Question 1

Simple interest on a certain sum of money for 3 years at 8% per annum is equal to half the compound interest on Rs 16,000 for 2 years at 10% per annum, compounded yearly. What is the sum of money placed on simple interest?

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Introduction / Context: This problem links a simple interest scenario with a compound interest scenario. The simple interest earned on an unknown principal for a certain time and rate is given to be half the compound interest earned on a known principal under different conditions. The goal is to determine the unknown principal in the simple interest case. Such mixed interest comparisons are popular in competitive exams because they test understanding of both types of interest and the ability to combine them in an equation. Given Data / Assumptions: Simple interest is on an unknown principal S at 8 percent per annum for 3 years Compound interest is on Rs 16,000 at 10 percent per annum for 2 years, compounded yearly Simple interest amount equals half of the compound interest amount We need to find the principal S used in the simple interest case Concept / Approach: First, compute the compound interest on Rs 16,000 at 10 percent per annum for 2 years using the formula A = P * (1 + r)^2 and then subtract the principal to get compound interest. Then take half of that compound interest. Next, express the simple interest on S at 8 percent per annum for 3 years as S * 0.08 * 3. Set this equal to half of the compound interest and solve for S. This gives a direct equation that can be solved algebraically. Step-by-Step Solution: Step 1: Compute compound amount on Rs 16,000 for 2 years at 10 percent. P1 = 16,000 and r1 = 10 percent = 0.10. Amount A1 = 16,000 * (1 + 0.10)^2 = 16,000 * (1.10)^2. (1.10)^2 = 1.21, so A1 = 16,000 * 1.21 = 19,360 rupees. Step 2: Compute compound interest on Rs 16,000. C.I. = A1 - P1 = 19,360 - 16,000 = 3,360 rupees. Step 3: Find half of this compound interest. Half of C.I. = 3,360 / 2 = 1,680 rupees. Step 4: Express simple interest on S at 8 percent for 3 years. Rate r2 = 8 percent = 0.08, time T2 = 3 years. Simple interest = S * 0.08 * 3 = S * 0.24. Step 5: Set simple interest equal to half the compound interest. S * 0.24 = 1,680. Step 6: Solve for S. S = 1,680 / 0.24 = 7,000 rupees. Verification / Alternative check: Verify by reapplying the formulas. Simple interest on Rs 7,000 at 8 percent for 3 years is 7,000 * 0.08 * 3 = 7,000 * 0.24 = 1,680 rupees. Compound interest on 16,000 at 10 percent for 2 years is 3,360 rupees, as calculated earlier, and half of that is 1,680 rupees. Because both sides equal 1,680 rupees, the equality required by the question is satisfied, confirming that S = 7,000 rupees is correct. Why Other Options Are Wrong: If S were Rs 14,000, the simple interest at 8 percent for 3 years would be 14,000 * 0.24 = 3,360 rupees, which equals the full compound interest, not half of it. If S were Rs 3,500, the simple interest would be only 840 rupees, which is one quarter of the compound interest. For S = 5,600, the simple interest would be 1,344 rupees, which does not match half of 3,360. Only S = 7,000 rupees produces simple interest equal to half the compound interest as required. Common Pitfalls: A typical mistake is to equate simple interest directly to the full compound interest instead of to half of it, which changes the resulting principal. Others may miscalculate the compound amount or incorrectly square 1.10. It is also easy to forget to convert percentages to decimal form when applying formulas. Careful reading of the phrase half the compound interest and precise arithmetic at each step are crucial. Final Answer: The sum of money placed on simple interest is Rs 7,000.

Question 2

In an aptitude question on simple interest with changing principal and rate, the simple interest on a certain principal for 16 years at a fixed annual rate is Rs. 10000. If instead the principal is doubled after every block of 4 years and, at the sa…

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Introduction / Context: This aptitude question is based on the concept of simple interest with a twist: the principal and the rate of interest change at regular intervals of time. Instead of keeping the principal and rate constant for the whole 16 years, we are told that both are modified after every 4 year block. The question asks us to compare this new scenario with the original one, in which the simple interest earned in 16 years is Rs. 10000, and then compute the new total simple interest over the same overall period of 16 years. Given Data / Assumptions: The simple interest on a certain principal for 16 years at a fixed annual rate is Rs. 10000. Let the principal be P and the annual rate be r percent. Time for the original situation = 16 years. In the new situation, the principal is doubled after every 4 years. At the same moments (every 4 years), the rate of interest is tripled. Interest throughout is simple interest, calculated separately for each 4 year block. Concept / Approach: The basic formula for simple interest is: Simple interest = (Principal * Rate * Time) / 100. From the original information we can find the product P * r, which will be useful. Then, for the modified case, we break the 16 years into four blocks of 4 years each. In every block, principal and rate are different, but time is the same (4 years). We compute the simple interest for each block in terms of P * r, add them up, and finally substitute the numerical value of P * r using the original simple interest information to obtain the new total interest. Step-by-Step Solution: Step 1: For the original situation, simple interest for 16 years is 10000, so using SI = (P * r * 16) / 100, we get (P * r * 16) / 100 = 10000. Step 2: Rearranging, P * r = (10000 * 100) / 16 = 1000000 / 16 = 62500. Step 3: Now consider the new situation. Divide the 16 years into four blocks of 4 years each: 0–4, 4–8, 8–12, and 12–16 years. Step 4: In years 0–4, principal = P and rate = r, so SI₁ = (P * r * 4) / 100 = (4/100) * (P * r). Step 5: After 4 years, principal doubles to 2P and rate triples to 3r. For years 4–8, SI₂ = (2P * 3r * 4) / 100 = (24/100) * P * r = (4/100) * 6 * P * r. Step 6: After 8 years, principal doubles again to 4P and rate triples again to 9r. For years 8–12, SI₃ = (4P * 9r * 4) / 100 = (144/100) * P * r = (4/100) * 36 * P * r. Step 7: After 12 years, principal becomes 8P and rate becomes 27r. For years 12–16, SI₄ = (8P * 27r * 4) / 100 = (864/100) * P * r = (4/100) * 216 * P * r. Step 8: Total new interest = SI₁ + SI₂ + SI₃ + SI₄ = (4/100) * P * r * (1 + 6 + 36 + 216) = (4/100) * P * r * 259. Step 9: Substitute P * r = 62500 to get total interest = (4/100) * 62500 * 259. Step 10: First compute (4/100) * 62500 = 2500. Then total interest = 2500 * 259 = 647500. Step 11: Therefore, the new simple interest over the entire 16 year period is Rs. 647500. Verification / Alternative check: We can quickly verify by comparing the factor of increase. Originally, over 16 years we had interest equal to (16/100) * P * r = 0.16 * P * r. In the new case, we effectively get (4/100) * P * r multiplied by 259, that is 0.04 * P * r * 259 = 10.36 * P * r. The ratio of new interest to original interest is 10.36 / 0.16 = 64.75, and 64.75 * 10000 = 647500, which matches the computed answer and confirms the calculation is consistent. Why Other Options Are Wrong: Rs. 632100 underestimates the effect of repeated doubling of principal and tripling of rate because it corresponds to using a smaller combined multiplier than 259. Rs. 689620 and Rs. 602260 come from incorrect intermediate sums or misapplied multipliers for each 4 year block. Rs. 615000 assumes a wrong factor for P * r or miscalculates one of the blocks. Only Rs. 647500 uses the correct sum of multipliers (1 + 6 + 36 + 216 = 259) and the correct value of P * r = 62500. Common Pitfalls: A common mistake is to treat this as a compound interest problem, but the question clearly specifies simple interest in each time segment. Another frequent error is to double the principal and triple the rate only once instead of after every 4 year block. Some students also forget to include all four blocks, or they fail to compute the factor sum correctly as 259. Finally, neglecting the initial step of finding P * r from the original simple interest relation can make the calculation unnecessarily complicated. Final Answer: The total simple interest earned when the principal is doubled and the rate is tripled after every 4 years over the full 16 year period is Rs. 647500.

Question 3

For a certain principal, the differences between compound interest and simple interest for 3 years and for 2 years are in the ratio 23 : 7 respectively. Based on this information about the gap between compound interest and simple interest, what is t…

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Introduction / Context: This question tests your understanding of how the difference between compound interest and simple interest behaves over time for the same principal and the same annual rate. Instead of giving the actual principal and interest amounts, the problem provides a ratio between the differences over 3 years and 2 years. Using algebra and standard compound interest relationships, we can deduce the rate of interest per annum in percent without ever knowing the principal explicitly. Given Data / Assumptions: Principal amount is some fixed sum P (not numerically specified). Annual rate of interest is r percent per annum. Time periods considered are 2 years and 3 years. The difference (Compound Interest − Simple Interest) for 3 years and for 2 years are in the ratio 23 : 7. Interest is compounded annually when calculating compound interest. Concept / Approach: For a principal P and annual rate r percent, we use x = 1 + r/100 for convenience. The formula for simple interest is SI = (P * r * t) / 100. The formula for compound interest is CI = P * (x^t − 1). The difference between CI and SI for t years, denoted Dₜ, can be simplified algebraically. For 2 years, D₂ becomes P * (x − 1)^2. For 3 years, D₃ becomes P * (x − 1)^2 * (x + 2). Because P and (x − 1)^2 are common factors, the ratio D₃ : D₂ simplifies nicely, allowing us to solve directly for x and hence for r. Step-by-Step Solution: Step 1: Let x = 1 + r/100. For t years, SIₜ = (P * r * t) / 100 = P * t * (x − 1). Step 2: For 2 years, CI₂ = P * (x^2 − 1). So D₂ = CI₂ − SI₂ = P * (x^2 − 1 − 2(x − 1)). Step 3: Simplify D₂: x^2 − 1 − 2x + 2 = x^2 − 2x + 1 = (x − 1)^2. So D₂ = P * (x − 1)^2. Step 4: For 3 years, CI₃ = P * (x^3 − 1). SI₃ = 3P * (x − 1). Then D₃ = P * (x^3 − 1 − 3(x − 1)). Step 5: Simplify inside: x^3 − 1 − 3x + 3 = x^3 − 3x + 2. This factors as (x − 1)^2 * (x + 2). So D₃ = P * (x − 1)^2 * (x + 2). Step 6: The ratio D₃ : D₂ is therefore [P * (x − 1)^2 * (x + 2)] : [P * (x − 1)^2] = x + 2. Step 7: Given that D₃ : D₂ = 23 : 7, we set x + 2 = 23/7. Step 8: Solving, x = 23/7 − 2 = (23 − 14) / 7 = 9/7. Step 9: Recall x = 1 + r/100, so 1 + r/100 = 9/7 implies r/100 = 9/7 − 1 = 2/7. Step 10: Therefore r = (2/7) * 100 = 200/7 percent per annum. Verification / Alternative check: As a quick check, r ≈ 200/7 ≈ 28.57 percent. Then x = 1 + r/100 ≈ 1.2857. Compute x + 2 ≈ 3.2857, which is exactly 23/7. Since the simplified ratio D₃ : D₂ equals x + 2, this confirms that the difference ratio is indeed 23 : 7 when r = 200/7 percent. This back substitution confirms the correctness of our algebraic reasoning. Why Other Options Are Wrong: 100/7 %, 300/7 %, 400/7 %, and 50/7 % are all obtained by guessing different fractional forms without respecting the relationship x + 2 = 23/7. For each of these alternative rates, if we compute x = 1 + r/100 and then x + 2, we do not obtain 23/7, so the given ratio 23 : 7 would not hold. Only 200/7 % satisfies the ratio condition derived from the exact formulas for the differences between compound and simple interest over 2 years and 3 years. Common Pitfalls: Many learners try to plug in numerical values for the principal or use trial and error with the given options directly in interest formulas, which is time consuming. Another pitfall is to attempt to compute actual amounts of compound and simple interest for 2 and 3 years separately and then compare, instead of using the neat algebraic simplifications. Forgetting that the principal P and the factor (x − 1)^2 cancel out in the ratio also leads to unnecessary complexity. Careful factorization and understanding of how the difference grows with time make this question much simpler and faster to solve. Final Answer: The annual rate of interest that makes the ratio of the differences between compound and simple interest for 3 years and 2 years equal to 23 : 7 is 200/7 %.

Question 4

A person lends Rs. 10000 to B for 3 years and Rs. 6000 to C for 4 years, both at the same annual simple interest rate. If the total interest received from B and C together is Rs. 5400 over these periods, what is the common rate of interest per annum…

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Introduction / Context: This is a straightforward simple interest problem that often appears in aptitude exams. A lender gives different amounts to two borrowers for different time periods, but at the same simple interest rate. The total interest from both loans is known, and we need to find the common rate of interest per annum. The question checks whether you can correctly set up and solve a linear equation using the basic simple interest formula. Given Data / Assumptions: Loan to B: Principal P₁ = Rs. 10000 for 3 years. Loan to C: Principal P₂ = Rs. 6000 for 4 years. Rate of simple interest is r percent per annum for both loans. Total interest from B and C together over the entire time is Rs. 5400. Interest is simple interest in both cases, not compound interest. Concept / Approach: For simple interest, the formula is: Simple interest = (Principal * Rate * Time) / 100. We will calculate the interest from each borrower separately in terms of r, then add them to match the given total interest of Rs. 5400. This will give us a single linear equation in r. Solving that equation will directly provide the annual rate of interest in percent. Because both loans are at the same rate, we can combine the terms neatly. Step-by-Step Solution: Step 1: Let the common rate of interest be r percent per annum. Step 2: Interest from B = (P₁ * r * t₁) / 100 = (10000 * r * 3) / 100. Step 3: Simplify B's interest: (10000 * 3 / 100) * r = 300 * r. Step 4: Interest from C = (P₂ * r * t₂) / 100 = (6000 * r * 4) / 100. Step 5: Simplify C's interest: (6000 * 4 / 100) * r = 240 * r. Step 6: Total interest from both loans = 300r + 240r = 540r. Step 7: We are told that this total interest equals Rs. 5400, so 540r = 5400. Step 8: Divide both sides by 540 to get r = 5400 / 540 = 10. Step 9: Therefore, the rate of simple interest is 10 percent per annum. Verification / Alternative check: We can quickly verify the answer by substituting r = 10 percent back into the calculations. Interest from B becomes (10000 * 10 * 3) / 100 = 3000. Interest from C becomes (6000 * 10 * 4) / 100 = 2400. Adding both, 3000 + 2400 = 5400, which exactly matches the total interest given in the problem. Thus, the computed rate is consistent with all given data. Why Other Options Are Wrong: At 12 % or 15 %, the total interest would exceed Rs. 5400, giving values higher than the target amount. At 20 % or 25 %, the interest amounts become even larger and clearly unrealistic compared with the stated total. Only 10 % produces exactly Rs. 5400 when the interest from both loans is combined, so the other options do not satisfy the equation 540r = 5400. Common Pitfalls: One common mistake is to average the time periods or principals incorrectly instead of using the formula on each loan separately. Another error is to forget to divide by 100 when applying the simple interest formula, which leads to very large or very small rates. Some students also miscalculate the combined coefficient (300 + 240) and set up the wrong equation. Being systematic about writing each interest term and then combining them avoids these errors. Final Answer: The common annual simple interest rate that makes the total interest from both loans equal to Rs. 5400 is 10 % per annum.

Question 5

A sum of Rs. 400 amounts to Rs. 448 at simple interest in 2 years. At the same simple interest rate, in how many years will a sum of Rs. 550 amount to Rs. 682 in this aptitude question on simple interest and time?

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Introduction / Context: This question examines your ability to work with simple interest by first determining the rate of interest from one scenario and then using that rate to find the time required in a second scenario. The principal and amount change between the two scenarios, but the rate of interest remains the same. This type of problem is common in banking and aptitude exams where quick manipulation of the simple interest formula is required. Given Data / Assumptions: First situation: Principal P₁ = Rs. 400, Amount A₁ = Rs. 448, Time t₁ = 2 years. Second situation: Principal P₂ = Rs. 550, Amount A₂ = Rs. 682, Time t₂ = ? years. The rate of simple interest r is the same in both situations. Interest is simple interest only, with no compounding. Concept / Approach: Simple interest (SI) is given by: SI = (Principal * Rate * Time) / 100. Amount (A) is related to principal (P) and simple interest (SI) as A = P + SI. We first use the information from the first situation to calculate the rate r. Once we know r, we use it in the second situation formula where principal and final amount are known, and solve for time t₂. This two-stage approach is standard for chained simple interest questions like this. Step-by-Step Solution: Step 1: In the first situation, simple interest SI₁ = A₁ − P₁ = 448 − 400 = Rs. 48. Step 2: Use SI₁ = (P₁ * r * t₁) / 100 to find r: 48 = (400 * r * 2) / 100. Step 3: Simplify: 48 = (800r) / 100 = 8r, so r = 48 / 8 = 6 percent per annum. Step 4: Now, in the second situation, simple interest SI₂ = A₂ − P₂ = 682 − 550 = Rs. 132. Step 5: Use SI₂ = (P₂ * r * t₂) / 100 with r = 6: 132 = (550 * 6 * t₂) / 100. Step 6: Simplify the right-hand side: (550 * 6 / 100) * t₂ = (33) * t₂. Step 7: Therefore, 132 = 33 * t₂, so t₂ = 132 / 33 = 4 years. Step 8: Hence, it will take 4 years for Rs. 550 to amount to Rs. 682 at the same simple interest rate. Verification / Alternative check: To verify, we recompute the amount for P₂ = Rs. 550, r = 6 percent, t₂ = 4 years. Simple interest = (550 * 6 * 4) / 100 = (550 * 24) / 100 = 132. The amount becomes 550 + 132 = 682, which matches the required amount. This confirms that our calculated time of 4 years is correct and consistent with the given data. Why Other Options Are Wrong: If t₂ were 2 or 3 years, the interest would be too small to reach Rs. 682. For example, at 3 years, interest would be (550 * 6 * 3) / 100 = 99, giving an amount of Rs. 649. At 3.5 years, the interest would be 115.5, still short of 132. For 5 years, the interest would be 165, giving an amount of 715, which is too high. Only 4 years produces exactly the required amount of Rs. 682. Common Pitfalls: Students sometimes miscalculate the rate from the first scenario by confusing amount and interest, or by forgetting to subtract principal from amount. Another common mistake is to plug values into the formula without simplifying step by step, which can lead to arithmetic errors. Some also mistakenly assume that the same time is used in both scenarios, instead of properly solving for the new time. Being methodical about computing interest and then using the same rate avoids these issues. Final Answer: At the same simple interest rate, Rs. 550 will amount to Rs. 682 in 4 years.

Question 6

For a certain amount, the simple interest at 12 percent per annum for 6 years is Rs. 25920. On this same principal, what will be the compound interest for 2 years at 8 percent per annum, compounded annually, in this aptitude question on comparing si…

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Introduction / Context: This problem links simple interest and compound interest on the same principal. First, the principal is used with simple interest at a given rate and time to determine its value. Then, the same principal is invested at a different rate under compound interest for a shorter period. The task is to calculate the compound interest earned in the second scenario. Such chained questions are very common in quantitative aptitude tests and bank exams. Given Data / Assumptions: Simple interest rate r₁ = 12 percent per annum. Time for simple interest t₁ = 6 years. Simple interest for this period is Rs. 25920. Same principal is then used for compound interest. Compound interest rate r₂ = 8 percent per annum. Time for compound interest t₂ = 2 years, compounded annually. Concept / Approach: First, we use the simple interest formula to find the principal P: SI = (P * r₁ * t₁) / 100. Once we know P, we apply the compound interest formula for 2 years at 8 percent per annum with annual compounding. The compound amount A is A = P * (1 + r₂/100)^t₂, and the compound interest CI is A − P. We then compare the computed value with the options provided. Step-by-Step Solution: Step 1: Let the principal be P. We are told that simple interest SI = 25920 at 12 percent for 6 years. Step 2: Use SI = (P * r₁ * t₁) / 100, so 25920 = (P * 12 * 6) / 100. Step 3: Simplify the denominator: 12 * 6 = 72, so 25920 = (72P) / 100. Step 4: Rearranging gives P = (25920 * 100) / 72 = 36000. Step 5: Now use compound interest for 2 years at 8 percent with P = 36000. Step 6: The amount after 2 years with annual compounding is A = P * (1 + 8/100)^2 = 36000 * (1.08)^2. Step 7: Compute (1.08)^2 = 1.1664, so A = 36000 * 1.1664 = 41990.4. Step 8: Compound interest CI = A − P = 41990.4 − 36000 = 5990.4. Step 9: Therefore, the compound interest for 2 years at 8 percent per annum is Rs. 5990.4. Verification / Alternative check: We can alternatively compute the compound interest directly using the formula CI = P * [(1 + r₂/100)^t₂ − 1]. With P = 36000, r₂ = 8, t₂ = 2, we get CI = 36000 * [(1.08)^2 − 1] = 36000 * (1.1664 − 1) = 36000 * 0.1664 = 5990.4. This matches the earlier calculation, confirming that our answer is consistent and correct. Why Other Options Are Wrong: Rs. 4326.3 and Rs. 5563.4 are lower than the correct interest and result from miscalculating the squared factor or mistakenly using one year instead of two. Rs. 5888.6 is close, but still incorrect, typically arising from rounding errors or approximating (1.08)^2 inaccurately. Rs. 6200.0 overshoots the correct amount and would correspond to a higher effective rate or misapplied principal. Only Rs. 5990.4 matches the exact computation of compound interest for the given data. Common Pitfalls: Some students incorrectly reuse the simple interest rate or time when computing the compound interest, or they forget to first find the principal from the simple interest information. Another error is to use 2 * 8 = 16 percent directly on the principal as if it were simple interest instead of using the compound factor (1.08)^2. Rounding (1.08)^2 too aggressively can also produce wrong answers close to, but not exactly, 5990.4. Careful step-by-step calculations help avoid these mistakes. Final Answer: The compound interest earned on the same principal for 2 years at 8 percent per annum, compounded annually, is Rs. 5990.4.

Question 7

A certain principal is invested at compound interest. The amount after 1 year is Rs. 2400 and the amount after 2 years is Rs. 2880. Based on these two consecutive yearly amounts, what is the annual rate of interest in percent for this compound inter…

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Introduction / Context: This question is a classic example of finding the rate of compound interest when successive yearly amounts are given. The key idea is that under compound interest with annual compounding, each year the amount is multiplied by the factor (1 + r/100). When we know the amount at the end of one year and at the end of the next year, the ratio of these amounts directly reveals the interest factor and hence the annual rate. This is a quick and important shortcut in competitive exams. Given Data / Assumptions: Amount after 1 year, A₁ = Rs. 2400. Amount after 2 years, A₂ = Rs. 2880. Principal P is the same underlying amount, but we are not directly asked to find it. Interest is compounded annually at rate r percent. We need to find the value of r in percent per annum. Concept / Approach: If interest is compounded annually at rate r percent, then: A₁ = P * (1 + r/100), A₂ = P * (1 + r/100)^2. Therefore, the ratio A₂ / A₁ equals (1 + r/100). This relationship allows us to find the yearly growth factor and hence the rate without first computing the principal. Once we compute A₂ / A₁, we can subtract 1 and convert to a percentage to get the annual rate of interest. Step-by-Step Solution: Step 1: Write the relationship between consecutive amounts under annual compounding: A₂ = A₁ * (1 + r/100). Step 2: Therefore, (1 + r/100) = A₂ / A₁. Step 3: Substitute the given values: (1 + r/100) = 2880 / 2400. Step 4: Simplify the ratio 2880 / 2400 = 1.2. Step 5: Hence, 1 + r/100 = 1.2. Step 6: Subtract 1 from both sides to get r/100 = 1.2 − 1 = 0.2. Step 7: Multiply by 100 to get r = 0.2 * 100 = 20 percent per annum. Step 8: Thus, the annual rate of compound interest is 20 percent. Verification / Alternative check: We can verify by assuming the principal P and checking both amounts. Suppose P is some value; after 1 year, A₁ = P * 1.2 = 2400, so P = 2400 / 1.2 = 2000. After 2 years, A₂ = 2000 * (1.2)^2 = 2000 * 1.44 = 2880, which matches the given amount. This confirms that both the data and calculated rate r = 20 percent are internally consistent. Why Other Options Are Wrong: At 15 % (factor 1.15), the second year amount would be 2400 * 1.15 = 2760, not 2880. At 10 % (factor 1.10), A₂ would be 2400 * 1.10 = 2640, again incorrect. At 12 % (factor 1.12), A₂ would be 2400 * 1.12 = 2688. At 25 % (factor 1.25), the amount would jump to 2400 * 1.25 = 3000. None of these match the given A₂ except the factor 1.2 corresponding to 20 percent per annum. Common Pitfalls: One common mistake is to treat the difference between 2880 and 2400 as simple interest for one year and base the rate on the original principal instead of on the amount. Others wrongly assume that the increase of Rs. 480 over Rs. 2400 is 48 percent rather than correctly computing 480/2400 = 0.2, that is 20 percent. Remembering that under compound interest with yearly compounding, consecutive amounts differ by the factor (1 + r/100) is essential. Final Answer: The annual rate of compound interest implied by the given amounts is 20 % per annum.

Question 8

A certain sum of money doubles itself in 7 years 8 months at simple interest. For this simple interest investment, what is the yearly rate of interest, expressed in percent per annum, that causes the amount to become twice the original principal in …

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Introduction / Context: This question checks your understanding of how to determine the rate of simple interest when the time required for a sum to become a multiple of itself is given. When a sum doubles, triples, or reaches some other multiple under simple interest, there is a very direct relationship between the rate, time, and the multiple. Here, the sum doubles in 7 years 8 months, and we must find the annual rate of simple interest in percent. Given Data / Assumptions: Initial principal is P (some positive amount). Final amount after 7 years 8 months is 2P (the sum doubles). Interest is simple interest at a constant rate r percent per annum. Time t is 7 years 8 months, which we must convert to a single value in years. Concept / Approach: Under simple interest, the amount A after time t years at rate r percent on principal P is given by: A = P * (1 + r * t / 100). Here, the amount becomes 2P, so we set 2P = P * (1 + r * t / 100). Cancelling P, we get a simple equation in terms of r and t. The time must be expressed entirely in years, so we convert 7 years 8 months to a fractional form. Solving for r gives the annual simple interest rate in percent. Step-by-Step Solution: Step 1: Convert 7 years 8 months to years. Eight months is 8/12 years = 2/3 years. Step 2: Therefore, t = 7 + 2/3 = (21/3) + (2/3) = 23/3 years. Step 3: Use the simple interest amount formula: A = P * (1 + r * t / 100). Step 4: Since the sum doubles, A = 2P. So 2P = P * (1 + r * t / 100). Step 5: Cancel P from both sides to get 2 = 1 + (r * t / 100). Step 6: Subtract 1 to obtain r * t / 100 = 1. Step 7: Therefore, r = 100 / t = 100 / (23/3) = 100 * (3/23) = 300/23. Step 8: Express 300/23 as a mixed fraction: 23 * 13 = 299, remainder 1, so r = 13 1/23 percent. Step 9: Hence, the annual simple interest rate is 13 1/23 percent per annum. Verification / Alternative check: We can check by computing the simple interest using r = 300/23 percent over t = 23/3 years. The total interest fraction is (r * t / 100) = (300/23) * (23/3) / 100 = (300/3) / 100 = 100/100 = 1. This means the interest equals the principal P, so the final amount equals P + P = 2P, which agrees with the given condition that the sum doubles. This confirms the correctness of the rate 13 1/23 percent. Why Other Options Are Wrong: 18 3/4 %, 26 2/23 %, 30 %, and 12 1/2 % correspond to different multiples or different times when substituted into the formula A = P * (1 + r * t / 100). None of them give exactly a doubling for t = 23/3 years. For example, 12 1/2 % (which is 12.5 %) over 23/3 years yields a total interest fraction less than 1, so the amount would be less than 2P. Only 13 1/23 % makes the sum exactly double in the specified time. Common Pitfalls: Students often mishandle the conversion of months into years, using 7.8 years instead of 7 years 8 months, or simplifying 8/12 incorrectly. Another common mistake is to forget to divide by 100 when rearranging the formula. Some may also try to use compound interest logic here, but the question explicitly states simple interest. Keeping track of fractions carefully and following the algebraic steps avoids these errors. Final Answer: The yearly rate of simple interest that causes the sum to double in 7 years 8 months is 13 1/23 % per annum.

Question 9

The difference between compound interest and simple interest on a certain sum for 2 years at 20 percent per annum is Rs. 200. If, instead, the interest is compounded half yearly, what is the difference between compound interest and simple interest f…

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Introduction / Context: This problem explores how the difference between compound interest (CI) and simple interest (SI) behaves under different compounding frequencies. First, we are told the difference between CI and SI for 2 years at 20 percent per annum with annual compounding. From this, we can deduce the principal. Then, the question asks us to switch to half yearly compounding and compute the difference between CI and SI for just the first year. Understanding the standard formulas for the CI–SI difference makes this type of question much faster to solve. Given Data / Assumptions: Annual rate r = 20 percent. Time t₁ = 2 years for the first scenario with annual compounding. Difference between CI and SI for 2 years is Rs. 200. Same principal P is used in both scenarios. In the second scenario, interest is compounded half yearly for 1 year. We need the difference between CI and SI for just the first year in the half yearly case. Concept / Approach: For 2 years at rate r percent, the difference between compound and simple interest is: D₂ = CI₂ − SI₂ = P * (r/100)^2. This formula is a standard shortcut. Using D₂ = 200 and r = 20, we can find P. For the second part, with half yearly compounding in 1 year, the rate per half year is r/2 = 10 percent and there are 2 periods. We compute SI for 1 year at 20 percent and CI for 1 year with two half yearly periods at 10 percent, then take the difference CI − SI for that year. Step-by-Step Solution: Step 1: Use the formula D₂ = P * (r/100)^2 for 2 years. Here, D₂ = 200 and r = 20. Step 2: Compute (r/100)^2 = (20/100)^2 = (0.2)^2 = 0.04. Step 3: So 200 = P * 0.04, giving P = 200 / 0.04 = 5000. Step 4: Now consider the second scenario: 1 year with half yearly compounding at a nominal rate of 20 percent per annum. Step 5: Rate per half year = 20/2 = 10 percent. Number of half years in 1 year = 2. Step 6: Simple interest for 1 year at 20 percent on P = 5000 is SI₁ = (5000 * 20 * 1) / 100 = 1000. Step 7: Compound interest for 1 year with two half yearly periods is computed from the amount: A = P * (1 + 10/100)^2 = 5000 * (1.1)^2 = 5000 * 1.21 = 6050. Step 8: CI₁ = A − P = 6050 − 5000 = 1050. Step 9: The required difference for the first year is CI₁ − SI₁ = 1050 − 1000 = 50. Step 10: Therefore, the difference between compound interest and simple interest for the first year with half yearly compounding is Rs. 50. Verification / Alternative check: We can verify by observing that the extra amount due to compounding more frequently within the same year can be approximated as P * (r/2/100)^2 times the number of periods. Here, P = 5000, half yearly rate = 10 percent, so per period factor squared is (0.1)^2 = 0.01. Roughly, the extra due to intra-year compounding is about 5000 * 0.01 = 50, which matches our exact calculation. This quick consistency check reassures us that our more detailed steps are correct. Why Other Options Are Wrong: Rs. 75, Rs. 100, Rs. 150, and Rs. 25 correspond to incorrect assumptions about the principal or about how many periods are compounding within the year. For example, Rs. 100 would result if we mistakenly doubled the extra interest or misapplied the 20 percent rate directly in a compound interest formula for the full year without recognizing the half yearly structure correctly. Only Rs. 50 fits the accurate calculation with P = 5000, SI = 1000, and CI = 1050 for the first year. Common Pitfalls: Many learners forget the standard shortcut D₂ = P * (r/100)^2 and instead perform longer calculations that are more prone to arithmetic mistakes. Another error is to think that the difference for 1 year with half yearly compounding is the same as the 2 year difference with annual compounding divided by 2, which is not true. Confusing nominal annual rate (20 percent) with per period rate (10 percent) is another common source of error. Careful attention to compounding frequency is essential. Final Answer: When interest is compounded half yearly, the difference between compound interest and simple interest for the first year on the same principal is Rs. 50.

Question 10

A certain sum of money becomes three times its original value in 5 years 4 months at simple interest. For this simple interest investment, what is the yearly rate of interest, expressed in percent per annum, that causes the amount to become three ti…

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Introduction / Context: This question extends the common idea of a sum doubling under simple interest to the case where a sum becomes three times its original value. Given the time required for tripling, we must determine the annual simple interest rate. This kind of problem appears frequently in banking and competitive exams and tests whether you understand the linear growth nature of simple interest with respect to time and rate. Given Data / Assumptions: Initial principal is P (some positive amount). Final amount after the given time is 3P (the sum becomes three times). Interest is at simple interest, not compound interest. Time t = 5 years 4 months. We need to find r, the annual rate of simple interest in percent. Concept / Approach: Under simple interest, the amount A after time t years at rate r percent is: A = P * (1 + r * t / 100). Here, A is given to be 3P. Thus, 3P = P * (1 + r * t / 100). Cancelling P leads to a simple equation in r and t. We must convert 5 years 4 months into an exact number of years as a fraction, then isolate r. Because simple interest grows linearly, the total interest fraction is r * t / 100, which in this case must equal 2 (since 3P = P + 2P). Step-by-Step Solution: Step 1: Convert 5 years 4 months to years. Four months is 4/12 years = 1/3 years. Step 2: Therefore, t = 5 + 1/3 = (15/3) + (1/3) = 16/3 years. Step 3: Use the simple interest amount formula: A = P * (1 + r * t / 100). Step 4: Since the amount becomes 3P, we have 3P = P * (1 + r * t / 100). Step 5: Cancel P from both sides to get 3 = 1 + r * t / 100. Step 6: Subtract 1 to obtain r * t / 100 = 2. Step 7: Therefore, r = (2 * 100) / t = 200 / (16/3) = 200 * (3/16) = 600 / 16. Step 8: Simplify 600 / 16 = 37.5, so r = 37.5 percent per annum. Step 9: Hence, the yearly simple interest rate is 37.5 percent. Verification / Alternative check: To verify, compute the total interest fraction using r = 37.5 and t = 16/3 years. The fraction r * t / 100 = (37.5 * 16/3) / 100. First compute 37.5 * 16 = 600, so the numerator is 600/3 = 200. Then 200 / 100 = 2. This means the interest is 2P, so the final amount is P + 2P = 3P, exactly as the question states. This confirms that 37.5 percent is indeed the correct annual rate for the given time and tripling effect. Why Other Options Are Wrong: Any rate less than 37.5 percent, such as 18.75 %, 25 %, or 27.5 %, results in r * t / 100 being less than 2, so the amount would be less than 3P. A larger rate like 42.25 % would make r * t / 100 exceed 2, leading to an amount greater than 3P. Only 37.5 % gives exactly the required threefold increase in the principal over 5 years 4 months. Common Pitfalls: Common mistakes include misreading the phrase "three times its original value" as meaning the interest equals the principal instead of twice the principal. Another frequent error is incorrectly converting 4 months into years, sometimes treating 5 years 4 months as 5.4 years, which is incorrect. Some students also forget to subtract 1 when going from 3 = 1 + r * t / 100, leading to r * t / 100 = 3 instead of 2. Being precise with conversions and algebraic steps avoids these issues. Final Answer: The yearly rate of simple interest that makes a sum become three times its original value in 5 years 4 months is 37.5 % per annum.

Question 11

A sum amounts to Rs. 7727.104 at 12 percent per annum, compounded annually, after 3 years. For this compound interest investment, what is the original principal amount in rupees that was invested?

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Introduction / Context: This question asks you to work backwards in a compound interest scenario. Instead of being given the principal and asked to find the amount, you are given the final amount after 3 years at a known compound rate and must find the original principal. Reverse calculations like this are very common in banking and finance aptitude questions and rely on inverting the compound interest growth factor. Given Data / Assumptions: Final amount after 3 years, A = Rs. 7727.104. Annual rate of compound interest r = 12 percent. Time t = 3 years. Interest is compounded annually. Principal P is unknown and must be found. Concept / Approach: The compound interest amount formula is: A = P * (1 + r/100)^t. Here, r = 12 and t = 3, so the growth factor is (1.12)^3. To find P, we rearrange the formula to P = A / (1 + r/100)^t. Once we compute (1.12)^3 and divide the final amount by this factor, we obtain the principal. We then compare the result with the given options. Step-by-Step Solution: Step 1: Write the compound amount formula: A = P * (1 + r/100)^t. Step 2: Substitute the given values: 7727.104 = P * (1 + 12/100)^3 = P * (1.12)^3. Step 3: Compute (1.12)^2 = 1.2544, then multiply by 1.12 again: (1.12)^3 = 1.404928. Step 4: Rearrange to solve for P: P = 7727.104 / 1.404928. Step 5: Perform the division: P ≈ 5500. Step 6: Therefore, the original principal invested is Rs. 5500. Step 7: We match this to the nearest option, which is exactly Rs. 5500. Verification / Alternative check: To verify, start with P = 5500 and compute the amount after 3 years at 12 percent with annual compounding. After 1 year: A₁ = 5500 * 1.12 = 6160. After 2 years: A₂ = 6160 * 1.12 = 6900.8. After 3 years: A₃ = 6900.8 * 1.12 = 7727.104, which matches the given amount exactly. This confirms that the principal must be Rs. 5500 and validates our reversed calculation. Why Other Options Are Wrong: Rs. 5000, Rs. 5200, Rs. 5350, and Rs. 6000, when grown at 12 percent compound interest for 3 years, do not reach 7727.104. For example, 5000 * (1.12)^3 is significantly less than 7727.104, while 6000 * (1.12)^3 is noticeably higher. Only 5500 reproduces the exact final amount. Thus, the other options represent incorrect guesses of the principal or rounding errors in reversing the growth factor. Common Pitfalls: Some students wrongly use the simple interest formula and set 7727.104 = P * (1 + r * t / 100), which leads to the wrong principal because the problem clearly mentions compounding. Others approximate (1.12)^3 too crudely or treat 12 percent over 3 years as 36 percent simple growth, which again is inappropriate. Precisely computing or carefully estimating the compound factor is important to arrive at the correct answer. Final Answer: The original principal that grows to Rs. 7727.104 at 12 percent per annum compounded annually over 3 years is Rs. 5500.

Question 12

Rohan borrows a certain sum of money at simple interest. The rate is 3 percent per annum for the first 3 years, 4 percent per annum for the next 5 years, and 6 percent per annum for the following 7 years. If he pays Rs. 2059 as total interest over t…

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Introduction / Context: This question involves simple interest with different rates applied over consecutive time intervals on the same principal. Rohan borrows a sum at 3 percent for the first 3 years, 4 percent for the next 5 years, and 6 percent for the next 7 years. The total interest is given, and we must find the original principal. Problems like this test your ability to handle piecewise interest rates and to sum the contributions from each time segment correctly under simple interest. Given Data / Assumptions: Principal P is unknown. Rate r₁ = 3 percent per annum for the first 3 years. Rate r₂ = 4 percent per annum for the next 5 years. Rate r₃ = 6 percent per annum for the following 7 years. Total interest paid over all these years is Rs. 2059. Interest is simple interest, calculated separately on the same principal for each segment. Concept / Approach: Under simple interest, for each segment the interest is: SI = (Principal * Rate * Time) / 100. Since the principal P is constant, we can calculate the interest expression for each sub-period in terms of P and then add all three. The sum of these interests equals Rs. 2059. This yields a single linear equation in P. Solving that equation gives the principal borrowed by Rohan. Step-by-Step Solution: Step 1: Let the principal be P rupees. Step 2: Interest for the first 3 years at 3 percent: SI₁ = (P * 3 * 3) / 100 = (9P) / 100. Step 3: Interest for the next 5 years at 4 percent: SI₂ = (P * 4 * 5) / 100 = (20P) / 100. Step 4: Interest for the next 7 years at 6 percent: SI₃ = (P * 6 * 7) / 100 = (42P) / 100. Step 5: Total simple interest = SI₁ + SI₂ + SI₃ = (9P/100) + (20P/100) + (42P/100) = (71P/100). Step 6: We are told that this total interest equals Rs. 2059, so (71P / 100) = 2059. Step 7: Rearrange to get P = (2059 * 100) / 71. Step 8: Compute 2059 / 71 = 29, so P = 29 * 100 = 2900. Step 9: Therefore, Rohan originally borrowed Rs. 2900. Verification / Alternative check: We can verify by computing the interest in each segment with P = 2900. First 3 years at 3 percent: SI₁ = 2900 * 3 * 3 / 100 = 261. Next 5 years at 4 percent: SI₂ = 2900 * 4 * 5 / 100 = 580. Next 7 years at 6 percent: SI₃ = 2900 * 6 * 7 / 100 = 1218. Summing them: 261 + 580 + 1218 = 2059, which matches the total interest given. This confirms the principal P = 2900 is correct. Why Other Options Are Wrong: At Rs. 2400, 2500, 2700, or 3100, the total interest calculated using the same rates and times would not equal Rs. 2059. For example, at P = 2500, the total interest becomes (71 * 2500) / 100 = 1775, which is much lower than 2059. At P = 3100, the total interest would be (71 * 3100) / 100 = 2201, higher than 2059. Only P = 2900 gives the correct total interest of 2059. Common Pitfalls: Students sometimes incorrectly apply each new rate to the accumulated amount instead of the principal, effectively mixing compound interest ideas into a simple interest problem. Another pitfall is miscalculating the combined coefficient 9 + 20 + 42 = 71, leading to an incorrect total interest expression. Others try to average the rates over the total time period without weighting for years, which is not needed in this direct algebraic approach. Carefully summing the segment-wise simple interest expressions avoids these mistakes. Final Answer: The sum borrowed by Rohan, given the varying simple interest rates over 15 years and a total interest of Rs. 2059, is Rs. 2900.

Question 13

The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at 9 percent per annum is Rs. 405. For this investment, what is the principal amount in rupees?

Accepted Answer

Introduction / Context: This question uses a well-known shortcut formula for the difference between compound interest (CI) and simple interest (SI) for 2 years at a given rate. When interest is compounded annually, the difference over 2 years depends only on the principal and the square of the rate. The problem gives us this difference and the rate, and asks us to find the principal. It is a classic formula-based question that saves time in competitive exams. Given Data / Assumptions: Annual rate of interest r = 9 percent. Time t = 2 years. Difference between CI and SI for 2 years, D₂ = Rs. 405. Interest is compounded annually when calculating CI. Principal P is unknown and must be determined. Concept / Approach: For 2 years at rate r percent per annum, the difference between compound interest and simple interest is: D₂ = P * (r/100)^2. This result comes from expanding the compound interest formula and comparing it to simple interest for 2 years on the same principal. Given D₂ and r, we can directly solve for P. This avoids computing CI and SI separately and is much faster. Step-by-Step Solution: Step 1: Use the shortcut formula for 2 years: D₂ = P * (r/100)^2. Step 2: Substitute D₂ = 405 and r = 9: 405 = P * (9/100)^2. Step 3: Compute (9/100)^2 = 81/10000 = 0.0081. Step 4: So 405 = P * 0.0081. Step 5: Rearranging, P = 405 / 0.0081. Step 6: Divide: 405 / 0.0081 = 50000. Step 7: Hence, the principal amount is Rs. 50000. Verification / Alternative check: We can verify by calculating SI and CI explicitly. For P = 50000, r = 9 percent, t = 2 years, simple interest SI = (50000 * 9 * 2) / 100 = 9000. For compound interest, the amount A = 50000 * (1.09)^2. Compute (1.09)^2 = 1.1881, so A = 50000 * 1.1881 = 59405. CI = A − P = 59405 − 50000 = 9405. The difference CI − SI = 9405 − 9000 = 405, matching the given value. This confirms the principal is indeed Rs. 50000. Why Other Options Are Wrong: If P were Rs. 25000, then D₂ would be 25000 * 0.0081 = 202.5, which is too small. For P = 100000, D₂ would be 810, which is too large. Similarly, 150000 and 200000 produce even larger differences. Only P = 50000 yields the exact difference of Rs. 405, so all other options are inconsistent with the given information. Common Pitfalls: Some students attempt to compute CI and SI separately from scratch without knowing the shortcut formula, which is more time consuming and prone to calculation errors. Others forget to square r/100 or mistakenly use r instead of r/100 in the formula, giving wildly incorrect values for P. Remembering that for 2 years, the difference depends on the squared rate factor is crucial for speed and accuracy. Final Answer: The principal amount on which the difference between compound and simple interest over 2 years at 9 percent per annum is Rs. 405 is Rs. 50000.

Question 14

The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at 12 percent per annum is Rs. 72. For this investment, what is the value of the principal sum in rupees?

Accepted Answer

Introduction / Context: This question is another application of the standard formula for the difference between compound interest (CI) and simple interest (SI) for 2 years at a given rate. The difference over 2 years is proportional to the principal and to the square of the rate. The question supplies the difference and the rate, and asks for the principal. This kind of formula-based problem is very common in interest-related aptitude tests. Given Data / Assumptions: Annual interest rate r = 12 percent. Time period t = 2 years. Difference between CI and SI for 2 years, D₂ = Rs. 72. Interest is compounded annually when computing CI. Principal P is unknown and needs to be determined. Concept / Approach: For 2 years at r percent per annum, the difference between compound interest and simple interest is given by: D₂ = P * (r/100)^2. This comes from comparing the expression for compound amount with the linear simple interest formula over 2 years. Once D₂ and r are known, we substitute them into the formula and solve for P directly, which is much faster than computing CI and SI separately. Step-by-Step Solution: Step 1: Use the standard formula D₂ = P * (r/100)^2. Step 2: Substitute D₂ = 72 and r = 12: 72 = P * (12/100)^2. Step 3: Compute (12/100)^2 = 144/10000 = 0.0144. Step 4: So 72 = P * 0.0144. Step 5: Rearranging, P = 72 / 0.0144. Step 6: Evaluate the division: 72 / 0.0144 = 5000. Step 7: Therefore, the principal sum is Rs. 5000. Verification / Alternative check: We can verify this result by computing CI and SI explicitly. For P = 5000, r = 12 percent, t = 2 years, simple interest SI = (5000 * 12 * 2) / 100 = 1200. The compound amount A under annual compounding is A = 5000 * (1.12)^2. Compute (1.12)^2 = 1.2544, so A = 5000 * 1.2544 = 6272. Compound interest CI = A − P = 6272 − 5000 = 1272. The difference CI − SI = 1272 − 1200 = 72, which matches the given value exactly. Why Other Options Are Wrong: If P were Rs. 10000, the difference would be 10000 * 0.0144 = 144, which is double the required 72. At P = 20000, the difference becomes 288, and at P = 15000 it becomes 216. For P = 2500, the difference is 36. None of these match the given difference of Rs. 72. Only a principal of Rs. 5000 produces exactly the specified difference between compound and simple interest. Common Pitfalls: Some candidates wrongly compute (r/100)^2 as r^2/100 or forget to square the fraction, leading to large errors. Others attempt a more complicated approach by computing CI and SI separately with incomplete arithmetic precision. Remembering and correctly applying the simple formula D₂ = P * (r/100)^2 makes such questions fast and reliable. Careful handling of decimals is also important to avoid calculation errors. Final Answer: The principal sum for which the difference between simple interest and compound interest over 2 years at 12 percent per annum is Rs. 72 is Rs. 5000.

Question 15

A bank offers compound interest at 20 percent per half year. A customer deposits Rs. 7600 on 1 January and another Rs. 7600 on 1 July of the same year. By the end of the year, how much total interest in rupees will the customer have earned from thes…

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Introduction / Context: This question focuses on compound interest with a compounding period of half a year. A customer makes two equal deposits at different times in the year, and the bank compounds the interest every six months at 20 percent per half year. We are asked to determine the total interest earned by the end of the year. This is a practical type of problem that tests your ability to handle multiple deposits and different effective durations under the same compounding scheme. Given Data / Assumptions: Rate per half year = 20 percent. First deposit: Rs. 7600 on 1 January. Second deposit: Rs. 7600 on 1 July of the same year. Compounding is half yearly, so compounding dates are effectively at the end of each half year. We consider the total interest earned by the end of the year. Concept / Approach: Under half yearly compounding, each half year the amount is multiplied by (1 + 20/100) = 1.2. The first deposit is invested for two half years (one full year), while the second deposit is invested for one half year only. We compute the amount from each deposit separately using the appropriate number of compounding periods, subtract the original principal for each, and then sum the two interest amounts. This gives the total interest earned by the end of the year. Step-by-Step Solution: Step 1: For the first deposit of Rs. 7600 on 1 January, there are two half yearly periods before year end. Step 2: Amount from first deposit after 2 half years: A₁ = 7600 * (1.2)^2. Step 3: Compute (1.2)^2 = 1.44, so A₁ = 7600 * 1.44 = 10944. Step 4: Interest from first deposit = A₁ − 7600 = 10944 − 7600 = 3344. Step 5: For the second deposit of Rs. 7600 on 1 July, there is only one half yearly period before year end. Step 6: Amount from second deposit after 1 half year: A₂ = 7600 * 1.2 = 9120. Step 7: Interest from second deposit = A₂ − 7600 = 9120 − 7600 = 1520. Step 8: Total interest by year end = 3344 + 1520 = 4864. Step 9: Therefore, the customer earns Rs. 4864 in total interest over the year. Verification / Alternative check: We can verify by computing the total amount and subtracting the total principal. Total principal = 7600 + 7600 = 15200. Total amount = A₁ + A₂ = 10944 + 9120 = 20064. Interest = Total amount − Total principal = 20064 − 15200 = 4864, which matches our earlier result. This confirms that the total interest has been computed correctly. Why Other Options Are Wrong: Rs. 2432 and Rs. 1216 are obtained if we mistakenly calculate interest for only one deposit or for only one half year period. Rs. 3648 corresponds roughly to taking interest for one and a half periods or mixing up the compounding effect. Rs. 9727 is far too high and would imply a much larger effective rate or longer investment duration. Only Rs. 4864 correctly represents the total interest on two deposits of Rs. 7600 each under the stated half yearly compounding scheme. Common Pitfalls: A common error is to treat the nominal rate of 20 percent per half year as 20 percent per annum and apply only one compounding factor for the full year, which underestimates the interest on the first deposit. Another mistake is to forget that the second deposit has only one half year to grow, not two. Some students simply double the interest from one deposit instead of accounting for different durations. Carefully tracking how long each deposit earns interest is crucial in such questions. Final Answer: By the end of the year, the total compound interest earned from the two Rs. 7600 deposits at 20 percent per half year is Rs. 4864.

Question 16

The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at 16 percent per annum is Rs. 320. For this investment, what is the principal sum in rupees?

Accepted Answer

Introduction / Context: This problem is yet another application of the shortcut formula for the difference between compound interest (CI) and simple interest (SI) over 2 years at a given rate. The question supplies the rate, the time, and the difference, and asks you to find the principal amount. Efficiently using this formula helps save time in quantitative aptitude sections that involve many interest-related questions. Given Data / Assumptions: Annual rate of interest r = 16 percent. Time period t = 2 years. Difference between CI and SI over 2 years, D₂ = Rs. 320. Interest for CI is compounded annually. Principal P is unknown. Concept / Approach: For a principal P invested at rate r percent per annum for 2 years, the difference between CI and SI is: D₂ = P * (r/100)^2. This result is derived by comparing the compound amount P * (1 + r/100)^2 with the simple interest amount P * (1 + 2r/100). Once D₂ and r are known, we substitute and solve for P directly. This method avoids long calculations of CI and SI separately. Step-by-Step Solution: Step 1: Use the formula D₂ = P * (r/100)^2. Step 2: Substitute D₂ = 320 and r = 16: 320 = P * (16/100)^2. Step 3: Compute (16/100)^2 = 256/10000 = 0.0256. Step 4: So 320 = P * 0.0256. Step 5: Rearranging for P, we get P = 320 / 0.0256. Step 6: Divide: 320 / 0.0256 = 12500. Step 7: Therefore, the principal sum is Rs. 12500. Verification / Alternative check: We can verify by explicitly computing CI and SI for P = 12500 at r = 16 percent over 2 years. Simple interest SI = (12500 * 16 * 2) / 100 = 4000. Compound amount A = 12500 * (1.16)^2. Compute (1.16)^2 = 1.3456, so A = 12500 * 1.3456 = 16820. Compound interest CI = A − P = 16820 − 12500 = 4320. The difference CI − SI = 4320 − 4000 = 320, matching the given difference. This confirms that P = 12500 is correct. Why Other Options Are Wrong: If P were Rs. 25000, then D₂ would be 25000 * 0.0256 = 640, which is double the required 320. For P = 50000, the difference becomes 1280, and for P = 37500 it is 960. At P = 10000, the difference is 256, smaller than 320. Only P = 12500 yields the correct difference of Rs. 320, making the other options inconsistent with the data and formula. Common Pitfalls: Common sources of error include miscalculating (r/100)^2, forgetting to convert the percentage to a fraction before squaring, or incorrectly rearranging the equation to solve for P. Some test-takers also mistakenly apply the simple interest formula instead of using the CI–SI difference shortcut. Carefully following the formula and checking the arithmetic helps avoid these mistakes. Final Answer: The principal sum on which the difference between simple interest and compound interest over 2 years at 16 percent per annum is Rs. 320 is Rs. 12500.

Question 17

On a certain principal, the simple interest for 2 years is Rs. 3000 and the compound interest for the same principal and the same 2 years with annual compounding is Rs. 3225. For this investment, what is the annual rate of interest in percent?

Accepted Answer

Introduction / Context: This question provides both the simple interest (SI) and compound interest (CI) for 2 years on the same principal and asks for the annual rate of interest. By combining the basic SI formula with the standard CI–SI difference relationship, we can find the rate without ever knowing the principal explicitly. This is a neat algebraic question that combines two closely related interest concepts. Given Data / Assumptions: Simple interest SI over 2 years = Rs. 3000. Compound interest CI over 2 years, compounded annually = Rs. 3225. Principal P is the same in both cases. Rate of interest r percent per annum is the same in both calculations. Time t = 2 years. Concept / Approach: Under simple interest: SI = (P * r * t) / 100. For t = 2 years, SI = (P * r * 2) / 100. Under compound interest over 2 years, the difference between CI and SI is D₂ = P * (r/100)^2. We know SI and CI, so D₂ = CI − SI = 3225 − 3000 = 225. Using SI, we find P * r. Using the difference formula, we find P * r^2. Then, dividing (P * r^2) by (P * r) gives r directly. Step-by-Step Solution: Step 1: Compute the difference between CI and SI for 2 years: D₂ = CI − SI = 3225 − 3000 = 225. Step 2: From simple interest, SI = (P * r * 2) / 100 = 3000. Step 3: Rearranging gives P * r = (3000 * 100) / 2 = 150000. Step 4: From the difference formula, D₂ = P * (r/100)^2 = 225. Step 5: Multiply both sides by 10000 to eliminate the denominator: P * r^2 = 225 * 10000 = 2250000. Step 6: We now have P * r = 150000 and P * r^2 = 2250000. Step 7: Divide P * r^2 by P * r to get r: (P * r^2) / (P * r) = r = 2250000 / 150000. Step 8: Compute 2250000 / 150000 = 15. Step 9: Hence, the annual rate of interest is 15 percent per annum. Verification / Alternative check: We can verify by picking P from the equation P * r = 150000. With r = 15, P = 150000 / 15 = 10000. For simple interest: SI = (10000 * 15 * 2) / 100 = 3000, which matches the given SI. For compound interest: amount A = 10000 * (1.15)^2. Compute (1.15)^2 = 1.3225, so A = 13225. CI = A − P = 13225 − 10000 = 3225, which matches the given CI. This confirms that r = 15 percent is correct. Why Other Options Are Wrong: At 7.5 percent, the SI and CI values would be much smaller and their difference would not match 225. At 10 percent, the difference CI − SI for 2 years is P * (0.10)^2 = 0.01P, which would not align with SI = 3000. Higher rates such as 22.5 percent or 30 percent would produce much larger differences and inconsistent SI values. Only 15 percent satisfies both the simple interest and compound interest conditions simultaneously. Common Pitfalls: Some learners try to solve directly from the CI formula without leveraging the neat ratio trick of using P * r and P * r^2. Others confuse the formulas for CI and SI or forget that the difference formula D₂ = P * (r/100)^2 applies for 2 years. Another frequent pitfall is mixing up the units when moving between r and r/100. Following a structured, algebraic approach helps avoid these errors. Final Answer: The annual rate of interest that makes the simple interest Rs. 3000 and the compound interest Rs. 3225 in 2 years on the same principal is 15 percent per annum.

Question 18

Ganesh invests an amount of Rs. x in a fixed deposit scheme that offers 5 percent per annum for the first year and 15 percent per annum for the second year, with interest compounded annually. If he receives Rs. 9660 after two years, what is the valu…

Accepted Answer

Introduction / Context: This question involves compound interest with different rates in each year. Ganesh invests an amount x, which earns 5 percent in the first year and 15 percent in the second year, with annual compounding. The final amount after 2 years is given, and we must find the original principal x. This is a practical type of banking and finance question that tests your understanding of compounding with changing yearly rates. Given Data / Assumptions: Initial principal = Rs. x. Rate in first year r₁ = 5 percent. Rate in second year r₂ = 15 percent. Interest is compounded annually, with the rate changing year by year. Final amount after 2 years A = Rs. 9660. Concept / Approach: With annual compounding and different rates each year, the amount after 2 years is obtained by multiplying the principal by successive growth factors. After the first year, the amount becomes x * (1 + 5/100). After the second year, 15 percent is applied on this new amount. Therefore, the overall growth factor across 2 years is (1 + 5/100) * (1 + 15/100) = 1.05 * 1.15. Setting x * 1.05 * 1.15 equal to 9660 allows us to solve for x by division. Step-by-Step Solution: Step 1: After the first year, amount A₁ = x * (1 + 5/100) = x * 1.05. Step 2: After the second year, interest at 15 percent is applied on A₁, giving A₂ = A₁ * (1 + 15/100) = x * 1.05 * 1.15. Step 3: We know that A₂ = 9660. Step 4: So x * 1.05 * 1.15 = 9660. Step 5: Compute the combined factor: 1.05 * 1.15 = 1.2075. Step 6: Therefore, x * 1.2075 = 9660. Step 7: Solve for x: x = 9660 / 1.2075. Step 8: Perform the division to get x = 8000. Step 9: Hence, the original principal invested by Ganesh is Rs. 8000. Verification / Alternative check: We can verify by directly recomputing the amount starting from x = 8000. After 1 year at 5 percent: A₁ = 8000 * 1.05 = 8400. After 2 years at 15 percent on A₁: A₂ = 8400 * 1.15 = 9660. This matches the given final amount, confirming that the principal must have been Rs. 8000. Why Other Options Are Wrong: For Rs. 7500, the final amount would be 7500 * 1.2075 = 9056.25, which is less than 9660. For Rs. 8200, the final amount would be 8200 * 1.2075 ≈ 9901.5, larger than 9660. Similarly, Rs. 8500 and Rs. 9000 give final amounts that do not match 9660. Only Rs. 8000, when multiplied by the combined factor 1.2075, reproduces the given final amount exactly. Common Pitfalls: Some students mistakenly average the two rates (5 percent and 15 percent) to 10 percent per year and then treat the situation as a 2 year investment at 10 percent compound interest, which is incorrect. Others apply 5 percent and 15 percent as simple interest instead of compounding. Forgetting to multiply the factors sequentially (1.05 and 1.15) is another frequent error. Working carefully with the compound growth factors avoids these mistakes. Final Answer: The original principal that Ganesh invested in the fixed deposit scheme is Rs. 8000.

Question 19

An amount earns a total simple interest of Rs. 3200 at the rate of 6.25 percent per annum in 4 years. For this simple interest investment, what is the principal amount in rupees that was originally invested?

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Introduction / Context: This question is a standard simple interest problem in which the total interest, rate, and time are given, and the principal must be found. It is a direct application of the simple interest formula and is commonly asked in introductory quantitative aptitude and banking exams. The rate is given as 6.25 percent per annum, which can also be expressed as a fraction to make calculations easier. Given Data / Assumptions: Total simple interest SI = Rs. 3200. Rate of interest r = 6.25 percent per annum. Time t = 4 years. Principal P is unknown. Interest is purely simple interest, not compounded. Concept / Approach: The simple interest formula is: SI = (P * r * t) / 100. We are given SI, r, and t, so we rearrange the formula to solve for P: P = (SI * 100) / (r * t). Because r = 6.25 percent, we may express it as 6.25 = 25/4 percent for easier multiplication and division. Substituting the numbers and simplifying gives us the principal amount. Step-by-Step Solution: Step 1: Write the simple interest formula: SI = (P * r * t) / 100. Step 2: Substitute the given data: 3200 = (P * 6.25 * 4) / 100. Step 3: Compute the denominator: 6.25 * 4 = 25. Step 4: So 3200 = (P * 25) / 100. Step 5: Rearranging for P: P = (3200 * 100) / 25. Step 6: Compute the numerator: 3200 * 100 = 320000. Step 7: Divide: 320000 / 25 = 12800. Step 8: Therefore, the principal amount invested is Rs. 12800. Verification / Alternative check: We can verify quickly by checking the interest generated by P = 12800 at 6.25 percent for 4 years. SI = (12800 * 6.25 * 4) / 100. First compute 6.25 * 4 = 25, so SI = (12800 * 25) / 100 = 12800 * 0.25 = 3200, which matches the given interest. This confirms that the principal must be Rs. 12800. Why Other Options Are Wrong: At P = 11800 or 13800, the resulting simple interest would not be exactly 3200. For example, with 11800, SI would be (11800 * 25) / 100 = 2950, which is too small. With 14800, SI would be 3700, and with 16000 it would be 4000. Only P = 12800 gives SI equal to Rs. 3200 at 6.25 percent over 4 years, making all other options incorrect. Common Pitfalls: One frequent mistake is mishandling the decimal rate 6.25 percent, such as treating it as 6.5 or 6.2. Another issue is forgetting to multiply by 100 when rearranging the formula. Some students also confuse simple interest with compound interest and attempt unnecessary compounding steps. Writing the formula clearly and simplifying step by step helps avoid these errors. Final Answer: The principal amount that earns Rs. 3200 as simple interest at 6.25 percent per annum in 4 years is Rs. 12800.

Question 20

The simple interest and compound interest that can be earned in 2 years at the same annual rate are Rs. 1500 and Rs. 1575, respectively. For this investment, what is the annual rate of interest in percent per annum?

Accepted Answer

Introduction / Context: This question provides both the simple interest (SI) and the compound interest (CI) for 2 years at the same rate and on the same principal, and asks us to determine the rate of interest per annum. The key idea is that the difference between CI and SI over 2 years is related to the principal and the square of the rate. By combining the facts SI = 1500 and CI = 1575, we can solve for both the principal–rate product and the rate itself. Given Data / Assumptions: Simple interest SI over 2 years = Rs. 1500. Compound interest CI over 2 years, compounded annually = Rs. 1575. Principal P is the same in both cases. Rate of interest r percent per annum is the same in both cases. Time t = 2 years. Concept / Approach: Under simple interest for 2 years, SI = (P * r * 2) / 100. Under compound interest, the difference between CI and SI for 2 years is D₂ = P * (r/100)^2. We know SI and CI, so the difference D₂ = 1575 − 1500 = 75. From SI we get P * r, and from D₂ we get P * r^2. Dividing P * r^2 by P * r yields r directly, which is the annual rate of interest. Step-by-Step Solution: Step 1: Compute the difference between CI and SI: D₂ = 1575 − 1500 = 75. Step 2: From simple interest, SI = (P * r * 2) / 100 = 1500. Step 3: Rearranging gives P * r = (1500 * 100) / 2 = 75000. Step 4: From the difference formula, D₂ = P * (r/100)^2 = 75. Step 5: Multiply both sides by 10000: P * r^2 = 75 * 10000 = 750000. Step 6: Now we have P * r = 75000 and P * r^2 = 750000. Step 7: Divide P * r^2 by P * r to isolate r: (P * r^2) / (P * r) = r = 750000 / 75000. Step 8: Compute 750000 / 75000 = 10. Step 9: Thus, the annual rate of interest is 10 percent per annum. Verification / Alternative check: To verify, we can find P using P * r = 75000 and r = 10. Then P = 75000 / 10 = 7500. For simple interest: SI = (7500 * 10 * 2) / 100 = 1500, which matches the given SI. For compound interest: amount A = 7500 * (1.10)^2. Compute (1.10)^2 = 1.21, so A = 7500 * 1.21 = 9075. CI = A − P = 9075 − 7500 = 1575, matching the given CI. This confirms r = 10 percent is correct. Why Other Options Are Wrong: At 5 %, SI over 2 years would be much smaller and CI − SI would also be much smaller than 75. At 8 % or 12 %, the product P * r and the difference P * (r/100)^2 would not match the given SI and CI values simultaneously. A rate as high as 15 % would produce a much larger difference between CI and SI than Rs. 75 for realistic principal values. Only 10 % satisfies both conditions exactly. Common Pitfalls: Some candidates misapply the CI formula directly without using the neat P * r and P * r^2 relationships, leading to more complicated algebra and potential mistakes. Others forget to multiply by 100 or 10000 at the right stage and end up with fractional or unrealistic rates. Being systematic in using SI to get P * r and the CI–SI difference to get P * r^2 makes this question straightforward. Final Answer: The annual rate of interest that leads to simple interest of Rs. 1500 and compound interest of Rs. 1575 in 2 years on the same principal is 10 % per annum.

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