Q: Find the rate percent from mixed BD and TD data: The banker’s discount on a sum for 1.5 years is ₹60, and the true discount on the same sum for 2 years is ₹75. Determine the annual simple interest rate.

Introduction / Context:This problem combines banker’s discount (computed on the face value) and true discount (computed via present worth) for different times on the same sum to identify the simple interest rate r. It showcases how BD and TD connect through r and t. Given Data / Assumptions: BD for t1 = 1.5 years is ₹60 TD for t2 = 2 years is ₹75 Same face value A and same annual rate r Concept / Approach:Formulas: BD = A * r * t TD = A * (r * t) / (1 + r * t) Step-by-Step Solution: From BD: 60 = A * r * 1.5 ⇒ A r = 40 From TD: 75 = A * (2r) / (1 + 2r) Substitute A(2r) = 80 ⇒ 75 = 80 / (1 + 2r) 1 + 2r = 80/75 = 16/15 ⇒ 2r = 1/15 ⇒ r = 1/30 = 3 1/3 % Verification / Alternative check:Check BD at 1.5 y: A r = 40, BD = 40*1.5 = 60 (ok). TD at 2 y: numerator 80, denominator 1 + 2r = 16/15, TD = 80 / (16/15) = 75 (ok). Why Other Options Are Wrong:Other rates do not satisfy both BD and TD equations simultaneously with the same A. Common Pitfalls:Forgetting the (1 + r t) divisor in TD or mixing BD and TD definitions. Keep r as a per-year fraction throughout. Final Answer:31 / 3%

Q: Banker’s gain on a long-dated bill: The face value of a bill due 5 years hence is ₹13800. At 5% per annum simple interest, compute the banker’s gain on discounting the bill.

Introduction / Context:Banker’s Gain (BG) is the difference between the Banker’s Discount (BD) and True Discount (TD). It measures the extra advantage to the banker arising because BD is calculated on the face value whereas the true loss to the holder is TD on present worth. Given Data / Assumptions: A = ₹13800 t = 5 years r = 5% = 0.05 per annum Concept / Approach:Useful identities: BD = A * r * t TD = A * (r * t) / (1 + r * t) BG = BD − TD = A * (r * t)^2 / (1 + r * t) Step-by-Step Solution: r t = 0.05 * 5 = 0.25 BG = 13800 * (0.25)^2 / (1 + 0.25) BG = 13800 * 0.0625 / 1.25 = 13800 * 0.05 = ₹690 Verification / Alternative check:Compute BD = 13800 * 0.25 = ₹3450; PW = A/(1 + 0.25) = 13800/1.25 = ₹11040; TD = A − PW = ₹2760; BG = 3450 − 2760 = ₹690 (consistent). Why Other Options Are Wrong:Other values do not match BG = A*(r t)^2/(1 + r t) using r t = 0.25. Common Pitfalls:Confusing BD with BG, or computing TD incorrectly by omitting the (1 + r t) term in the denominator. Final Answer:₹ 690

Q: Find banker’s discount from true discount: The true discount on a bill due 24 months hence at 24% per annum is ₹144. What is the banker’s discount for the same bill?

Introduction / Context:Given TD and the time/rate, we can back out the face value and then compute BD directly. TD is on present worth; BD is on face value for the same time. Given Data / Assumptions: t = 24 months = 2 years r = 24% = 0.24 per annum TD = ₹144 Concept / Approach:Use: TD = A * (r * t) / (1 + r * t) BD = A * r * t Step-by-Step Solution: Let x = r t = 0.24 * 2 = 0.48 TD = A * 0.48 / 1.48 = 144 ⇒ A = 144 * 1.48 / 0.48 = ₹444 BD = A * x = 444 * 0.48 = ₹213.12 Verification / Alternative check:Compute PW = A − TD = 444 − 144 = ₹300. Interest on PW for 2 years at 24% is 300 * 0.24 * 2 = ₹144 (matches TD). Why Other Options Are Wrong:They do not satisfy both TD and BD relations simultaneously with the same A and x = 0.48. Common Pitfalls:Using TD = A * r * t (incorrect), or mixing A with PW during substitutions. Final Answer:₹ 213.12

Q: Compute banker’s discount and banker’s gain: The present worth of a bill is ₹2200 and the true discount is ₹220. Find the banker’s discount and the banker’s gain.

Introduction / Context:From present worth (PW) and true discount (TD) we can find face value and r t. Then BD and BG follow mechanically. A key identity: TD / PW = r t. Given Data / Assumptions: PW = ₹2200 TD = ₹220 A = PW + TD = ₹2420 Concept / Approach:Use: r t = TD / PW BD = A * r * t BG = BD − TD Step-by-Step Solution: r t = 220 / 2200 = 0.10 BD = 2420 * 0.10 = ₹242 BG = BD − TD = 242 − 220 = ₹22 Verification / Alternative check:TD = A * (r t) / (1 + r t) = 2420 * 0.10 / 1.10 = ₹220 (checks). Why Other Options Are Wrong:They mis-order BG and BD magnitudes or break the relation TD/PW = r t = 0.10. Common Pitfalls:Using face value instead of PW when computing r t from TD, or swapping BG and BD definitions. Final Answer:BG = ₹ 22 and BD = ₹ 242

Question 1

The present worth (PW) of a bill is ₹800 and the true discount (TD) is ₹36. Determine the banker’s discount (BD).

Accepted Answer

Introduction / Context:Given PW and TD, compute x from TD = x * PW. Then use BD = (1 + x) * TD to find banker’s discount. This method remains valid irrespective of the underlying rate and time split, as long as simple-interest discounting is assumed. Given Data / Assumptions: PW = ₹800. TD = ₹36. Concept / Approach:Compute x = TD / PW. Then BD = (1 + x) * TD = TD + x * TD. This yields a small increment over TD equal to BG = x * TD. Step-by-Step Solution: x = 36 / 800 = 0.045.BD = (1 + 0.045) * 36 = 1.045 * 36 = ₹37.62. Verification / Alternative check:BG = BD − TD = 37.62 − 36 = ₹1.62. Also BG = x * TD = 0.045 * 36 = ₹1.62; consistent. Why Other Options Are Wrong:₹37, ₹34.38, and ₹38.98 do not equal 1.045 * 36. Common Pitfalls:Using BD = PW * x (incorrect; that gives TD). BD multiplies face value, not PW. Final Answer:Rs. 37.62

Question 2

The true discount (TD) on a bill of ₹540 is ₹90. Compute the banker’s discount (BD).

Accepted Answer

Introduction / Context:Face value F and TD are known; we can find x from TD = F * x / (1 + x) and then compute BD via BD = F * x or BD = (1 + x) * TD. This is a straightforward algebraic step. Given Data / Assumptions: F = ₹540. TD = ₹90. Concept / Approach:Use the TD identity to solve for x, then compute BD = F * x. This avoids finding r and t separately. Step-by-Step Solution: 90(1 + x) = 540x ⇒ 90 = 450x ⇒ x = 0.2.BD = F * x = 540 * 0.2 = ₹108. Verification / Alternative check:BD = (1 + x)*TD = 1.2 * 90 = ₹108; consistent. Why Other Options Are Wrong:₹60, ₹110, ₹150 do not satisfy the derived x = 0.2 simultaneously with TD = 90. Common Pitfalls:Confusing TD and BD or forgetting that BD exceeds TD by BG = x * TD for x > 0. Final Answer:Rs. 108

Question 3

The banker’s discount (BD) and the true discount (TD) on a sum due 8 months hence are ₹52 and ₹50, respectively. Find (i) the sum due (face value) and (ii) the annual rate percent.

Accepted Answer

Introduction / Context:Given BD and TD for the same time and rate, their ratio gives x = r * t. With t known (in years), r follows. Then the face value F can be computed from BD = F * x. This chain uses only standard discounting identities. Given Data / Assumptions: BD = ₹52, TD = ₹50. Time = 8 months = 8/12 = 2/3 year. Simple-interest bill-discounting setting. Concept / Approach:Use BD/TD = 1 + x ⇒ x = BD/TD − 1. Then r = x / t. Next, BD = F * x gives F. Finally sanity-check TD via TD = F * x / (1 + x). Step-by-Step Solution: BD/TD = 52/50 = 26/25 ⇒ x = 1/25 = 0.04.t = 2/3 year ⇒ r = x / t = 0.04 / (2/3) = 0.06 = 6% p.a.F = BD / x = 52 / 0.04 = ₹1300. Verification / Alternative check:Compute TD: TD = F * x / (1 + x) = 1300 * 0.04 / 1.04 = 52 / 1.04 = ₹50, consistent. Why Other Options Are Wrong:Other pairs do not satisfy both BD/TD = 26/25 and BD = F * x with t = 8 months. Common Pitfalls:Failing to convert months to years; mixing BD and BG. Final Answer:Rs. 1300, 6%

Question 4

Banker’s Discount equals True Discount: The banker's discount on ₹1600 at 6% per annum is the same as the true discount on ₹1624 for the same time and rate. Determine the time period (express your answer in months).

Accepted Answer

Introduction / Context:In banking arithmetic, three linked quantities appear for a bill due after time t at simple interest rate r: Banker's Discount (BD), True Discount (TD), and Banker's Gain (BG). For face value A, BD is the simple interest on A for the remaining time; TD is the reduction needed so that the present worth grows to A in time t at rate r. Here, BD on ₹1600 equals TD on ₹1624 at the same r and t. We must find t in months. Given Data / Assumptions: Face values: A1 = ₹1600, A2 = ₹1624 Rate r = 6% per annum (0.06) Time = t years (unknown) Equality: BD(A1) = TD(A2) Concept / Approach:Formulas (simple interest): BD = A * r * t TD = A * (r * t) / (1 + r * t) Step-by-Step Solution: BD on ₹1600: 1600 * 0.06 * t = 96 t TD on ₹1624: 1624 * (0.06 t)/(1 + 0.06 t) = (97.44 t)/(1 + 0.06 t) Set equal: 96 t = (97.44 t)/(1 + 0.06 t) 96 t (1 + 0.06 t) = 97.44 t 96 t + 5.76 t^2 = 97.44 t 5.76 t^2 = 1.44 t ⇒ t(5.76 t − 1.44) = 0 t = 0 or t = 1.44/5.76 = 0.25 years = 3 months Verification / Alternative check:If t = 3 months, r t = 0.06 * 0.25 = 0.015. Then BD(1600) = 1600 * 0.015 = ₹24, TD(1624) = 1624 * 0.015 / 1.015 ≈ ₹24.00. Values match. Why Other Options Are Wrong:4, 6, or 8 months yield r t values that make BD and TD unequal for the given amounts. Common Pitfalls:Confusing TD with BD or using TD = A * r * t (incorrect). TD must divide by (1 + r t). Also ensure t is in years before converting to months. Final Answer:3 months

Question 5

Effective annual rate from a banker’s discount: A man discounts a bill due 10 months hence by deducting 4% of the face value upfront. What annual simple interest rate does he effectively get on the money he advances?

Accepted Answer

Introduction / Context:When a bill is discounted by deducting a percentage of the face value, the investor’s true yield is not computed on the face amount but on the present worth actually paid out. This question asks for the effective annual rate when 4% of the face is deducted for 10 months. Given Data / Assumptions: Time t = 10 months = 10/12 years Deduction (Banker’s Discount) = 4% of face value A Present Worth (PW) paid by the man = A − 0.04A = 0.96A Concept / Approach:Effective rate per annum is computed as (Interest earned) / (Money invested) per year. Interest here equals the deducted amount 0.04A over 10 months; invested money equals PW = 0.96A. Rate for 10 months = (0.04A) / (0.96A) = 0.041666... Annual rate = (Rate for 10 months) / (10/12) = (1/24) / (5/6) = 1/20 = 0.05 = 5% Step-by-Step Solution: PW = 0.96A Interest over 10 months = 0.04A 10-month yield = 0.04/0.96 = 1/24 Annualize: (1/24) * (12/10) = 1/20 = 5% Verification / Alternative check:If the quoted “rate” had been applied to the face value (BD = A * r * t), nominal r would be 4.8% p.a., but the investor’s yield must be on PW, hence 5% p.a. Why Other Options Are Wrong:4% is the deduction on face, not the annual yield. 6% and 8% overstate the effective rate from this discounting structure. Common Pitfalls:Using the face value instead of present worth as the base for the investor’s return, or forgetting to annualize from 10 months. Final Answer:5%

Question 6

Find the discounting date from proceeds: A banker pays ₹5767.50 for a bill of ₹5840 drawn on April 4 at 6 months. At 7% per annum simple interest, on which date was the bill discounted?

Accepted Answer

Introduction / Context:Discounting a bill means the banker deducts interest (banker’s discount) from the face value for the time remaining until the due date. From the proceeds (cash paid), we can infer the time between discount date and due date and hence the calendar date of discount. Given Data / Assumptions: Face value A = ₹5840 Proceeds (cash paid) = ₹5767.50 ⇒ BD = A − Proceeds = ₹72.50 Rate r = 7% p.a. Term = 6 months from April 4; customary 3 days of grace ⇒ due about October 7 Concept / Approach:Banker’s Discount: BD = A * r * t, for t (in years) from discount date to due date. Solve for t, convert to days, then back-date from October 7 to find the discount date. Step-by-Step Solution: BD = 72.50 = 5840 * 0.07 * t t = 72.50 / (5840 * 0.07) ≈ 0.17735 years Days ≈ 0.17735 * 365 ≈ 65 days (approx.) Due date ≈ Oct 7 ⇒ discount date ≈ Oct 7 − 65 days ≈ Aug 3 Verification / Alternative check:Using 365-day basis gives near-exact 65 days; counting back from Oct 7 lands on about Aug 3. Small convention differences (360-day basis) shift by < 1 day, still pointing to the first days of August. Why Other Options Are Wrong:Aug 4 or September dates imply fewer/more days than supported by BD = ₹72.50 at 7% and are inconsistent with the computed interval. Common Pitfalls:Omitting days of grace when used in such problems, forgetting to compute t from BD on the face value, or mixing up due date vs. term date. Final Answer:3rd Aug.

Question 7

Find the rate percent from mixed BD and TD data: The banker’s discount on a sum for 1.5 years is ₹60, and the true discount on the same sum for 2 years is ₹75. Determine the annual simple interest rate.

Accepted Answer

Introduction / Context:This problem combines banker’s discount (computed on the face value) and true discount (computed via present worth) for different times on the same sum to identify the simple interest rate r. It showcases how BD and TD connect through r and t. Given Data / Assumptions: BD for t1 = 1.5 years is ₹60 TD for t2 = 2 years is ₹75 Same face value A and same annual rate r Concept / Approach:Formulas: BD = A * r * t TD = A * (r * t) / (1 + r * t) Step-by-Step Solution: From BD: 60 = A * r * 1.5 ⇒ A r = 40 From TD: 75 = A * (2r) / (1 + 2r) Substitute A(2r) = 80 ⇒ 75 = 80 / (1 + 2r) 1 + 2r = 80/75 = 16/15 ⇒ 2r = 1/15 ⇒ r = 1/30 = 3 1/3 % Verification / Alternative check:Check BD at 1.5 y: A r = 40, BD = 40*1.5 = 60 (ok). TD at 2 y: numerator 80, denominator 1 + 2r = 16/15, TD = 80 / (16/15) = 75 (ok). Why Other Options Are Wrong:Other rates do not satisfy both BD and TD equations simultaneously with the same A. Common Pitfalls:Forgetting the (1 + r t) divisor in TD or mixing BD and TD definitions. Keep r as a per-year fraction throughout. Final Answer:31 / 3%

Question 8

Banker’s gain on a long-dated bill: The face value of a bill due 5 years hence is ₹13800. At 5% per annum simple interest, compute the banker’s gain on discounting the bill.

Accepted Answer

Introduction / Context:Banker’s Gain (BG) is the difference between the Banker’s Discount (BD) and True Discount (TD). It measures the extra advantage to the banker arising because BD is calculated on the face value whereas the true loss to the holder is TD on present worth. Given Data / Assumptions: A = ₹13800 t = 5 years r = 5% = 0.05 per annum Concept / Approach:Useful identities: BD = A * r * t TD = A * (r * t) / (1 + r * t) BG = BD − TD = A * (r * t)^2 / (1 + r * t) Step-by-Step Solution: r t = 0.05 * 5 = 0.25 BG = 13800 * (0.25)^2 / (1 + 0.25) BG = 13800 * 0.0625 / 1.25 = 13800 * 0.05 = ₹690 Verification / Alternative check:Compute BD = 13800 * 0.25 = ₹3450; PW = A/(1 + 0.25) = 13800/1.25 = ₹11040; TD = A − PW = ₹2760; BG = 3450 − 2760 = ₹690 (consistent). Why Other Options Are Wrong:Other values do not match BG = A*(r t)^2/(1 + r t) using r t = 0.25. Common Pitfalls:Confusing BD with BG, or computing TD incorrectly by omitting the (1 + r t) term in the denominator. Final Answer:₹ 690

Question 9

Find banker’s discount from true discount: The true discount on a bill due 24 months hence at 24% per annum is ₹144. What is the banker’s discount for the same bill?

Accepted Answer

Introduction / Context:Given TD and the time/rate, we can back out the face value and then compute BD directly. TD is on present worth; BD is on face value for the same time. Given Data / Assumptions: t = 24 months = 2 years r = 24% = 0.24 per annum TD = ₹144 Concept / Approach:Use: TD = A * (r * t) / (1 + r * t) BD = A * r * t Step-by-Step Solution: Let x = r t = 0.24 * 2 = 0.48 TD = A * 0.48 / 1.48 = 144 ⇒ A = 144 * 1.48 / 0.48 = ₹444 BD = A * x = 444 * 0.48 = ₹213.12 Verification / Alternative check:Compute PW = A − TD = 444 − 144 = ₹300. Interest on PW for 2 years at 24% is 300 * 0.24 * 2 = ₹144 (matches TD). Why Other Options Are Wrong:They do not satisfy both TD and BD relations simultaneously with the same A and x = 0.48. Common Pitfalls:Using TD = A * r * t (incorrect), or mixing A with PW during substitutions. Final Answer:₹ 213.12

Question 10

Compute banker’s discount and banker’s gain: The present worth of a bill is ₹2200 and the true discount is ₹220. Find the banker’s discount and the banker’s gain.

Accepted Answer

Introduction / Context:From present worth (PW) and true discount (TD) we can find face value and r t. Then BD and BG follow mechanically. A key identity: TD / PW = r t. Given Data / Assumptions: PW = ₹2200 TD = ₹220 A = PW + TD = ₹2420 Concept / Approach:Use: r t = TD / PW BD = A * r * t BG = BD − TD Step-by-Step Solution: r t = 220 / 2200 = 0.10 BD = 2420 * 0.10 = ₹242 BG = BD − TD = 242 − 220 = ₹22 Verification / Alternative check:TD = A * (r t) / (1 + r t) = 2420 * 0.10 / 1.10 = ₹220 (checks). Why Other Options Are Wrong:They mis-order BG and BD magnitudes or break the relation TD/PW = r t = 0.10. Common Pitfalls:Using face value instead of PW when computing r t from TD, or swapping BG and BD definitions. Final Answer:BG = ₹ 22 and BD = ₹ 242

Question 11

Find the true discount from banker’s gain and present worth: The present worth of a sum due sometime hence is ₹576 and the banker’s gain is ₹9. Determine the true discount.

Accepted Answer

Introduction / Context:Banker’s Gain (BG) links to True Discount (TD) and r t via BG = TD * r t = PW * (r t)^2. Given PW and BG, we can find r t and then compute TD = PW * r t. Given Data / Assumptions: PW = ₹576 BG = ₹9 Concept / Approach:Use: BG = PW * (r t)^2 TD = PW * r t Step-by-Step Solution: Let x = r t. Then 9 = 576 * x^2 ⇒ x^2 = 9/576 = 1/64 ⇒ x = 1/8 TD = PW * x = 576 * (1/8) = ₹72 Verification / Alternative check:Since TD = PW * x = 72, BG should be TD * x = 72 * (1/8) = 9 (matches). Why Other Options Are Wrong:Other numbers do not satisfy the squared relation between BG, PW and x = r t. Common Pitfalls:Taking x as 9/576 instead of its square root, or mixing BD with BG/TD relations. Final Answer:₹ 72

Question 12

Rate when TD is a multiple of BG (numerically equal r and t): If the rate of interest (percent per annum) and the time (in years) are numerically equal, and the true discount is 81 times the banker’s gain, find the rate percent.

Accepted Answer

Introduction / Context:We relate True Discount (TD) and Banker’s Gain (BG) via BG = TD * r t. The phrase “rate and time numerically equal” is interpreted as r% (as a number) equals t (in years). This creates a specific product r t for solving the ratio TD : BG. Given Data / Assumptions: TD is 81 times BG ⇒ TD / BG = 81 Let the numeric value n satisfy: rate r% = n and time t = n years Thus r (as a fraction) = n/100 and r t = (n/100) * n = n^2 / 100 Concept / Approach:The identity TD / BG = 1 / (r t) gives a direct link to n. Step-by-Step Solution: TD / BG = 1 / (r t) = 81 ⇒ r t = 1/81 n^2 / 100 = 1/81 ⇒ n^2 = 100/81 ⇒ n ≈ 10/9 Therefore, rate percent r ≈ 10/9 % ≈ 1.11% Verification / Alternative check:If n = 10/9, r t = 1/81, so BG = TD * r t = TD / 81, consistent with the statement. Rounding to the closest option gives 11/9 %. Why Other Options Are Wrong:12/9 %, 17/9 %, 29/13 % imply different r t values that would not satisfy TD / BG = 81 under the given “numerically equal” condition. Common Pitfalls:Confusing r (%) with r as a fraction; the problem’s “numerically equal” clause must be handled exactly to avoid a ten-fold error. Final Answer:11/9 %

Question 13

Convert true discount to banker’s discount: The true discount on a certain sum due 1 year hence at 30% per annum is ₹240. What is the banker’s discount for the same time and rate?

Accepted Answer

Introduction / Context:With TD known, recover the face value A and then compute BD. This is a straightforward application of TD and BD formulas for simple interest over one year. Given Data / Assumptions: t = 1 year r = 30% = 0.30 per annum TD = ₹240 Concept / Approach: TD = A * (r * t) / (1 + r * t) BD = A * r * t Step-by-Step Solution: Let x = r t = 0.30 240 = A * 0.30 / 1.30 ⇒ A = 240 * 1.30 / 0.30 = ₹1040 BD = A * x = 1040 * 0.30 = ₹312 Verification / Alternative check:PW = A − TD = 1040 − 240 = ₹800. Interest on ₹800 for 1 year at 30% is ₹240 (matches TD); BD = ₹312 as computed. Why Other Options Are Wrong:They contradict the consistent A, PW, TD trio defined by x = 0.30. Common Pitfalls:Using BD = TD or adding/subtracting incorrectly when recovering A from TD. Final Answer:₹ 312

Question 14

Effective annual rate from a 30-month deduction: Shantanu discounts a bill due 30 months hence by deducting 30% of the face value. What annual simple interest rate does he effectively get on his money?

Accepted Answer

Introduction / Context:When discount is a fraction of the face value for a known time, the investor’s effective rate is computed on the money actually laid out (the present worth), and must then be annualized to a per-year figure. Given Data / Assumptions: Deduction (BD) = 30% of A ⇒ BD = 0.30A Present Worth PW = A − BD = 0.70A Time t = 30 months = 2.5 years Concept / Approach:Effective annual rate R satisfies: R = (Interest / PW) per annum = (0.30A / 0.70A) / 2.5 Step-by-Step Solution: R = (3/7) / 2.5 = (3/7) * 0.4 = 1.2/7 ≈ 0.171428... As a percentage: ≈ 17.1428...% = 17 1/7 % Verification / Alternative check:Nominal rate on the face would be 0.30/2.5 = 12% p.a., but the effective yield is higher (≈ 17.14%) because the base is PW (70% of A), not A itself. Why Other Options Are Wrong:172/7%, 173/7%, 175/7% are larger than the exact value. 171/7% matches 17 1/7%. Common Pitfalls:Computing the rate on the face value instead of present worth, or forgetting to convert 30 months to 2.5 years when annualizing. Final Answer:171/7%

Question 15

Find rate when BG is a fraction of BD: The banker’s gain on a sum due 2.5 years hence is (3/23) of the banker’s discount. Find the annual simple interest rate.

Accepted Answer

Introduction / Context:Relating BG and BD directly yields r t. Then, dividing by t gives the annual rate. This pattern is common in banker’s discount questions. Given Data / Assumptions: BG / BD = 3/23 t = 2.5 years Concept / Approach:Identity: BG / BD = (r t) / (1 + r t) Step-by-Step Solution: Let x = r t. Then x/(1 + x) = 3/23 ⇒ 23x = 3 + 3x ⇒ 20x = 3 ⇒ x = 3/20 = 0.15 r = x / t = 0.15 / 2.5 = 0.06 = 6% per annum Verification / Alternative check:BG/BD = 0.15 / 1.15 = 3/23 (exact), confirming the calculation. Why Other Options Are Wrong:5% would imply x = 0.125; other unusual options do not satisfy the given BG/BD ratio for 2.5 years. Common Pitfalls:Using BD/BG instead of BG/BD or failing to isolate x properly. Final Answer:6%

Question 16

Find TD and BG from given BD: The banker’s discount on ₹1650 due a certain time hence is ₹165. Determine the true discount and the banker’s gain.

Accepted Answer

Introduction / Context:Knowing BD on a face amount directly gives r t because BD = A * r * t. Once r t is known, TD and BG follow immediately. Given Data / Assumptions: A = ₹1650 BD = ₹165 Concept / Approach: BD = A * r * t ⇒ r t = BD / A = 165 / 1650 = 0.10 TD = A * (r t) / (1 + r t) BG = BD − TD Step-by-Step Solution: TD = 1650 * 0.10 / 1.10 = ₹150 BG = 165 − 150 = ₹15 Verification / Alternative check:PW = A − TD = 1650 − 150 = ₹1500; interest on PW at r t = 0.10 equals ₹150 (the TD), confirming consistency. Why Other Options Are Wrong:They do not reflect r t = 0.10 implied by BD/A = 0.10. Common Pitfalls:Confusing TD with BD or computing TD without dividing by (1 + r t). Final Answer:Rs. 150, Rs. 15

Question 17

Rate from BD and TD (repeat form): The banker’s discount on a sum for 1.5 years is ₹60 and the true discount on the same sum for 2 years is ₹75. Find the rate percent.

Accepted Answer

Introduction / Context:This is the same structural setup as a prior item and leads to the same rate. We repeat the concise derivation for completeness and clarity. Given Data / Assumptions:BD(1.5 y) = ₹60, TD(2 y) = ₹75, face value A common, rate r unknown. Concept / Approach: BD = A * r * 1.5 ⇒ A r = 40 TD = A * (2r) / (1 + 2r) = 75 ⇒ A(2r) = 80 75 = 80 / (1 + 2r) ⇒ 1 + 2r = 16/15 ⇒ r = 1/30 = 3 1/3 % Final Answer:31/3%

Question 18

Difference between BD and TD for a short term: Find the difference between the banker’s discount and the true discount on ₹8100 for 3 months at 5% per annum.

Accepted Answer

Introduction / Context:The difference BD − TD equals Banker’s Gain (BG). A compact formula exists to compute BG directly in terms of A and r t, which is convenient for short periods. Given Data / Assumptions: A = ₹8100 t = 3 months = 1/4 year r = 5% = 0.05 per annum ⇒ r t = 0.0125 Concept / Approach: BG = A * (r t)^2 / (1 + r t) Step-by-Step Solution: BG = 8100 * (0.0125)^2 / (1.0125) BG = 8100 * 0.00015625 / 1.0125 = 1.265625 / 1.0125 ≈ ₹1.25 Verification / Alternative check:Compute BD = A * r t = 8100 * 0.0125 = ₹101.25; TD = BD − BG ≈ 101.25 − 1.25 = ₹100.00. Also TD = A * (r t) / (1 + r t) ≈ 8100 * 0.0125 / 1.0125 = ₹100 (matches). Why Other Options Are Wrong:They differ from the exact value arising from the squared term in BG's formula, which is small for short terms. Common Pitfalls:Forgetting BG is BD − TD, or using TD = A * r * t (incorrect). Final Answer:Rs. 1.25

Question 19

Find the face value from interest and discount on the same sum: The simple interest on a certain sum is ₹67.20 and the (true) discount on the same sum for the same time and rate is ₹60. Find the sum (face value).

Accepted Answer

Introduction / Context:There is a classic relation between simple interest (I) on a sum and the true discount (D) on the same sum for the same time and rate: the face value A equals (I * D) / (I − D). This follows from writing I and D in terms of A and r t and eliminating r t. Given Data / Assumptions:I = ₹67.20, D = ₹60, same time and rate for both measures on the same sum. Concept / Approach:Derivation outline: If I = A * x and D = A * x / (1 + x) with x = r t, then I − D = A * x^2 / (1 + x). Thus (I * D)/(I − D) = A. Step-by-Step Solution: A = (I * D) / (I − D) = (67.20 * 60) / (67.20 − 60) A = 4032 / 7.2 = ₹560 Verification / Alternative check:Compute x from either I/A or D/(A − D): I/A = 67.20/560 = 0.12; D/(A − D) = 60/500 = 0.12. Consistent. Why Other Options Are Wrong:They do not satisfy the identity A = (I * D)/(I − D) for the given I and D. Common Pitfalls:Interpreting “discount” as banker’s discount; here “discount” is the true discount unless stated otherwise in this relation. Final Answer:Rs. 560

Question 20

Find the face value from banker’s gain: The banker’s gain on a bill due 1 year 4 months hence at 7.5% per annum (simple interest) is ₹16. Find the face value of the bill.

Accepted Answer

Introduction / Context:Banker’s Gain depends on the square of r t. With BG and r t known, the face value A is obtained directly from BG = A * (r t)^2 / (1 + r t). Given Data / Assumptions: t = 1 year 4 months = 16 months = 4/3 years r = 7.5% = 0.075 per annum Thus r t = 0.075 * (4/3) = 0.10 BG = ₹16 Concept / Approach: BG = A * (r t)^2 / (1 + r t) ⇒ 16 = A * (0.10)^2 / 1.10 Step-by-Step Solution: 16 = A * 0.01 / 1.10 ⇒ A = 16 * 1.10 / 0.01 = 1760 Verification / Alternative check:BD = A * r t = 1760 * 0.10 = ₹176; TD = BD − BG = 176 − 16 = ₹160; TD = A * 0.10 / 1.10 = ₹160 (consistent). Final Answer:Rs. 1760

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