Question 1

Simple Interest — What equal annual payment will discharge a debt of ₹ 1696 in 4 years at 4% per annum simple interest?

Accepted Answer

Introduction / Context:The original stem omitted the number of years. By the Recovery-First policy, we minimally repair the stem to the standard 4-year annuity-at-simple-interest version that is consistent with the answer set. Under SI, each installment is increased linearly to the focal date and the totals are equated to the amount due. Given Data / Assumptions (Repaired): Debt amount to be cleared at end of year 4: ₹ 1696. Equal annual payment A at ends of years 1–4. Rate r = 4% p.a., simple interest. Concept / Approach:Accumulate each A to end of year 4: factors are 1 + r*(4 − k) for k = 1..4. Sum of factors = 4 + r*(1 + 2 + 3) = 4 + 0.04*6 = 4.24. Step-by-Step Solution:(Accumulated total) = A * 4.24 = 1696.A = 1696 / 4.24 = ₹ 400. Verification / Alternative check:Check: 400*(1.12 + 1.08 + 1.04 + 1.00) = 400*4.24 = 1696 (matches). Why Other Options Are Wrong:₹ 525, ₹ 425, ₹ 350, ₹ 325 do not satisfy the 4.24 multiplier relation. Common Pitfalls:Using compound accumulation or forgetting to use the arithmetic series of SI factors for the buildup. Final Answer:₹ 400

Question 2

Simple Interest — Harsha deposits ₹ 20,000 at 13% per annum simple interest (interest reckoned half-yearly but still simple). What total amount will he receive after 42 months (3.5 years)?

Accepted Answer

Introduction / Context:In simple interest, interest grows linearly with time regardless of sub-year reckoning. If the stated rate is “per annum, charged half-yearly” but there is no compounding, we simply pro-rate the annual rate by the exact time in years. Given Data / Assumptions: P = ₹ 20,000. r = 13% per annum (simple), reckoned half-yearly but not compounded. t = 42 months = 3.5 years. Concept / Approach:Simple interest SI = P * r * t. Amount A = P + SI. Step-by-Step Solution:SI = 20,000 * 0.13 * 3.5 = 20,000 * 0.455 = ₹ 9,100.Amount A = 20,000 + 9,100 = ₹ 29,100. Verification / Alternative check:13% per year for 3 years gives 39% (= ₹ 7,800); for 0.5 year gives 6.5% (= ₹ 1,300). Total 45.5% of 20,000 = 9,100; A = 29,100. Why Other Options Are Wrong:₹ 27,800 and ₹ 28,100 undercount interest; ₹ 30,000 assumes 50% return; ₹ 28,900 is an in-between distractor. Common Pitfalls:Confusing “charged half-yearly” with compounding; here it does not change linear SI. Final Answer:₹ 29,100

Question 3

Simple Interest — A sum amounts to ₹ 1350 in 5 years and to ₹ 1620 in 8 years at the same simple interest rate. Find the principal (original sum).

Accepted Answer

Introduction / Context:Two different future amounts at the same simple interest rate allow us to extract the yearly interest from their difference and then deduce the principal. Given Data / Assumptions: A5 = ₹ 1350 at t = 5 years. A8 = ₹ 1620 at t = 8 years. Simple interest; principal remains constant. Concept / Approach:Difference between amounts over 3 years equals 3 years of interest: (A8 − A5) = 3 * I_per_year. Step-by-Step Solution:A8 − A5 = 1620 − 1350 = ₹ 270.Therefore I_per_year = 270 / 3 = ₹ 90.A5 = P + 5 * 90 = 1350 ⇒ P = 1350 − 450 = ₹ 900. Verification / Alternative check:Check A8: P + 8 * 90 = 900 + 720 = 1620, matches. Why Other Options Are Wrong:₹ 700, ₹ 800, ₹ 850, ₹ 1000 do not reproduce both amounts when forward-checked. Common Pitfalls:Using compounding or changing principal; SI keeps principal constant and adds equal yearly interest. Final Answer:₹ 900

Question 4

Simple Interest — ₹ 6000 becomes ₹ 8000 in 4 years at simple interest. In how many years at the same rate will ₹ 525 become ₹ 700?

Accepted Answer

Introduction / Context:The first pair (₹ 6000 to ₹ 8000 in 4 years) reveals the annual rate under simple interest. That same rate is then applied to a different principal to find the time needed to reach a new amount. Given Data / Assumptions: From ₹ 6000 to ₹ 8000 in 4 years. Find t such that ₹ 525 becomes ₹ 700 at the same rate. Concept / Approach:From the first pair: SI = 2000 in 4 years ⇒ yearly interest = 500 ⇒ rate r = 500 / 6000 = 1/12 ≈ 8.333% p.a. Then use SI = P * r * t for the second principal. Step-by-Step Solution:Needed SI on ₹ 525: 700 − 525 = ₹ 175.P * r * t = 175 ⇒ 525 * (1/12) * t = 175.t = 175 * 12 / 525 = 4 years. Verification / Alternative check:At ~8.333% p.a., yearly SI on 525 is 43.75; 4 years produce 175, giving 700. Why Other Options Are Wrong:2, 3, 5, 6 years do not generate exactly ₹ 175 interest at this rate. Common Pitfalls:Using proportionality with amounts instead of SI; in SI, interest is proportional to time for a fixed rate. Final Answer:4 yr

Question 5

Simple Interest — A principal amounts to ₹ 944 in 3 years and ₹ 1040 in 5 years at the same simple interest rate. Find the principal.

Accepted Answer

Introduction / Context:Two amounts at different times under the same simple interest rate differ by a multiple of the annual interest. That difference allows quick extraction of the yearly interest and hence the principal. Given Data / Assumptions: A3 = ₹ 944 at 3 years. A5 = ₹ 1040 at 5 years. Concept / Approach:A5 − A3 = 2 years of interest. Let I be annual interest; then 2I = 1040 − 944 = 96 ⇒ I = 48. Now use A3 = P + 3I to get P. Step-by-Step Solution:I = 48 per year.P = 944 − 3*48 = 944 − 144 = ₹ 800. Verification / Alternative check:At 5 years: 800 + 5*48 = 1040, matches. Why Other Options Are Wrong:Other choices fail the forward check against both 3-year and 5-year amounts. Common Pitfalls:Mixing SI with compounding; only linear increments apply here. Final Answer:₹ 800

Question 6

Simple Interest — ₹ 1550 is lent partly at 5% and partly at 8% per annum (simple). Over 4 years, the total interest received is ₹ 400. Find the ratio of the amounts lent at 5% to 8%.

Accepted Answer

Introduction / Context:Splitting a principal into two parts at different simple interest rates over the same time yields a linear equation in the split amounts. The ratio follows directly from that equation. Given Data / Assumptions: Total principal = ₹ 1550, time = 4 years. Part at 5% = x; part at 8% = 1550 − x. Total SI in 4 years = ₹ 400. Concept / Approach:Total SI = 4 * (0.05x + 0.08(1550 − x)) = 400. Solve for x and express the split as a ratio. Step-by-Step Solution:0.20x + 0.32(1550 − x) = 400.0.20x + 496 − 0.32x = 400 ⇒ −0.12x = −96 ⇒ x = 800.Thus amounts: at 5% = 800, at 8% = 750 ⇒ ratio = 800 : 750 = 16 : 15. Verification / Alternative check:Interest: 800*0.05*4 = 160; 750*0.08*4 = 240; total = 400 as given. Why Other Options Are Wrong:Other ratios do not reproduce the total ₹ 400 interest. Common Pitfalls:Forgetting to multiply by time (4 years) or mishandling the algebraic signs. Final Answer:16 : 15

Question 7

Simple Interest — Neeta borrows at tiered simple interest: 6% p.a. for the first 3 years, 9% p.a. for the next 5 years, and 13% p.a. for the period beyond 8 years. If the total interest paid in 11 years is ₹ 8160, find the principal borrowed.

Accepted Answer

Introduction / Context:When the simple interest rate changes across disjoint periods, total interest equals principal times the sum of periodwise rates * years. The principal can be factored out. Given Data / Assumptions: Years 1–3: 6% p.a. Years 4–8: 9% p.a. Years 9–11: 13% p.a. Total SI over 11 years = ₹ 8160. Concept / Approach:Let P be principal. Then SI_total = P*(0.06*3 + 0.09*5 + 0.13*3) = P*(0.18 + 0.45 + 0.39) = P*1.02. Step-by-Step Solution:P * 1.02 = 8160 ⇒ P = 8160 / 1.02 = ₹ 8,000. Verification / Alternative check:Annualized interest: 6%*3=18%, 9%*5=45%, 13%*3=39%; total 102% of P = ₹ 8160 ⇒ P = 8000. Why Other Options Are Wrong:₹ 12,000, ₹ 10,000, ₹ 9,000 do not satisfy the 1.02 multiplier. Common Pitfalls:Confusing percentage points with percent of P; under SI, add (rate * years) linearly. Final Answer:₹ 8,000

Question 8

Simple Interest — Reena has ₹ 10,000. She lends ₹ x to Akshay for 2 years at 15% p.a. and the rest to Brijesh for 2 years at 18% p.a. After 2 years, Akshay’s interest exceeds Brijesh’s by ₹ 360. How much did Reena lend to Brijesh?

Accepted Answer

Introduction / Context:A split investment under simple interest with a specified difference in interest lets us write one linear equation in the split variable. Then the asked amount follows directly. Given Data / Assumptions: Total funds = ₹ 10,000. Akshay receives ₹ x at 15% for 2 years. Brijesh receives ₹ (10,000 − x) at 18% for 2 years. Akshay’s interest is ₹ 360 more than Brijesh’s. Concept / Approach:Interest_A = x * 0.15 * 2 = 0.30x. Interest_B = (10,000 − x) * 0.18 * 2 = 0.36(10,000 − x). Equation: 0.30x = 0.36(10,000 − x) + 360. Step-by-Step Solution:0.30x = 3600 − 0.36x + 360 ⇒ 0.66x = 3960 ⇒ x = 6000.Thus Brijesh’s share = 10,000 − 6,000 = ₹ 4,000. Verification / Alternative check:Interests: Akshay = 6000*0.30 = 1800; Brijesh = 4000*0.36 = 1440; difference = 360, as stated. Why Other Options Are Wrong:₹ 2,500, ₹ 3,500, ₹ 4,200, ₹ 5,000 fail the interest-difference check. Common Pitfalls:Sign mistakes when moving terms; always keep track of which interest is larger. Final Answer:₹ 4,000

Question 9

Simple Interest — Mr. Pawan invests ₹ 2,42,000 at 4% p.a. for 6 years (simple). He then invests the accumulated amount (principal + interest) for another 4 years at the same rate, again under simple interest. Find the simple interest earned in the l…

Accepted Answer

Introduction / Context:This is a two-stage simple-interest investment. Stage 1 grows the principal linearly; Stage 2 uses that higher base (principal + first-stage interest) to earn further simple interest. The question specifically asks for the interest of the second stage only. Given Data / Assumptions: Initial principal P = ₹ 2,42,000. Rate r = 4% p.a. throughout; simple interest only. Stage 1 time t1 = 6 years; Stage 2 time t2 = 4 years. Concept / Approach:Stage 1 interest: SI1 = P * r * t1. Amount after Stage 1: A1 = P + SI1. Stage 2 interest on A1: SI2 = A1 * r * t2. Step-by-Step Solution:SI1 = 2,42,000 * 0.04 * 6 = ₹ 58,080.A1 = 2,42,000 + 58,080 = ₹ 3,00,080.SI2 = 3,00,080 * 0.04 * 4 = 3,00,080 * 0.16 = ₹ 48,012.80. Verification / Alternative check:Orderly separation of stages ensures no compounding is assumed; each segment is linear SI on its own base. Why Other Options Are Wrong:₹ 48,000 rounds off A1; ₹ 47,000 and ₹ 46,800 are underestimates; ₹ 48,501.28 uses an incorrect base or rate. Common Pitfalls:Accidentally compounding or taking Stage 2 interest on the original P; the correct base is A1. Final Answer:₹ 48,012.80

Question 10

Simple Interest — ₹ 12,000 is placed at 10% p.a. simple interest. Because the depositor withdraws after 3 years (instead of keeping it for 5 years at 10%), the bank pays a lower simple-interest rate for the 3-year period. If his interest is ₹ 3,320 …

Accepted Answer

Introduction / Context:Under simple interest, interest is proportional to time and rate. The question contrasts the interest that would have been earned over 5 years at 10% with the interest actually paid for 3 years at an unknown lower rate. The difference between these totals is known, allowing us to solve for the unknown rate. Given Data / Assumptions: Principal P = ₹ 12,000. Benchmark (5 years at 10% SI): I_5y = P * 0.10 * 5 = ₹ 6,000. Actual (3 years at rate r SI): I_3y = P * r * 3. Shortfall relative to benchmark: I_5y − I_3y = ₹ 3,320. Concept / Approach:Set up the equation 12,000 * 0.10 * 5 − 12,000 * r * 3 = 3,320 and solve for r as a percentage. Step-by-Step Solution:6,000 − 36,000 r = 3,320.36,000 r = 6,000 − 3,320 = 2,680.r = 2,680 / 36,000 = 0.074444… = 7.444…% = 67/9%. Verification / Alternative check:Check: I_3y = 12,000 * 0.074444… * 3 = 2,680; shortfall = 6,000 − 2,680 = 3,320 (as given). Why Other Options Are Wrong:78/9%, 77/9%, 87/9% are around 8.6–9.7% and do not satisfy the shortfall equation; 7.25% also fails the equality. Common Pitfalls:Interpreting “less than” backward; ensure the equation subtracts actual from benchmark, not vice versa. Final Answer:67/9%

Question 11

Simple Interest — Ajay borrows at 5% p.a. simple interest and repays ₹ 8,800 after 2 years, clearing the loan. Find the interest paid by Ajay.

Accepted Answer

Introduction / Context:For simple interest, the amount after t years is A = P * (1 + r * t). Given the amount and rate-time product, the principal and interest are straightforward to compute. Given Data / Assumptions: A = ₹ 8,800 after 2 years. r = 5% p.a. (0.05), simple interest. Concept / Approach:A = P * (1 + 0.05 * 2) = 1.10 P ⇒ P = A / 1.10. Interest paid = A − P. Step-by-Step Solution:P = 8,800 / 1.10 = ₹ 8,000.Interest = 8,800 − 8,000 = ₹ 800. Verification / Alternative check:Direct SI: 8,000 * 0.05 * 2 = 800; A = 8,800. Why Other Options Are Wrong:₹ 825, ₹ 850, ₹ 975, ₹ 880 do not match 10% of principal over 2 years. Common Pitfalls:Misapplying compounding or forgetting that SI over 2 years at 5% totals 10% of principal. Final Answer:₹ 800

Question 12

The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at the rate of 4% per annum is Rs 1. What is the principal (sum of money in rupees)?

Accepted Answer

Introduction:This is another short question where the difference between compound interest and simple interest over 2 years is used to back calculate the principal. The rate is small (4%), and a standard formula makes this calculation very quick in competitive exams. Given Data / Assumptions: Rate of interest (r) = 4% per annum. Time period (t) = 2 years. Difference between simple interest and compound interest for 2 years is Rs 1 (in magnitude). Compounding is annual in the CI case. We need to determine the principal P. Concept / Approach:For 2 years and annual compounding, the difference between CI and SI is given by Difference = P * (r/100)^2. This formula captures the interest on interest that simple interest does not include. By substituting the given values, we can immediately find P. Step-by-Step Solution:Difference = P * (r/100)^2Here r = 4%, so r/100 = 0.04Thus Difference = P * (0.04)^2 = P * 0.0016Given that the difference is Rs 1, we have P * 0.0016 = 1So P = 1 / 0.0016P = 625 Verification / Alternative Check:If P = Rs 625, compute SI and CI for 2 years. Simple interest: SI2 = 625 * 4 * 2 / 100 = 625 * 0.08 = Rs 50. Compound amount: A = 625 * (1.04)^2 = 625 * 1.0816 = Rs 676.0. CI2 = 676 − 625 = Rs 51. Difference CI2 − SI2 = 51 − 50 = Rs 1, matching the statement. Why Other Options Are Wrong:Rs 620, Rs 630, Rs 640 and Rs 600: For each of these principals, P * 0.0016 gives differences not equal to 1. For example, 600 * 0.0016 = 0.96, and 640 * 0.0016 = 1.024. Common Pitfalls:Students sometimes forget to square the rate fraction and instead multiply only by r/100, which leads to a wrong principal. Others may attempt to calculate SI and CI separately without realizing the availability of the shortcut formula, making the computation longer and more error prone. Final Answer:The principal (sum of money) is Rs 625.

Question 13

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

Accepted Answer

Introduction / Context:This question connects two different interest situations: simple interest on an unknown principal and compound interest on a known sum. The simple interest for 3 years at 8% per annum is half of the compound interest on Rs. 16,000 for 2 years at 10% per annum. You must carefully compute the compound interest first, then relate it to the simple interest to find the unknown principal. Given Data / Assumptions:- Simple interest is on an unknown principal P at 8% per annum for 3 years.- Compound interest is on Rs. 16,000 at 10% per annum for 2 years, compounded annually.- S.I. for 3 years at 8% = ½ × C.I. for 2 years at 10% on Rs. 16,000.- We must find P, the sum placed on simple interest. Concept / Approach:The approach is in two stages:1) Calculate the compound interest on Rs. 16,000 for 2 years at 10%.2) Take half of that amount to get the simple interest on P for 3 years at 8%.Then, using the simple interest formula S.I. = P * R * T / 100, we can solve for P. This illustrates the interplay between C.I. and S.I. in exam problems. Step-by-Step Solution:Step 1: Compute compound amount on Rs. 16,000 at 10% for 2 years.Step 2: A = 16000 * (1.10)^2 = 16000 * 1.21 = 19360.Step 3: Compound interest on Rs. 16,000 = 19360 - 16000 = Rs. 3360.Step 4: Given that S.I. = half of this C.I. = 3360 / 2 = Rs. 1680.Step 5: Simple interest formula: 1680 = P * 8 * 3 / 100.Step 6: Simplify: 1680 = P * 24 / 100 = 0.24P.Step 7: Solve for P: P = 1680 / 0.24 = 7000.Step 8: Hence, the sum placed on simple interest is Rs. 7000. Verification / Alternative check:Check S.I. on Rs. 7000 at 8% for 3 years: S.I. = 7000 * 8 * 3 / 100 = 7000 * 24 / 100 = 1680. Half of C.I. on Rs. 16,000 (which we calculated as 3360) is also 1680, so the condition is satisfied exactly. Why Other Options Are Wrong:Rs. 14,000 would give S.I. of 3360, which equals full C.I., not half. Rs. 3,500, Rs. 5,600 and Rs. 9,000 give simple interest values that do not match 1680. They arise from misusing the factor ½ or from arithmetic errors in computing the compound interest. Common Pitfalls:Some students forget that the relationship is “half the compound interest”, not equal to it, and set S.I. = C.I. Others miscalculate the compound amount on Rs. 16,000 or mistakenly treat it as simple interest. Always compute the compound interest carefully, then apply the proportion given in the problem. Final Answer:The principal placed on simple interest is Rs. 7,000.

Question 14

The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at 16% per annum is Rs. 320. What is the value of the principal (sum invested) in rupees?

Accepted Answer

Introduction / Context: This problem compares simple interest and compound interest on the same principal, at the same rate and time. The difference between compound interest and simple interest over 2 years has a neat formula that can be used to find the principal quickly when the rate and difference are known. Given Data / Assumptions: Rate of interest r = 16% per annum. Time t = 2 years. Difference between compound interest and simple interest = Rs. 320. Interest is compounded annually for the CI calculation. Principal is P and must be found. Concept / Approach: For 2 years at rate r% per annum, the difference between compound interest (CI) and simple interest (SI) on principal P is: CI − SI = P * (r^2 / 100^2). This arises because CI for 2 years equals SI for 2 years plus an extra interest on the first year's interest. Knowing r and the difference, we can directly solve for P. Step-by-Step Solution: Step 1: Write the formula for the difference. CI − SI = P * r^2 / 100^2. Given CI − SI = 320 and r = 16. Step 2: Substitute values. 320 = P * 16^2 / 100^2. 16^2 = 256 and 100^2 = 10,000. So 320 = P * 256 / 10,000. Step 3: Solve for P. P = 320 * 10,000 / 256. P = 3,200,000 / 256 = 12,500. Verification / Alternative check: Compute SI and CI explicitly. SI for 2 years: SI = P * r * t / 100 = 12,500 * 16 * 2 / 100 = 4,000. CI amount: Amount A = 12,500 * (1.16)^2 = 12,500 * 1.3456 = 16,820. CI = A − P = 16,820 − 12,500 = 4,320. Difference CI − SI = 4,320 − 4,000 = 320, matching the given value. Why Other Options Are Wrong: If P were 25,000, the difference would be double, 640. For 37,500 it would be 960, and for 50,000 it would be 1,280. None of these match the specified difference of 320, so those options must be rejected. Common Pitfalls: Some candidates try to compute SI and CI with an unknown P separately, leading to more algebra than necessary. Others forget or mishandle the r^2 / 100^2 term. Remembering the direct formula for the difference over 2 years saves time and reduces the likelihood of algebraic errors. Final Answer: The principal (sum invested) was Rs. 12,500.

Question 15

At what annual rate of compound interest, compounded annually, will a sum of Rs. 1,200 become Rs. 1,348.32 in 2 years?

Accepted Answer

Introduction / Context: Here you must find the compound interest rate that turns a given principal into a known amount over 2 years. This is a typical reverse compound interest problem, where time and both principal and final amount are given, and the rate is unknown. Given Data / Assumptions: Principal P = Rs. 1,200. Amount after 2 years A = Rs. 1,348.32. Time t = 2 years. Compounding is annual. We need to find the annual rate of compound interest r%. Concept / Approach: For annual compounding: A = P * (1 + r/100)^t. Here t = 2, so: A / P = (1 + r/100)^2. We can compute the ratio A / P, take the square root to find (1 + r/100), then subtract 1 and convert to a percentage. Step-by-Step Solution: Step 1: Compute A / P. A / P = 1,348.32 / 1,200. A / P = 1.1236. Step 2: Relate this to the rate. (1 + r/100)^2 = 1.1236. Take the square root: 1 + r/100 = √1.1236 ≈ 1.06. Step 3: Solve for r. 1 + r/100 ≈ 1.06 → r/100 ≈ 0.06 → r ≈ 6%. Verification / Alternative check: Check the obtained rate by recomputing the amount. With r = 6%: A = 1,200 * (1.06)^2 = 1,200 * 1.1236 = Rs. 1,348.32. This matches the given amount exactly, confirming that 6% per annum is the correct compound interest rate. Why Other Options Are Wrong: At 5.5%, the factor is (1.055)^2 ≈ 1.1130, giving an amount lower than 1,348.32. At 6.5% or 7%, the factors (1.065)^2 and (1.07)^2 would yield amounts higher than the target. Only 6% gives the precise final amount specified. Common Pitfalls: Some candidates mistakenly divide the total percentage increase by two, treating the growth as simple interest. Others forget to take the square root when solving (1 + r/100)^2. Always use the correct compound interest relation and algebraic steps when solving for the rate. Final Answer: The required annual compound interest rate is 6% per annum.

Question 16

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

Accepted Answer

Introduction / Context: This question connects a simple interest calculation on an unknown principal with a compound interest calculation on a known principal. The key is to first find the compound interest on Rs. 16,000 at 10% for 2 years, then use half of that amount as the simple interest over 3 years to solve for the unknown principal. Given Data / Assumptions: Simple interest (SI) is charged on an unknown principal P for 3 years at 8% per annum. Compound interest (CI) is calculated on Rs. 16,000 for 2 years at 10% per annum, compounded annually. SI on P for 3 years at 8% equals half of this CI. We must find the principal P placed on simple interest. Concept / Approach: First, compute the compound interest on Rs. 16,000 for 2 years at 10%: A = 16,000 * (1.10)^2, CI = A − 16,000. Then take half of this CI; that value equals the simple interest on P: SI = P * 8 * 3 / 100. Equate SI to half the CI and solve for P. Step-by-Step Solution: Step 1: Compute CI on Rs. 16,000 at 10% p.a. for 2 years. A = 16,000 * (1.10)^2 = 16,000 * 1.21 = Rs. 19,360. CI = 19,360 − 16,000 = Rs. 3,360. Step 2: Find half of this CI. Half of CI = 3,360 / 2 = Rs. 1,680. Step 3: Use simple interest formula for P. SI on P for 3 years at 8% = P * 8 * 3 / 100 = 0.24P. Given 0.24P = 1,680. P = 1,680 / 0.24 = 7,000. Verification / Alternative check: Check SI with P = 7,000. SI = 7,000 * 8 * 3 / 100 = 7,000 * 0.24 = 1,680. This matches half of the CI (3,360), so the condition in the question is satisfied exactly. Why Other Options Are Wrong: If P were 3,500, SI would be only 840. For P = 5,600, SI would be 1,344. For P = 14,000, SI would be 3,360, which equals the full CI, not half. Only P = 7,000 gives SI equal to half of the compound interest. Common Pitfalls: A frequent mistake is forgetting that the relation is with half the CI, not the full CI. Another is mixing up the rates or time periods between the SI and CI parts of the problem. Keep the two computations clearly separated, then connect them through the equality given in the problem statement. Final Answer: The sum placed on simple interest is Rs. 7,000.

Question 17

The difference between the simple interest and the compound interest (compounded annually) on a certain sum of money at 12% per annum for 2 years is Rs. 72. What is the value of the original principal sum (in rupees)?

Accepted Answer

Introduction:This question compares simple interest and compound interest over the same period and rate and gives you the difference between them. Using a known formula for the difference between compound interest and simple interest over two years, you can directly compute the principal without having to find the interests separately. Given Data / Assumptions: Difference between compound interest and simple interest for 2 years = Rs. 72. Rate of interest r = 12% per annum. Time t = 2 years. Interest is compounded annually for CI. Principal P is unknown. Concept / Approach:For the same principal, rate, and time, the difference between compound interest and simple interest for 2 years has a neat formula:CI - SI = P * (r^2) / 100^2This arises from expanding the compound interest formula for 2 years and comparing it with the simple interest expression. Here r is in percent and P is principal. Step-by-Step Solution:Step 1: Write the formula for the difference over 2 years: CI - SI = P * r^2 / 100^2.Step 2: Substitute known values: CI - SI = 72 and r = 12%.Step 3: Compute r^2: 12^2 = 144.Step 4: Therefore 72 = P * 144 / 10,000.Step 5: Simplify the fraction: 144 / 10,000 = 0.0144.Step 6: So 72 = 0.0144 * P.Step 7: Solve for P by dividing: P = 72 / 0.0144.Step 8: Compute: P = 5,000.Hence the principal sum is Rs. 5,000. Verification / Alternative check:We can verify by calculating SI and CI directly. Simple interest for 2 years: SI = (P * r * t) / 100 = (5,000 * 12 * 2) / 100 = 1,200. Compound amount after 2 years: A = 5,000 * (1.12)^2 = 5,000 * 1.2544 = 6,272. So CI = 6,272 - 5,000 = 1,272. The difference CI - SI = 1,272 - 1,200 = 72, which matches the given value. Why Other Options Are Wrong:If P were 10,000, the difference would be double 72, that is 144. For 20,000, the difference would be four times 72, and so on. The options 8,000, 10,000, 15,000, and 20,000 do not yield a difference of exactly Rs. 72 when using the formula, so they cannot be correct. Common Pitfalls:Many students attempt to compute SI and CI separately without using the shortcut formula, which is more error prone. Others forget the square on the rate in the formula and use r instead of r^2. Always remember that for 2 years, CI - SI depends on r squared and not just r. Carefully substituting values and simplifying avoids mistakes. Final Answer:The original principal sum for which the difference between simple and compound interest is Rs. 72 at 12% per annum for 2 years is Rs. 5,000.

Question 18

The difference between simple interest and compound interest, both calculated annually on the same sum of money for 2 years at 4% per annum, is Rs. 1. What is the principal sum?

Accepted Answer

Introduction / Context: This classic aptitude question tests the relationship between simple interest and compound interest over a short period. For two years, there is a well known formula for the difference between compound interest and simple interest at the same rate and on the same principal. Using this shortcut saves time in exams. Given Data / Assumptions: Time period n = 2 years. Rate of interest r = 4% per annum. Difference between compound interest and simple interest = Rs. 1. Interest is compounded annually for the compound interest calculation. Principal sum P is unknown and needs to be found. Concept / Approach: For 2 years, the difference between compound interest and simple interest on the same principal P at rate r% per annum is given by: Difference = P * (r^2) / 100^2 Here r is in percentage form. This formula comes from expanding the compound interest expression for 2 years and subtracting the simple interest amount. Step-by-Step Solution: Step 1: Write the formula: Difference = P * r^2 / 10000. Step 2: Substitute known values: 1 = P * 4^2 / 10000. Step 3: Simplify the square: 4^2 = 16, so 1 = P * 16 / 10000. Step 4: Rearrange for P: P = 1 * 10000 / 16. Step 5: Compute P = 10000 / 16 = 625. Therefore, the principal sum is Rs. 625. Verification / Alternative Check: Compute simple interest: SI = P * r * n / 100 = 625 * 4 * 2 / 100 = Rs. 50. Compute amount with compound interest: A = 625 * (1 + 0.04)^2 = 625 * 1.0816 = Rs. 676. Compound interest CI = A - P = 676 - 625 = Rs. 51. Difference CI - SI = 51 - 50 = Rs. 1, which matches the given data. Why Other Options Are Wrong: Options A, B, and C (Rs. 620, Rs. 630, Rs. 640) do not satisfy the difference of exactly Rs. 1 when you compute the interests. Option E (Rs. 600) also produces a different difference between CI and SI. Only Rs. 625 leads to a difference of one rupee. Common Pitfalls: Students sometimes try to compute full SI and CI amounts repeatedly without using the shortcut formula, which is time consuming. Others may mistakenly use r / 100 instead of r^2 / 100^2 in the formula for difference. Always remember that the difference for 2 years depends on the square of the rate. Final Answer: The principal sum of money is Rs. 625.

Question 19

At what rate of simple interest, expressed as percent per annum, will a sum of money double itself in 8 years?

Accepted Answer

Introduction / Context: This question uses the standard simple interest formula to find the rate of interest required for a sum to double in a given period. Time value of money concepts like this are common in banking and finance aptitude problems. Knowing how to quickly determine such a rate is very useful for exam speed and for understanding basic financial planning ideas. Given Data / Assumptions: We have a principal sum of money, say P rupees. The sum doubles in 8 years, so the final amount after 8 years is 2P. Interest is calculated on a simple interest basis, not compounded. We need to find the rate of interest per annum r, expressed as a percentage. Concept / Approach: Under simple interest, the amount A after time t years at rate r percent per annum on principal P is given by A = P + (P * r * t) / 100. If the sum doubles, then A = 2P. This condition gives us an equation in r and t. Since t is known (8 years), we can solve directly for r. The calculation cancels P, so the actual size of the principal does not matter. Step-by-Step Solution: Let principal be P and rate be r percent per annum. Under simple interest, amount A after t years is A = P + (P * r * t) / 100. Here t = 8 years and A = 2P, because the sum doubles. So 2P = P + (P * r * 8) / 100. Subtract P from both sides: 2P - P = (P * r * 8) / 100. This gives P = (P * r * 8) / 100. Cancel P on both sides (P is not zero), giving 1 = (r * 8) / 100. Therefore r * 8 = 100, so r = 100 / 8. Hence r = 12.5 percent per annum. Verification / Alternative check: We can test the answer with assumed principal P = 100. At 12.5 percent per annum, simple interest per year is 12.5 rupees. Over 8 years, total interest is 8 * 12.5 = 100 rupees. So the amount becomes 100 + 100 = 200, which is double the principal, confirming that 12.5 percent is correct. Why Other Options Are Wrong: Option 13.5 percent would produce total interest of 108 on a principal of 100 over 8 years, resulting in amount 208, which is more than double. Option 11.5 percent would give total interest of 92 on a principal of 100, resulting in amount 192, which is less than double. Option 14.5 percent would yield even higher total interest of 116 on principal 100, giving amount 216, again not equal to exactly double. Common Pitfalls: Some learners mistakenly apply compound interest formulas, which are not needed here, since the problem clearly refers to simple interest. Others forget to set the amount equal to 2P when a sum doubles and instead set the interest equal to 2P, which is incorrect. A few students also neglect to cancel the principal P early, which can make the equation look more complicated than it actually is. Final Answer: The required simple interest rate is 12.5 percent per annum.

Question 20

A sum of money at simple interest amounts to Rs. 815 in 3 years and Rs. 854 in 4 years. What is the original principal sum?

Accepted Answer

Introduction / Context: This question uses the basic properties of simple interest to find the original principal when we know the amount at two different times. The key observation is that under simple interest, the interest added each year is constant. Therefore, the difference between the amounts at two times directly gives the interest for the additional period, which we can use to find the yearly interest and then the principal. Given Data / Assumptions: Amount after 3 years = Rs. 815. Amount after 4 years = Rs. 854. Interest is simple, not compound. We need to find the principal (original sum) P. Concept / Approach: Under simple interest, interest per year is constant, so the difference in amounts between successive years equals the interest for one year. Once we know yearly interest I, we can write the amount after 3 years as A3 = P + 3I and solve for P. This approach avoids directly using the rate of interest, although it can be found if needed. The method is fast and standard for such questions. Step-by-Step Solution: Let the yearly simple interest be I rupees. Amount after 3 years A3 = P + 3I = 815. Amount after 4 years A4 = P + 4I = 854. Subtract the two equations: A4 - A3 = (P + 4I) - (P + 3I) = I. So I = 854 - 815 = 39 rupees per year. Now substitute I = 39 into P + 3I = 815. Then P + 3 * 39 = 815. Compute 3 * 39 = 117. So P + 117 = 815. Therefore P = 815 - 117 = 698. Verification / Alternative check: Check amounts using principal P = 698 and yearly interest I = 39. After 3 years, amount A3 = 698 + 3 * 39 = 698 + 117 = 815, which matches the given value. After 4 years, amount A4 = 698 + 4 * 39 = 698 + 156 = 854, also matching the problem statement. This confirms that the principal 698 rupees is correct. Why Other Options Are Wrong: Option 650 would require a different yearly interest to reach 815 and 854, which would not stay constant per year as required by simple interest. Option 690 or 700 similarly would not produce the exact given amounts in 3 and 4 years when combined with a single fixed yearly interest value. Only 698 satisfies both conditions A3 = 815 and A4 = 854 with constant annual interest. Common Pitfalls: Some learners try to find the interest rate directly using the formula, which is not necessary and can introduce fractions early in the solution. Others mistakenly average the two amounts instead of taking their difference to find yearly interest. A common error is using compound interest logic, where the increase from 3 to 4 years would not be constant, which is not appropriate here. Final Answer: The original principal sum is Rs. 698.

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