (7700R + 7200R) / 100 = (14900R) / 100So (14900R) / 100 = 1788149R = 1788R = 1788 / 149R = 12Therefore, the annual rate of simple interest is 12% per annum Verification / Alternative check: Check total interest using R = 12%. For the first loan: SI1 = (1540 * 12 * 5) / 100 = 1540 * 0.6 = Rs. 924. For the second loan: SI2 = (1800 * 12 * 4) / 100 = 1800 * 0.48 = Rs. 864. Total interest = 924 + 864 = Rs. 1,788, which matches the given total. This confirms that 12% is correct. Why Other Options Are Wrong: At 14.5%, 11% or 10.5%, the calculated total interest would not be Rs. 1,788. Each would give a different sum either higher or lower than 1,788. Only 12% yields exactly the required amount. Common Pitfalls: Students may forget to divide by 100 in the simple interest formula or incorrectly add the two interest expressions. Another mistake is to substitute numbers too early and lose track of the common factor R. Keeping R symbolic until the final step makes the algebra straightforward and avoids arithmetic errors. Final Answer: The annual rate of simple interest is 12% per annum.

Question 6

Two equal sums of money are lent at simple interest: one sum is lent at 11% per annum for 5 years and the other equal sum is lent at 8% per annum for 6 years. If the difference between the simple interest earned on the two sums is Rs 1008, what is t…

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Introduction / Context: This question checks your understanding of simple interest when two equal principals are invested at different rates and for different time periods. Instead of directly giving the principal, the problem gives the difference between the interests earned on the two sums. By forming and solving a simple algebraic equation using the standard simple interest formula, we can determine the value of each original sum of money. Given Data / Assumptions: Two equal sums are lent on simple interest. First sum: rate = 11% per annum, time = 5 years. Second sum: rate = 8% per annum, time = 6 years. Difference between the simple interests on the two sums = Rs 1008. Simple interest formula: SI = (P * R * T) / 100, where P is principal, R is rate, T is time in years. Concept / Approach: Let each original sum be P. We compute the simple interest on each sum using SI = (P * R * T) / 100. The problem states that the difference between these two interests is Rs 1008, which leads to a linear equation in P. Solving that equation gives the principal. Finally, we compare the numerical value of P with the options to identify the correct answer. Step-by-Step Solution: Step 1: Let each original sum = P rupees. Step 2: Simple interest on the first sum at 11% for 5 years: SI1 = (P * 11 * 5) / 100 = (55P) / 100. Step 3: Simple interest on the second sum at 8% for 6 years: SI2 = (P * 8 * 6) / 100 = (48P) / 100. Step 4: The difference between the two interests is given as SI1 - SI2 = 1008. Step 5: So, (55P / 100) - (48P / 100) = 1008. Step 6: This simplifies to (7P / 100) = 1008. Step 7: Therefore, P = 1008 * 100 / 7 = 100800 / 7 = 14400. Step 8: Hence, each original sum lent is Rs 14,400. Verification / Alternative check: Quickly verify by computing both interests with P = 14,400. First sum: SI1 = 14400 * 11 * 5 / 100 = 14400 * 55 / 100 = 7920. Second sum: SI2 = 14400 * 8 * 6 / 100 = 14400 * 48 / 100 = 6912. The difference = 7920 - 6912 = 1008, which matches the given difference. This confirms that P = 14,400 is correct. Why Other Options Are Wrong: For Rs 15,600, the interest difference is greater than 1008. For Rs 14,850, the difference is not exactly 1008. For Rs 15,220 and Rs 16,000, the computed differences also do not match 1008. Only Rs 14,400 satisfies the given condition precisely. Common Pitfalls: A frequent mistake is to add the rates or times incorrectly or to forget that the sums are equal and not separate unknowns. Another common error is to subtract interests in the wrong order, which can lead to a negative or incorrect value of P. Always carefully set up the equation SI1 - SI2 = given difference and then solve. Final Answer: The correct value of each original sum lent is Rs 14,400.

Question 7

A certain principal earns a simple interest of Rs 2260 in 3 years at some unknown rate of interest. If the rate of interest per annum had been 1% higher than the actual rate, by how much additional simple interest would the amount have increased ove…

Accepted Answer

Introduction / Context: This question tests your understanding of the relationship between principal, rate, time, and interest in simple interest problems. It also focuses on whether the given data are sufficient to compute a unique numerical answer. Many aptitude questions are designed to see if you correctly identify that there are too many unknowns compared to the number of equations provided. Given Data / Assumptions: A principal amount P earns simple interest of Rs 2260 in 3 years. The rate of interest per annum is r%, which is not directly given. Simple interest formula: SI = (P * r * T) / 100. Time T in this problem is 3 years. We are asked to consider what happens if the rate becomes (r + 1)% per annum for the same 3 years and to find the additional interest or decide if it cannot be uniquely determined. Concept / Approach: From the given information, we can form one equation involving P and r using the simple interest formula. However, there are two unknowns and only one equation, so P and r individually cannot be obtained. To find the extra interest when the rate increases by 1%, we need the value of P. Since P depends on both r and the known interest, and r is unknown, there will be infinitely many possible values for P that satisfy the equation. Therefore the additional interest cannot be uniquely fixed. Step-by-Step Solution: Step 1: Use SI = (P * r * T) / 100 with SI = 2260 and T = 3. Step 2: 2260 = (P * r * 3) / 100. Step 3: Rearranging gives P * r = (2260 * 100) / 3. Step 4: Thus, P * r is known numerically, but P alone and r alone are not known. Step 5: If the rate were (r + 1)%, the new simple interest in 3 years would be SI_new = (P * (r + 1) * 3) / 100. Step 6: The additional interest would be SI_new - SI_original = (P * 3 / 100) * (r + 1 - r) = (3P) / 100. Step 7: To compute (3P) / 100 exactly, we need P. However, we only know P * r, not P. Step 8: Different combinations of P and r can give the same product P * r, so multiple answers for the extra interest are possible. Verification / Alternative check: Assume some trial value of r that satisfies the original equation and compute P from P = (2260 * 100) / (3 * r). For a different value of r that also satisfies the equation, a different P will result, giving a different value of (3P) / 100. This shows there is no single unique amount for the additional interest, confirming that the data are insufficient. Why Other Options Are Wrong: Any numerical option such as Rs 175, Rs 220.75, Rs 126, or Rs 150 assumes a specific fixed principal, which is not determined by the information given. Since P is not uniquely determined, none of these fixed values can be guaranteed to be correct in every consistent scenario. Hence, all numerical options are unjustified. Common Pitfalls: A common mistake is to try to assume a convenient rate and principal without realizing that many different combinations satisfy the original simple interest equation. Another error is to treat P * r as if it directly gives P or r. Always check the number of independent equations versus the number of unknowns before concluding you have a unique answer. Final Answer: The additional interest cannot be uniquely determined, so the correct choice is Cannot be determined with the given data.

Question 8

The simple interest on a certain sum of money is Rs 600 after 10 years. If, after the first 5 years, the principal is trebled and the new larger principal is continued at the same rate of simple interest for the remaining 5 years, what will be the t…

Accepted Answer

Introduction / Context: This question mixes a basic simple interest calculation with a change in principal partway through the total time period. First, you are told how much interest a certain sum earns in 10 years, which allows you to relate principal and rate. Then the question asks what happens if, after 5 years, the principal is trebled and the same rate is used for the remaining period. This tests your ability to use proportional reasoning and handle changes in principal over different intervals. Given Data / Assumptions: A sum of money earns simple interest of Rs 600 in 10 years. Principal is P rupees and rate is r% per annum. Simple interest formula: SI = (P * r * T) / 100. After the first 5 years, the principal becomes 3P and the same rate r continues for the next 5 years. Total time considered is 10 years (5 years + 5 years). Concept / Approach: From the original information, we know the interest on principal P for 10 years is Rs 600. That gives us a relation between P and r. Using proportionality of simple interest with respect to principal and time, we can then compute interest for P over 5 years and for 3P over 5 years, without explicitly finding r. Adding these two parts gives the total interest when the principal is trebled after 5 years. Step-by-Step Solution: Step 1: From the original condition, SI_original = 600 for 10 years on principal P at rate r. Step 2: Using SI = (P * r * T) / 100, 600 = (P * r * 10) / 100. Step 3: Rearranging, P * r = 600 * 100 / 10 = 6000. Step 4: Interest on P for the first 5 years: SI_first = (P * r * 5) / 100. Step 5: Since P * r = 6000, SI_first = (6000 * 5) / 100 = 30000 / 100 = 300. Step 6: After 5 years, principal becomes 3P, so interest for the next 5 years: SI_second = (3P * r * 5) / 100. Step 7: 3P * r = 3 * (P * r) = 3 * 6000 = 18000. Step 8: Thus, SI_second = (18000 * 5) / 100 = 90000 / 100 = 900. Step 9: Total interest with changed principal = SI_first + SI_second = 300 + 900 = 1200. Verification / Alternative check: A quick verification is to compute the average principal over each period and use proportionality. For the first 5 years, interest is 300, and for the next 5 years with triple principal, interest should be triple that of the first segment over equal time, that is 3 * 300 = 900. The sum 300 + 900 reproduces 1200, confirming the result. Why Other Options Are Wrong: Rs 1,100, Rs 1,000, Rs 1,840, and Rs 900 do not correctly reflect the proportional scaling of interest when principal is tripled for half the total time. They either underestimate or overestimate the contribution from the larger principal segment. Only Rs 1,200 matches the correctly computed total interest. Common Pitfalls: Learners sometimes mistakenly triple the entire 10-year interest or forget that the principal changes only after 5 years, not from the beginning. Another common error is to try to find the exact rate r unnecessarily, which can introduce arithmetic mistakes. It is often simpler to work with the product P * r, as shown in this solution. Final Answer: The total simple interest earned at the end of 10 years under the changed principal condition is Rs 1,200.

Question 9

Find the simple interest on Rs 3000 invested at 18% per annum for a period of 73 days, assuming a 365-day year and simple interest throughout, and express the interest in rupees correct to the nearest rupee.

Accepted Answer

Introduction / Context: This question involves calculating simple interest for a fractional part of a year. Instead of time being given in whole years, it is provided in days, and you are told to assume a 365-day year. Such problems are common in banking and finance aptitude tests because they reflect how interest is often computed for short-term investments or loans. Given Data / Assumptions: Principal P = Rs 3000. Rate of interest R = 18% per annum. Time T = 73 days, with a 365-day year assumption. Simple interest formula: SI = (P * R * T) / 100, where T is in years. We must convert 73 days into its equivalent in years using 365 days per year. Concept / Approach: The key idea is to express the time in years before substituting into the simple interest formula. Once T is expressed as 73 / 365 years, we plug P, R, and T into SI = (P * R * T) / 100. After computing, we round the result to the nearest whole rupee, as the options are given in rupees without paise. Step-by-Step Solution: Step 1: Convert days to years. T = 73 days = 73 / 365 years. Step 2: Write the simple interest formula: SI = (P * R * T) / 100. Step 3: Substitute P = 3000, R = 18, T = 73 / 365. Step 4: SI = (3000 * 18 * 73 / 365) / 100. Step 5: Simplify the expression: 73 / 365 = 1 / 5. So SI = (3000 * 18 * 1 / 5) / 100. Step 6: Compute 3000 * 18 = 54000. Step 7: Then 54000 * (1 / 5) = 10800. Step 8: Finally, divide by 100: SI = 10800 / 100 = 108. Step 9: Therefore, the simple interest is Rs 108. Verification / Alternative check: Another way to see this is to first compute the annual interest and then scale by the fraction of the year. Annual interest at 18% on Rs 3000 is 3000 * 18 / 100 = 540. For 73 days out of 365, interest = 540 * 73 / 365. Since 73 / 365 = 1 / 5, we get 540 * 1 / 5 = 108, consistent with the detailed calculation. Why Other Options Are Wrong: Values such as Rs 118, Rs 105, Rs 110, and Rs 112 result from incorrect handling of the fraction of the year, using 360 days instead of 365 days, or rounding at intermediate steps. Only Rs 108 exactly matches the correct computation based on a 365-day year and the given rate. Common Pitfalls: Common mistakes include treating 73 as 73 months, using 12 months or 360 days incorrectly, or forgetting to divide by 100 after multiplying P, R, and T. Another error is rounding too early, rather than after completing the full calculation. Always convert days to years carefully and perform the operations in the correct order. Final Answer: The simple interest on the investment is Rs 108.

Question 10

Arun took a loan of Rs 29,000 at simple interest for as many years as the numerical value of the rate of interest. If he paid Rs 10,440 as interest at the end of the loan period, what was the rate of interest per annum under this simple interest arr…

Accepted Answer

Introduction / Context: This question is a classic aptitude problem where the rate of interest and the time period in years are numerically equal. It uses the simple interest formula to form a quadratic-like relation in the rate and then solves for that rate. Understanding how to translate verbal conditions such as "for as many years as the rate" into algebra is important in many similar questions. Given Data / Assumptions: Principal P = Rs 29,000. Rate of interest per annum = r%. Time T in years is numerically equal to r (so T = r years). Simple interest paid at the end of the period = Rs 10,440. Formula: SI = (P * R * T) / 100. Concept / Approach: Since T = r, the simple interest formula becomes SI = (P * r * r) / 100. That is, SI depends on r squared. We substitute the known values of P and SI into this relation and solve for r. Because r represents a rate of interest, we look for a positive value that matches one of the given options. Step-by-Step Solution: Step 1: Use SI = (P * r * T) / 100 with T = r and P = 29000. Step 2: Then SI = (29000 * r * r) / 100 = (29000 * r^2) / 100. Step 3: Given SI = 10440, set up the equation 10440 = (29000 * r^2) / 100. Step 4: Multiply both sides by 100 to remove the denominator: 10440 * 100 = 29000 * r^2. Step 5: So, 1044000 = 29000 * r^2. Step 6: Divide both sides by 29000: r^2 = 1044000 / 29000 = 36. Step 7: Take the positive square root since rate cannot be negative: r = 6. Step 8: Therefore, the rate of interest per annum is 6%. Verification / Alternative check: Check by computing the interest directly. If r = 6%, then T = 6 years. SI = (29000 * 6 * 6) / 100 = (29000 * 36) / 100 = 1044000 / 100 = 10440. This matches the given simple interest, confirming that the rate must be 6% per annum. Why Other Options Are Wrong: If r = 5.5%, the interest becomes (29000 * 5.5 * 5.5) / 100, which is not 10440. Similarly, substituting 6.5%, 7%, or 4.5% into the formula produces interest amounts different from Rs 10,440. Thus, none of these alternative rates satisfy the equation, so they are incorrect. Common Pitfalls: Students sometimes mistakenly set T equal to r% instead of r years, or they forget to square r when T = r. Another error is failing to multiply the given simple interest by 100 while clearing the denominator. Careful algebra and a quick verification step help avoid such mistakes. Final Answer: The correct rate of interest per annum is 6%.

Question 11

A certain sum of money amounts to Rs 2500 in 5 years and to Rs 3000 in 7 years at the same rate of simple interest. What is the value of the original principal (the sum initially invested) before interest is added?

Accepted Answer

Introduction / Context: This problem uses two different future amounts of the same principal at the same simple interest rate but different time periods. From these two conditions, you can determine the annual simple interest and the original principal. Such questions reinforce the idea that under simple interest, the interest increases linearly with time, so the difference in amounts over different years reveals the yearly interest. Given Data / Assumptions: Principal P is unknown. Amount after 5 years, A1 = Rs 2500. Amount after 7 years, A2 = Rs 3000. Same simple interest rate r% per annum for both periods. Amount formula in simple interest: A = P + SI, and SI = (P * r * T) / 100. Concept / Approach: The difference between the two amounts over 2 extra years (from year 5 to year 7) equals the simple interest earned in those 2 years. From this, we can find the interest per year. Once we know yearly interest, we subtract 5 years of interest from the 5-year amount to recover the principal P. This approach avoids the need to explicitly find the rate r before finding P. Step-by-Step Solution: Step 1: Write amounts in terms of principal and interest. Step 2: A1 = P + SI_5, where SI_5 is interest for 5 years. Step 3: A2 = P + SI_7, where SI_7 is interest for 7 years. Step 4: Difference in amounts A2 - A1 = SI_7 - SI_5 = interest for additional 2 years. Step 5: Compute A2 - A1 = 3000 - 2500 = 500. Step 6: So, interest in 2 years is Rs 500, giving yearly interest = 500 / 2 = Rs 250. Step 7: Interest in 5 years is 5 * 250 = Rs 1250. Step 8: Since A1 = P + 5 years interest, we have 2500 = P + 1250. Step 9: Therefore, P = 2500 - 1250 = 1250. Verification / Alternative check: Check with the 7-year amount. If principal P is 1250 and yearly interest is 250, then interest for 7 years is 7 * 250 = 1750. Amount after 7 years = 1250 + 1750 = 3000, which matches A2. This confirms that the principal is correct. Why Other Options Are Wrong: If P were 1800, 2000, 1400, or 1500, the implied yearly interest would not be consistent with both given amounts at 5 and 7 years. These values either overshoot or undershoot one of the amounts, so they cannot satisfy both conditions simultaneously. Common Pitfalls: A common mistake is to try to form two separate equations with P and r and solve them simultaneously, which is possible but more tedious. Another frequent error is misinterpreting the difference in amounts as the total interest rather than only the additional interest for the extra years. Remember that in simple interest, interest per year is constant, so difference in amounts over extra years equals interest for those extra years only. Final Answer: The original principal invested is Rs 1,250.

Question 12

A sum of money amounts to Rs 9,800 after 5 years and to Rs 12,005 after 8 years at the same rate of simple interest. What is the rate of interest per annum implied by these two amounts?

Accepted Answer

Introduction / Context: Here, two amounts of the same principal at the same simple interest rate but for different times are given. From the change in amount over the extra years, you can find the yearly interest and then compute the rate of interest. This is a standard technique in simple interest questions where principal is unknown but the rate is required. Given Data / Assumptions: Amount after 5 years, A1 = Rs 9,800. Amount after 8 years, A2 = Rs 12,005. Principal P is the same in both cases. Rate of interest per annum is r%. Simple interest: SI = (P * r * T) / 100 and A = P + SI. Concept / Approach: Just as in similar problems, the difference in amounts over the extra 3 years equals the simple interest for those 3 years. From that difference we find yearly interest and then relate it to the principal. The principal can be found from either amount, and then the rate is obtained by using SI for a known number of years. Step-by-Step Solution: Step 1: Let principal be P. Then A1 = P + SI_5 and A2 = P + SI_8. Step 2: The difference A2 - A1 = SI_8 - SI_5 = interest for 3 extra years. Step 3: Compute A2 - A1 = 12005 - 9800 = 2205. Step 4: Thus, interest for 3 years is Rs 2205, so yearly interest = 2205 / 3 = Rs 735. Step 5: Interest for 5 years is 5 * 735 = Rs 3675. Step 6: Since A1 = P + interest for 5 years, we have 9800 = P + 3675. Step 7: So, P = 9800 - 3675 = 6125. Step 8: Now find the rate using yearly interest: SI_1year = (P * r * 1) / 100 = P * r / 100 = 735. Step 9: Therefore, P * r = 735 * 100 = 73500, and r = 73500 / 6125. Step 10: Compute r = 73500 / 6125 = 12. Step 11: Hence, the rate of interest per annum is 12%. Verification / Alternative check: Check with the 8-year amount. Interest for 8 years at 12% on principal 6125 is SI_8 = (6125 * 12 * 8) / 100 = 6125 * 0.96 = 5880. Amount after 8 years is 6125 + 5880 = 12005, which matches A2. This confirms that r = 12% is correct. Why Other Options Are Wrong: Rates like 5%, 8%, 10%, or 15% give yearly interests that, when applied for 5 and 8 years, cannot produce the given pair of amounts 9800 and 12005. Only 12% produces the correct annual interest and hence the correct amounts. Common Pitfalls: A frequent mistake is to compute the rate directly from only one of the amounts without first finding the principal, which often leads to algebraic confusion. Another mistake is to take the difference in amounts and divide by 8 instead of by 3, forgetting that the extra period is 3 years. Always focus on the extra time when taking the difference between two simple interest amounts. Final Answer: The rate of interest per annum is 12%.

Question 13

The total simple interest earned on a certain sum of money increases by Rs 600 when the rate of interest is increased by 2% per annum. If the investment is made for 5 years at simple interest, what is the principal (sum of money) originally invested?

Accepted Answer

Introduction / Context: This question is about how changes in the rate of simple interest affect the total interest over a fixed time. When the rate increases by a certain percentage, the interest increases proportionally to the principal and time. The problem tells you the increase in interest due to an increase in rate, and asks you to find the principal, which is a direct application of the simple interest formula. Given Data / Assumptions: Principal P is unknown. Time T = 5 years. Rate increases by 2% per annum (that is, from r% to (r + 2)% per annum). Increase in total simple interest over 5 years = Rs 600. Simple interest formula: SI = (P * R * T) / 100. Concept / Approach: When the rate increases from r% to (r + 2)%, the change in interest over the same principal P and time T is determined only by the additional 2%. The extra simple interest can therefore be computed using just the increment in rate, the principal, and the time. We treat the increase as simple interest at 2% per annum for 5 years on the same principal, and equate it to Rs 600, then solve for P. Step-by-Step Solution: Step 1: Let original principal be P. Step 2: Additional rate of interest = 2% per annum. Step 3: Time T = 5 years. Step 4: Extra simple interest caused by the extra 2% rate is given as Rs 600. Step 5: Using SI = (P * R * T) / 100 for this extra part, we have 600 = (P * 2 * 5) / 100. Step 6: Simplify the right side: (P * 2 * 5) / 100 = (10P) / 100 = P / 10. Step 7: Therefore, P / 10 = 600. Step 8: Multiply both sides by 10 to get P = 600 * 10 = 6000. Step 9: Hence, the principal invested is Rs 6,000. Verification / Alternative check: As a quick check, consider the extra interest as if principal is 6000, rate increase is 2%, and time is 5 years. Extra SI = (6000 * 2 * 5) / 100 = (6000 * 10) / 100 = 60000 / 100 = 600, which matches the given increase. This confirms that P = 6000 is consistent. Why Other Options Are Wrong: If P were 5550, 7500, 6580, or 7200, the extra interest at 2% for 5 years would be different from Rs 600. For example, with P = 7500, extra SI would be (7500 * 2 * 5) / 100 = 750, not 600. Thus, those options do not satisfy the condition provided. Common Pitfalls: Many learners unnecessarily introduce the original rate r into the calculation, which is not required. Others may forget that the increase in rate applies for the entire period of 5 years, not just for one year. Focus on the fact that extra interest is directly due to the extra 2% rate over the full time on the same principal. Final Answer: The principal originally invested is Rs 6,000.

Question 14

Gopal borrowed a sum of money at 12% simple interest per annum. If he had to pay back Rs 1,280 after 5 years in order to clear the loan, what amount did he originally borrow as principal?

Accepted Answer

Introduction / Context: This is a straightforward simple interest problem where the final amount after a given time period is known, and you must work backward to find the original principal. Such problems are common in banking aptitude tests and help you practice reversing the simple interest formula and understanding the relation between amount, principal, and interest. Given Data / Assumptions: Amount paid after 5 years, A = Rs 1,280. Rate of interest R = 12% per annum (simple interest). Time T = 5 years. Principal P is unknown. Simple interest SI = (P * R * T) / 100, and Amount A = P + SI. Concept / Approach: We first express the amount in terms of the principal using the simple interest formula. Because amount equals principal plus simple interest, we can write A = P * (1 + (R * T) / 100). Substituting the known values of A, R, and T results in a simple equation in one variable P, which we then solve to find the principal Gopal borrowed. Step-by-Step Solution: Step 1: Use the relationship A = P + SI where SI = (P * R * T) / 100. Step 2: Substitute R = 12 and T = 5 into SI: SI = (P * 12 * 5) / 100 = (P * 60) / 100 = 0.6P. Step 3: Therefore, A = P + 0.6P = 1.6P. Step 4: Given A = 1280, so 1.6P = 1280. Step 5: Solve for P by dividing both sides by 1.6: P = 1280 / 1.6. Step 6: 1280 / 1.6 = 800. Step 7: Hence, Gopal originally borrowed Rs 800. Verification / Alternative check: Compute the simple interest on Rs 800 at 12% for 5 years. SI = (800 * 12 * 5) / 100 = (800 * 60) / 100 = 480. Amount A = P + SI = 800 + 480 = 1280, which matches the given repayment amount. This confirms that the principal is correctly calculated as Rs 800. Why Other Options Are Wrong: If the principal were Rs 620, Rs 560, Rs 480, or Rs 700, the amount after 5 years at 12% simple interest would be different from Rs 1280. For example, for P = 700, interest would be (700 * 12 * 5) / 100 = 420 and amount would be 1120, not 1280. Thus, these options do not satisfy the given condition. Common Pitfalls: Some test takers compute only the interest and forget to add it to or subtract it from the amount appropriately when solving backward. Others may miscalculate the factor (1 + R * T / 100), or mistakenly use compound interest formulas. Always verify that the method aligns with simple interest and that the final amount is consistent with the given data. Final Answer: The original principal borrowed by Gopal is Rs 800.

Question 15

A sum of Rs 12,500 amounts to Rs 15,500 in 4 years at the rate of simple interest. What is the rate of interest per annum for this simple interest investment?

Accepted Answer

Introduction / Context: This problem is a direct application of the simple interest formula to find the rate of interest when the principal, final amount, and time are given. It reinforces the idea that interest under simple interest grows linearly with time and that the rate can be computed by comparing the interest earned to the principal over the given period. Given Data / Assumptions: Principal P = Rs 12,500. Amount after 4 years, A = Rs 15,500. Time T = 4 years. Rate of interest per annum is r%, unknown. Simple interest SI = A - P. Formula: SI = (P * r * T) / 100. Concept / Approach: First, we find the total simple interest by subtracting the principal from the final amount. Then we use the simple interest formula to relate SI, P, r, and T, and solve for r. Because the values are neat, the resulting rate will be an exact percentage that matches one of the options. Step-by-Step Solution: Step 1: Compute simple interest: SI = A - P = 15500 - 12500 = 3000. Step 2: Use SI = (P * r * T) / 100. Step 3: Substitute SI = 3000, P = 12500, T = 4. Step 4: 3000 = (12500 * r * 4) / 100. Step 5: Simplify numerator: 12500 * 4 = 50000. Step 6: So, 3000 = (50000 * r) / 100 = 500r. Step 7: Therefore, r = 3000 / 500 = 6. Step 8: Hence, the rate of interest per annum is 6%. Verification / Alternative check: Verify by recomputing the amount using r = 6%. SI = (12500 * 6 * 4) / 100 = (12500 * 24) / 100 = 3000. Then A = P + SI = 12500 + 3000 = 15500, which matches the given amount. This confirms that the rate is correct. Why Other Options Are Wrong: If the rate were 3%, 4%, 5%, or 7%, the interest would differ from 3000. For example, at 5% for 4 years, SI would be (12500 * 5 * 4) / 100 = 2500, and at 7% it would be 3500. These do not match the required 3000, so those options are not correct. Common Pitfalls: Typical errors include confusing amount with interest, or forgetting to subtract the principal before computing the rate. Some also mix up simple and compound interest formulas. Carefully identify SI = A - P first, then apply the simple interest formula to avoid confusion. Final Answer: The rate of simple interest per annum is 6%.

Question 16

The simple interest on a certain deposit at 4.5% per annum is Rs 202.50 in one year. How much additional interest will the same deposit earn in one year if the rate is increased to 5% per annum while keeping the principal and time the same?

Accepted Answer

Introduction / Context: This question focuses on how the simple interest changes when the rate of interest increases while the principal and time remain the same. It is a quick calculation problem that shows that the additional interest depends only on the increase in rate, the principal, and the time period, not on the old rate itself. Given Data / Assumptions: Interest at 4.5% per annum for 1 year is Rs 202.50. Principal P is the same in both scenarios and is unknown initially. New rate of interest = 5% per annum. Time T = 1 year in both cases. Simple interest formula: SI = (P * R * T) / 100. We must find the additional interest in one year when the rate rises from 4.5% to 5% per annum. Concept / Approach: First, we determine the principal from the given interest at 4.5% for one year. Once we know P, the extra interest due to the 0.5% rate increase is simply the interest at an additional 0.5% on that principal for 1 year. We can then compare the result with the options. Step-by-Step Solution: Step 1: Use SI = (P * R * T) / 100 with SI = 202.50, R = 4.5, T = 1. Step 2: 202.50 = (P * 4.5 * 1) / 100. Step 3: Multiply both sides by 100: 20250 = 4.5P. Step 4: Therefore, P = 20250 / 4.5 = 4500. Step 5: Increase in rate = 5% - 4.5% = 0.5%. Step 6: Extra interest for 1 year at 0.5% = (P * 0.5 * 1) / 100. Step 7: Substitute P = 4500: Extra SI = (4500 * 0.5) / 100 = 2250 / 100 = 22.5. Step 8: Hence, the additional interest is Rs 22.50. Verification / Alternative check: Compute interest at 5% directly and subtract the interest at 4.5%. At 5%, SI_new = (4500 * 5 * 1) / 100 = 225. At 4.5%, SI_old = 202.5. Difference = 225 - 202.5 = 22.5, which matches our earlier result. Why Other Options Are Wrong: Rs 2.50 and Rs 20 correspond to much smaller increments than 0.5% on Rs 4500. Rs 25 is slightly larger than the correct difference, and Rs 202.50 represents the full interest at 4.5%, not the extra part. Only Rs 22.50 matches the correct increase in interest. Common Pitfalls: Students sometimes wrongly compute the extra interest as 0.5% of the interest instead of 0.5% of the principal. Another error is to misinterpret the percentage step from 4.5 to 5 as a much larger relative increase. Always apply the changed rate directly to the principal when dealing with simple interest differences. Final Answer: The deposit will earn an additional interest of Rs 22.50 in one year at 5% compared to 4.5%.

Question 17

Ronika deposited Rs 7,000 at simple interest and the amount grew to Rs 9,200 after 3 years. If the rate of interest had been 2% per annum higher than the actual rate, what amount would she have received after the same period of 3 years?

Accepted Answer

Introduction / Context: This question asks you to first determine the actual simple interest rate from the original deposit and amount, and then consider what would happen if the rate were 2% higher. It is an excellent example of how to adjust an existing simple interest scenario when the rate changes, without altering the principal or time. Given Data / Assumptions: Principal P = Rs 7,000. Amount after 3 years, A_original = Rs 9,200. Time T = 3 years. Rate of interest per annum is r% initially, then (r + 2)% in the hypothetical case. Simple interest SI = A - P, and SI = (P * r * T) / 100. Concept / Approach: First, we compute the simple interest earned in 3 years using the original data, then find the rate r. Once we know r, we increase it by 2% to obtain the new rate. Using this new rate, we calculate the new simple interest for 3 years on the same principal and add it to the principal to get the revised amount Ronika would have received. Step-by-Step Solution: Step 1: Compute original simple interest: SI_original = A_original - P = 9200 - 7000 = 2200. Step 2: Use SI = (P * r * T) / 100 with SI_original = 2200, P = 7000, T = 3. Step 3: 2200 = (7000 * r * 3) / 100. Step 4: Multiply both sides by 100: 220000 = 21000r. Step 5: Therefore, r = 220000 / 21000. Step 6: Simplify r = 220000 / 21000 = 220 / 21 ≈ 10.47619%. Step 7: The increased rate is r_new = r + 2 ≈ 12.47619%. Step 8: Instead of working with decimals, note that extra interest due to the extra 2% over 3 years is (P * 2 * 3) / 100. Step 9: Extra SI = (7000 * 2 * 3) / 100 = (7000 * 6) / 100 = 420. Step 10: New total interest = SI_original + extra SI = 2200 + 420 = 2620. Step 11: New amount A_new = P + new total interest = 7000 + 2620 = 9620. Verification / Alternative check: You can also compute the interest directly at the increased rate. At r_new ≈ 12.47619%, SI_new = (7000 * r_new * 3) / 100, which numerically equals 2620 when computed precisely. Adding to principal 7000 gives 9620, consistent with the stepwise method that separates original and extra interest. Why Other Options Are Wrong: Amounts such as Rs 9,500, Rs 9,760, or Rs 9,700 result from miscalculating the rate or the extra interest. Rs 9,850 is too large for an increase of only 2% in the rate over 3 years. Only Rs 9,620 correctly represents principal plus the higher total interest. Common Pitfalls: Many students recompute the entire rate numerically and approximate too early, leading to rounding errors. A cleaner approach is to separate the original interest and the extra interest due to the 2% increase. Always focus on the extra interest as P * extra_rate * time / 100, then add it to the original amount. Final Answer: If the rate had been 2% higher, Ronika would have received Rs 9,620 after 3 years.

Question 18

A man invested a sum of money at 10% per annum on compound interest. After 2 years, he invested an additional amount equal to half of the initial principal at the same rate. If his total interest after 3 years is Rs 457.20, what amount would he have…

Accepted Answer

Introduction / Context: This question combines compound interest and simple interest in two different scenarios. In the first scenario, the investor uses compound interest at 10% with an additional investment after 2 years, and the total interest after 3 years is given. In the second scenario, you must determine what amount he would have after 3 years if he instead invested only the original principal at 15% simple interest. The problem tests the ability to handle compounding over different periods and then compare with an equivalent simple interest investment. Given Data / Assumptions: Initial principal in the compound interest scenario is P rupees. Rate under compound interest scenario = 10% per annum, compounded yearly. After 2 years, an additional amount equal to half of P (that is, P / 2) is invested at the same rate. Total interest earned in the 3-year compound interest scenario = Rs 457.20. In the comparison scenario, only P is invested at 15% simple interest for 3 years. We must find the final amount in the simple interest scenario. Concept / Approach: First, express total interest in the compound interest scenario in terms of P. The original principal P grows for 3 years at 10% compound interest, while the additional P / 2 grows for only 1 year at 10%. Summing the interest from these parts and equating to 457.20 allows us to find P. Once P is known, we compute the amount in the alternative simple interest scenario using A = P + SI with SI = (P * 15 * 3) / 100. Step-by-Step Solution: Step 1: For the original principal P at 10% compound interest, amount after 3 years is A1 = P * (1.1)^3. Step 2: Interest on this part is I1 = P * (1.1^3 - 1) = P * (1.331 - 1) = 0.331P. Step 3: After 2 years, an additional amount equal to P / 2 is invested. This extra amount earns compound interest only for the last year, so its interest is I2 = (P / 2) * 10 / 100 = 0.05P. Step 4: Total interest in the compound interest scenario is I_total = I1 + I2 = 0.331P + 0.05P = 0.381P. Step 5: Given I_total = 457.20, we have 0.381P = 457.20. Step 6: Therefore, P = 457.20 / 0.381 = 1200. Step 7: In the comparison scenario, only P = 1200 is invested at 15% simple interest for 3 years. Step 8: Compute simple interest: SI_simple = (P * 15 * 3) / 100 = (1200 * 45) / 100 = 540. Step 9: Amount in the simple interest scenario A_simple = P + SI_simple = 1200 + 540 = 1740. Verification / Alternative check: Reconfirm the compound interest side. For P = 1200, interest on P for 3 years is 1200 * (1.1^3 - 1) = 1200 * 0.331 = 397.2. Extra principal P / 2 = 600 earns interest for 1 year: 600 * 10 / 100 = 60. Total interest = 397.2 + 60 = 457.2, matching the given 457.20. This confirms P = 1200. The computed simple interest amount 1740 is therefore reliable. Why Other Options Are Wrong: Amounts such as Rs 1,720, Rs 1,760, Rs 1,780, and Rs 1,800 correspond to incorrect calculations of either the compound interest part or the simple interest part. They may arise from using 10% instead of 15%, using 2 years instead of 3 years, or misapplying the formula for compound interest. Only Rs 1,740 is consistent with the correctly derived principal and rate. Common Pitfalls: A common mistake is to assume that the additional amount P / 2 was invested from the beginning, which would cause it to compound for 3 years instead of 1. Another error is to confuse simple interest and compound interest formulas. Always carefully separate each time segment and ensure the correct formula is applied to each principal for the correct duration. Final Answer: Under the simple interest alternative at 15% for 3 years, he would have received Rs 1,740.

Question 19

An automobile financier claims to lend money at simple interest, but he effectively adds the interest to the principal every six months for calculating interest for the next period. If he charges 10% interest per annum in this way, what is his effec…

Accepted Answer

Introduction / Context: This question explores the concept of effective interest rate when interest is added to principal more than once per year. The financier claims to use simple interest, but by adding interest to the principal every six months, he is effectively compounding the interest half yearly. The given nominal annual rate is 10%, and you must find the effective annual rate when compounding happens twice a year. Given Data / Assumptions: Nominal rate of interest per annum = 10%. Interest is added to principal every six months, so compounding is half yearly. Number of compounding periods per year n = 2. Effective annual rate is the percentage increase in the principal after one full year. We use the compound interest factor (1 + i / n)^n to find the effective rate. Concept / Approach: If the nominal annual rate is R% and interest is compounded n times per year, the effective annual factor is (1 + R / (100n))^n. The effective annual rate Reff is then (this factor - 1) * 100%. In this question, R = 10 and n = 2, so each half year the rate is 10 / 2 = 5%. We apply the factor (1 + 0.05)^2 and find the effective rate. Step-by-Step Solution: Step 1: Nominal annual rate R = 10%. Step 2: Number of compounding periods per year n = 2 (every 6 months). Step 3: Rate per half year = R / n = 10 / 2 = 5% = 0.05 in decimal form. Step 4: Annual multiplication factor under half-yearly compounding is (1 + 0.05)^2. Step 5: Compute (1 + 0.05)^2 = 1.05^2 = 1.1025. Step 6: Effective annual rate Reff is given by (factor - 1) * 100% = (1.1025 - 1) * 100%. Step 7: Reff = 0.1025 * 100% = 10.25%. Step 8: Thus, the financier effectively charges an annual rate of 10.25%. Verification / Alternative check: Think in terms of a Rs 100 principal. After the first 6 months at 5%, it becomes 100 * 1.05 = 105. After the next 6 months, interest is again 5%, now on 105, so amount = 105 * 1.05 = 110.25. The increase from 100 to 110.25 is Rs 10.25 on Rs 100, corresponding to 10.25%, confirming the effective rate. Why Other Options Are Wrong: 10% would be correct only if there were no compounding. 10.5% arises if one incorrectly multiplies the nominal rate by 1.05 or uses an approximate method. 9.75% is less than the nominal rate and does not account for compounding. Therefore, none of these match the correct effective rate of 10.25%, so 10.25% is the right answer and option "None of these" is not needed here. Common Pitfalls: Students sometimes confuse nominal and effective rates or forget that compounding more frequently than annually increases the effective rate. Another common error is to multiply 10% by the number of compounding periods, which is incorrect. Always use the compounding factor (1 + R / (100n))^n to find the effective annual rate. Final Answer: The effective annual rate of interest charged by the financier is 10.25%.

Question 20

What will be the ratio of the simple interest earned on a fixed sum of money at the same rate of interest for 6 years and for 9 years under simple interest conditions?

Accepted Answer

Introduction / Context: This question examines your understanding of how simple interest depends on time when the principal and rate are fixed. Under simple interest, the interest earned is directly proportional to the time period. Therefore, the ratio of interests for different time periods is simply the ratio of the corresponding times, assuming principal and rate remain the same. Given Data / Assumptions: Principal P is constant in both cases. Rate of interest per annum r% is the same in both cases. Two time periods: T1 = 6 years and T2 = 9 years. Simple interest formula: SI = (P * r * T) / 100. We must find the ratio SI_6years : SI_9years. Concept / Approach: Since P and r are fixed, simple interest is proportional only to T. Therefore, SI_1 / SI_2 = T1 / T2. By substituting T1 = 6 and T2 = 9, we find the ratio of the interests and simplify it to the lowest terms. There is no need to know the actual values of P or r. Step-by-Step Solution: Step 1: Express simple interest for a time T as SI_T = (P * r * T) / 100. Step 2: For 6 years, SI_6 = (P * r * 6) / 100. Step 3: For 9 years, SI_9 = (P * r * 9) / 100. Step 4: Compute the ratio SI_6 : SI_9. Step 5: SI_6 : SI_9 = (P * r * 6 / 100) : (P * r * 9 / 100). Step 6: Cancel common factors P, r, and 1 / 100. Step 7: The ratio reduces to 6 : 9. Step 8: Simplify 6 : 9 by dividing both sides by 3 to get 2 : 3. Verification / Alternative check: Assume a simple numerical example to verify. Let P = 100 and r = 10% per annum. Then SI_6 = (100 * 10 * 6) / 100 = 60 and SI_9 = (100 * 10 * 9) / 100 = 90. The ratio 60 : 90 simplifies to 2 : 3, confirming the general reasoning. Why Other Options Are Wrong: Ratios like 1 : 3, 1 : 4, or 3 : 4 do not match the direct proportionality between simple interest and time for 6 and 9 years. The option that says "Data is inadequate" is incorrect because we do not need actual values of principal or rate to determine the ratio. Only the time periods matter under simple interest, so the data given are completely sufficient. Common Pitfalls: Some learners mistakenly think that without knowing principal or rate, they cannot solve such problems. Others may confuse simple interest with compound interest, where the relationship between time and interest is not strictly linear. For simple interest, always remember that interest is directly proportional to time when other factors are fixed. Final Answer: The ratio of the simple interest for 6 years and 9 years is 2 : 3.

Simple Interest Questions