Question 1

The banker's discount on a sum due 2 years hence is 11/10 of the true discount. What is the annual rate of interest (simple)?

Accepted Answer

Given data Time t = 2 years; BD = (11÷10) × TD. Concept/Approach For simple interest, BD ÷ TD = 1 + r t where r is the annual rate (in decimals). Step-by-Step calculation 1 + r t = 11 ÷ 10 r × 2 = 1÷10 ⇒ r = 1÷20 = 0.05 Rate = 5% per annum Verification/Alternative True discount formula TD = S r t ÷ (1 + r t); Banker's discount BD = S r t. Their ratio is indeed 1 + r t. Common pitfalls Using compound interest relations; the problem uses simple discount/interest. Final Answer 5% per annum

Question 2

The banker's discount on ₹1600 at 15% p.a. equals the true discount on ₹1680 for the same time and rate. What is the time?

Accepted Answer

Given data BD on ₹1600 at r = 15% for time t equals TD on ₹1680 at same r and t. Concept/Approach Use BD = S r t and TD = S r t ÷ (1 + r t) (simple interest/discount). Step-by-Step calculation BD = 1600 × 0.15 × t = 240 t TD = 1680 × 0.15 × t ÷ (1 + 0.15 t) = 252 t ÷ (1 + 0.15 t) Set 240 t = 252 t ÷ (1 + 0.15 t) ⇒ 240 = 252 ÷ (1 + 0.15 t) 1 + 0.15 t = 252 ÷ 240 = 21 ÷ 20 = 1.05 0.15 t = 0.05 ⇒ t = 1÷3 year = 4 months Verification/Alternative At t = 1÷3, BD = 1600 × 0.15 × 1÷3 = ₹80; TD on 1680 = 1680 × 0.15 × 1÷3 ÷ (1 + 0.05) = 84 ÷ 1.05 = ₹80. Final Answer 4 months

Question 3

On a bill due 4 months hence at 15% p.a., the banker's discount is ₹420. What is the true discount?

Accepted Answer

Given data Time t = 4 months = 1÷3 year; rate r = 15% p.a. BD = ₹420. Concept/Approach BD = S r t ⇒ find face value S, then TD = S r t ÷ (1 + r t). Step-by-Step calculation BD = S × 0.15 × (1÷3) = S × 0.05 = 420 S = 420 ÷ 0.05 = ₹8400 TD = S × 0.05 ÷ (1 + 0.05) = 420 ÷ 1.05 = ₹400 Verification/Alternative Relation BD ÷ TD = 1 + r t = 1.05 ⇒ TD = BD ÷ 1.05 = 420 ÷ 1.05 = ₹400. Final Answer ₹400

Question 4

Present worth is Rs. 576 and banker's gain is Rs. 16. Find the true discount.

Accepted Answer

Given data Present worth (PW) = Rs. 576 Banker's gain (BG) = Rs. 16 True discount (TD) = ? Concept/Approach Key relation: BG = TD2 / PW ⇒ TD = √(BG × PW) Step-by-step calculation TD = √(16 × 576)= √(9216) = Rs. 96 Verification/Alternative BG should equal TD2 / PW = 962 / 576 = 9216 / 576 = 16 (matches). Common pitfalls Confusing banker's discount (BD) with banker's gain (BG); BG = BD − TD, not BD itself. Final Answer Rs. 96

Question 5

Banker's discount for 1½ years is Rs. 558 and true discount for 2 years is Rs. 600. Find the annual rate of interest.

Accepted Answer

Given data BD (1.5 years) = Rs. 558 TD (2 years) = Rs. 600 Rate r = ? per annum; Sum due (A) = unknown Concept/Approach Use definitions under simple interest:BD = A × r × t / 100TD = A × r × t / (100 + r t)Form two equations and eliminate A. Step-by-step calculation BD: A × r × 1.5 / 100 = 558 ⇒ A r = 558 × 100 / 1.5 = 37,200TD: A × (2r) / (100 + 2r) = 600 ⇒ 600(100 + 2r) = 2 A r600(100 + 2r) = 2 × 37,200 = 74,40060,000 + 1,200 r = 74,400 ⇒ r = 12% Verification/Alternative With r = 12%, TD (2y): A × 24 / (100 + 24) = A × 24/124 = 600 ⇒ A = 3,100; BD (1.5y): 3100 × 18/100 = 558 (checks). Common pitfalls Plugging BD formula for TD or mixing the times 1.5 and 2 incorrectly. Final Answer 12%

Question 6

The banker's gain on a sum due 1½ years hence is 3/25 of the banker's discount. Find the rate of interest per annum.

Accepted Answer

Given data Time t = 1½ years (assumption clarified from the wording) BG = (3/25) × BD Rate r = ? Concept/Approach Let x = r t (in percent-years). A standard relation is:BG / BD = x / (100 + x)because BD = A x/100 and BG = BD − TD with TD = A x/(100 + x). Step-by-step calculation x / (100 + x) = 3/2525x = 300 + 3x ⇒ 22x = 300 ⇒ x = 150/11r = x / t = (150/11) / 1.5 = (150/11) × (2/3) = 100/11% ≈ 9.09% Verification/Alternative Compute BG/BD with r = 100/11% and t = 1.5 ⇒ x = 150/11; BG/BD = x/(100+x) = (150/11) / (1250/11) = 3/25 (checks). Common pitfalls Using BG/TD instead of BG/BD. Note BG/TD = x/100 while BG/BD = x/(100 + x). Final Answer 9.09% per annum

Question 7

Present worth is Rs. 1600 and true discount is Rs. 160. Find the banker's gain.

Accepted Answer

Given data PW = Rs. 1600 TD = Rs. 160 BG = ? Concept/Approach Use the identity BG = TD2 / PW. Step-by-step calculation BG = 1602 / 1600 = 25,600 / 1600 = Rs. 16 Verification/Alternative Also, BD = TD + BG = 160 + 16 = 176; consistency checks can be made with the implied rate-time if needed. Common pitfalls Applying BG = BD − TD without first finding BD; the squared relation is faster here. Final Answer Rs. 16

Question 8

Banker's gain for a sum due 2 years hence at 10% p.a. is Rs. 24. Find the present worth.

Accepted Answer

Given data t = 2 years, r = 10% p.a. ⇒ r t = 20 BG = Rs. 24 PW = ? Concept/Approach Since TD / PW = r t / 100 = 20/100 = 0.2 we have TD = 0.2 × PW. Also BG = TD2 / PW. Step-by-step calculation BG = (0.2 PW)2 / PW = 0.04 PW0.04 PW = 24 ⇒ PW = Rs. 600 Verification/Alternative Then TD = 0.2 × 600 = 120; BG = TD2 / PW = 14,400 / 600 = 24 (checks). Common pitfalls Confusing PW with sum due (face value A). Here we only need PW. Final Answer Rs. 600

Question 9

Banker's gain on a bill due 1 year hence at 12% p.a. is Rs. 6. Find the true discount.

Accepted Answer

Given data t = 1 year, r = 12% ⇒ r t = 12 BG = Rs. 6 TD = ? Concept/Approach Use the relation BG / TD = r t / 100 ⇒ TD = BG × 100 / (r t). Step-by-step calculation TD = 6 × 100 / 12 = Rs. 50 Verification/Alternative With TD = 50 and r t = 12, BG should be 50 × 12/100 = 6 (matches). Common pitfalls Using BD instead of BG in the ratio with TD; only BG/TD equals r t / 100. Final Answer Rs. 50

Question 10

Present worth is Rs. 800 and true discount is Rs. 36. Find the banker's discount.

Accepted Answer

Given data PW = Rs. 800 TD = Rs. 36 BD = ? Concept/Approach First find banker's gain using BG = TD2 / PW, then BD = TD + BG. Step-by-step calculation BG = 362 / 800 = 1296 / 800 = 1.62BD = 36 + 1.62 = Rs. 37.62 Verification/Alternative Implied sum due A = PW + TD = 836; check that BD − TD = BG = 1.62 as computed. Common pitfalls Taking BD = TD; remember BD exceeds TD by the banker's gain. Final Answer Rs. 37.62

Question 11

Banker's gain on a sum due 3 years hence at 12% p.a. is Rs. 270. Find the banker's discount.

Accepted Answer

Given data t = 3 years, r = 12% ⇒ r t = 36 BG = Rs. 270 BD = ? Concept/Approach Use BG = A (r t)2 / (100(100 + r t)) to find the sum due A, then BD = A × r t / 100. Step-by-step calculation BG = A × 362 / (100 × 136) = A × 1296 / 13,600 = A × 81 / 850270 = A × 81 / 850 ⇒ A = 270 × 850 / 81 = 8500 / 3BD = A × 36 / 100 = (8500 / 3) × 0.36 = Rs. 1,020 Verification/Alternative Compute TD = BD − BG = 1020 − 270 = 750; then PW = A − TD = 8500/3 − 750 = 1600/3; check BG = TD2/PW = 270. Common pitfalls Using PW instead of A in the BD formula; BD is computed on the face value A. Final Answer Rs. 1,020

Question 12

True discount on a bill of Rs. 540 is Rs. 90. Find the banker's discount.

Accepted Answer

Given data Face value A = Rs. 540 TD = Rs. 90 BD = ? Concept/Approach Present worth PW = A − TD. Also TD / PW = r t / 100. Once r t is known, BD = A × r t / 100. Step-by-step calculation PW = 540 − 90 = 450r t / 100 = TD / PW = 90 / 450 = 0.2 ⇒ r t = 20BD = A × 20 / 100 = 540 × 0.20 = Rs. 108 Verification/Alternative BG = BD − TD = 108 − 90 = 18; also BG = TD2 / PW = 8100 / 450 = 18 (checks). Common pitfalls Computing BD on PW rather than on A; BD is always on the face value. Final Answer Rs. 108

Question 13

Banker's discount is Rs. 72 and true discount is Rs. 60 for the same sum and time. Find the sum due (face value).

Accepted Answer

Given data BD = Rs. 72 TD = Rs. 60 Sum due A = ? Concept/Approach Ratio BD / TD = (100 + r t) / 100 ⇒ find r t. Then use TD = A × r t / (100 + r t) to get A. Step-by-step calculation BD / TD = 72 / 60 = 6 / 5 = 1.2 ⇒ (100 + r t) / 100 = 1.2 ⇒ r t = 20TD = A × 20 / 120 = A / 660 = A / 6 ⇒ A = Rs. 360 Verification/Alternative BD = A × 20 / 100 = 360 × 0.2 = 72 (matches). Common pitfalls Subtracting BD − TD to get A; remember A = PW + TD, and formulas above are cleaner. Final Answer Rs. 360

Question 14

Banker’s Gain → Present Worth (simple interest): The banker's gain on a certain sum due 2 years hence at 5% per annum is Rs. 80. What is the present worth (i.e., the discounted value today) of the bill?

Accepted Answer

Introduction / Context: Banker's Discount (BD), True Discount (TD), Banker's Gain (BG), Face Value (F), and Present Worth (P) are classic simple-interest notions used for short-term bill discounting. The banker usually deducts BD = F * r * t upfront, while the mathematically correct reduction is TD = F * r * t / (1 + r * t). The difference BG = BD − TD quantifies the banker's extra gain relative to true discounting. Given Data / Assumptions: Rate r = 5% per annum; time t = 2 years ⇒ r * t = 0.10. Banker's Gain BG = Rs. 80. Simple interest framework; P = present worth; F = face value at maturity. Concept / Approach: For simple interest, a key identity is BG = F * (r * t)^2 / (1 + r * t). Also, P = F / (1 + r * t). We first compute F from BG, then obtain P. Step-by-Step Solution: Let x = r * t = 0.10.BG = F * x^2 / (1 + x) = F * 0.01 / 1.10 = F / 110.Given BG = 80 ⇒ F = 80 * 110 = Rs. 8800.Present Worth P = F / (1 + x) = 8800 / 1.10 = Rs. 8000. Verification / Alternative check: BD = F * x = 8800 * 0.10 = 880. TD = F * x / (1 + x) = 880 / 1.10 = 800/1.0? Actually 880/1.10 = 800. Then BG = 880 − 800 = 80, matching the given. Why Other Options Are Wrong: Rs. 8800 is the face value (F), not the present worth. Rs. 1600 and Rs. 1200 are far too small; Rs. 880 is the banker's discount in this case. Common Pitfalls: Confusing BD with TD or P; using compound interest instead of simple interest; forgetting that BG = BD − TD. Final Answer: Rs. 8000

Question 15

Present Worth & True Discount known → Banker’s Gain: The present worth of a sum due sometime hence is Rs. 1600 and the true discount is Rs. 160. What is the banker's gain?

Accepted Answer

Introduction / Context: Given present worth (P) and true discount (TD), we can reconstruct face value F = P + TD and the combined interest factor (1 + r * t). From these, Banker's Discount (BD) and hence Banker's Gain (BG = BD − TD) follow directly. Given Data / Assumptions: P = Rs. 1600. TD = Rs. 160. Simple interest; time and rate combined as x = r * t (unknown). Concept / Approach: We use P = F / (1 + x) and TD = F − P = F * x / (1 + x). Knowing P and TD gives F and 1 + x immediately, then BD = F * x and BG = BD − TD. Step-by-Step Solution: F = P + TD = 1600 + 160 = Rs. 1760.Since P = F / (1 + x) ⇒ 1600 = 1760 / (1 + x) ⇒ 1 + x = 1760 / 1600 = 11/10 ⇒ x = 0.1.BD = F * x = 1760 * 0.1 = Rs. 176.BG = BD − TD = 176 − 160 = Rs. 16. Verification / Alternative check: TD formula: F * x / (1 + x) = 1760 * 0.1 / 1.1 = 176 / 1.1 = 160, consistent. Why Other Options Are Wrong: 10, 20, 24, 14 do not match BG after computing the correct BD and TD relation. Common Pitfalls: Mixing up TD with BD; forgetting that x = r * t is dimensionless; rounding prematurely. Final Answer: Rs. 16

Question 16

Banker’s Gain known → Banker’s Discount (3 years @ 5%): The banker's gain on a sum due 3 years hence at 5% per annum is Rs. 90. What is the banker's discount?

Accepted Answer

Introduction / Context: With simple interest, BG relates to face value as BG = F * (r * t)^2 / (1 + r * t). Banker's Discount is BD = F * r * t. Given BG and r, t, we can recover F, then BD. Given Data / Assumptions: r = 5% per annum; t = 3 years ⇒ x = r * t = 0.15. BG = Rs. 90. Concept / Approach: Compute F from BG = F * x^2 / (1 + x); then BD = F * x. Step-by-Step Solution: BG = F * (0.15)^2 / 1.15 = F * 0.0225 / 1.15 ≈ F * 0.0195652.F ≈ BG / 0.0195652 ≈ 90 / 0.0195652 ≈ Rs. 4600.BD = F * x = 4600 * 0.15 = Rs. 690. Verification / Alternative check: TD = BD − BG = 690 − 90 = 600. Check TD formula: F * x / (1 + x) = 4600 * 0.15 / 1.15 = 690 / 1.15 = 600, consistent. Why Other Options Are Wrong: 720 and 810 assume different x or F; 150 is the interest on Rs. 1000 at 5% for 3 years, unrelated; 600 is TD, not BD. Common Pitfalls: Using compound interest; rounding F before computing BD; mistaking TD for BD. Final Answer: Rs. 690

Question 17

Banker’s Discount given (6 months @ 6%) → True Discount: The banker's discount on a bill due 6 months hence at 6% per annum is Rs. 37.08. What is the true discount?

Accepted Answer

Introduction / Context: Banker's Discount (BD) is computed on the face value F: BD = F * r * t. True Discount (TD) discounts only to the present worth: TD = F * r * t / (1 + r * t). With BD known and r * t known, we can recover F and then TD. Given Data / Assumptions: t = 6 months = 0.5 year; r = 6% ⇒ x = r * t = 0.06 * 0.5 = 0.03. BD = Rs. 37.08. Concept / Approach: Find F from BD = F * x ⇒ F = BD / x. Then TD = F * x / (1 + x) = BD / (1 + x). Step-by-Step Solution: F = 37.08 / 0.03 = Rs. 1236.TD = F * 0.03 / 1.03 = 37.08 / 1.03 = Rs. 36.00. Verification / Alternative check: BD − TD = BG = 37.08 − 36.00 = 1.08, which also equals F * x^2 / (1 + x) = 1236 * 0.0009 / 1.03 ≈ 1.08. Why Other Options Are Wrong: 6.18, 12.36, 48, 37.80 don't match TD once x = 0.03 is applied; 48 corresponds to a different rate/time. Common Pitfalls: Forgetting to divide by (1 + x) when moving from BD to TD; mixing months and years. Final Answer: Rs. 36

Question 18

True Discount known (8 months) → Banker’s Discount: The true discount on a bill of Rs. 1860 due after 8 months is Rs. 60. Find the banker's discount.

Accepted Answer

Introduction / Context: Given True Discount (TD) on a known face value F for a given time, we can back out x = r * t using TD = F * x / (1 + x). With x, we find Banker's Discount via BD = F * x. Given Data / Assumptions: F = Rs. 1860. t = 8 months ⇒ t = 8/12 years = 2/3 years. TD = Rs. 60; simple interest. Concept / Approach: Let x = r * t. From TD relation, solve for x and compute BD. Step-by-Step Solution: TD = F * x / (1 + x) ⇒ 60 = 1860 x / (1 + x).1860 x = 60 + 60 x ⇒ 1800 x = 60 ⇒ x = 1/30.BD = F * x = 1860 * (1/30) = Rs. 62. Verification / Alternative check: P = F − TD = 1860 − 60 = 1800; also P = F / (1 + x) = 1860 / (1 + 1/30) = 1860 / (31/30) = 1800, consistent. Why Other Options Are Wrong: Rs. 52 and Rs. 60 do not satisfy BD = F * x with x = 1/30; "None of these" is unnecessary. Common Pitfalls: Solving for r instead of x = r * t; arithmetic slips when isolating x. Final Answer: Rs . 62

Question 19

Present Worth and Banker’s Gain known → True Discount: The present worth of a sum due some time hence is Rs. 576 and the banker's gain is Re. 1. What is the true discount?

Accepted Answer

Introduction / Context: With simple interest, BG = F * x^2 / (1 + x) and P = F / (1 + x), where x = r * t. If P and BG are known, x can be determined first, then TD = F * x / (1 + x) or simply TD = F − P. Given Data / Assumptions: P = Rs. 576. BG = Re. 1. x = r * t unknown; simple interest applies. Concept / Approach: From P = F / (1 + x) ⇒ F = P * (1 + x). Then BG = F * x^2 / (1 + x) = P * (1 + x) * x^2 / (1 + x) = P * x^2. Thus x^2 = BG / P ⇒ x directly. Step-by-Step Solution: x^2 = BG / P = 1 / 576 ⇒ x = 1/24 (positive branch).F = P * (1 + x) = 576 * (1 + 1/24) = 576 * (25/24) = 600.TD = F − P = 600 − 576 = Rs. 24. Verification / Alternative check: TD also equals F * x / (1 + x) = 600 * (1/24) / (25/24) = 600 / 25 = 24, consistent. Why Other Options Are Wrong: 16, 18, 32, 12 do not match once x = 1/24 is established. Common Pitfalls: Taking x = √(P / BG) instead of √(BG / P); forgetting F = P + TD. Final Answer: Rs. 24

Question 20

BG as a fraction of BD (time = 2.5 years) → Rate %: The banker's gain on a certain sum due 2 1/2 years hence is 2/23 of the banker's discount on it for the same time and rate. Find the rate of interest (per annum).

Accepted Answer

Introduction / Context: This problem ties BG and BD via a proportion. Using the simple-interest identities BD = F * x and BG = F * x^2 / (1 + x) with x = r * t, we can solve for x from the ratio, then obtain r by dividing x by t. Given Data / Assumptions: BG = (2/23) * BD. t = 2.5 years; r unknown; x = r * t. Concept / Approach: Set F * x^2 / (1 + x) = (2/23) * F * x ⇒ cancel F * x (x ≠ 0) and solve for x. Then r = x / t. Step-by-Step Solution: x / (1 + x) = 2 / 23 ⇒ 23x = 2 + 2x ⇒ 21x = 2 ⇒ x = 2/21.r = x / t = (2/21) / 2.5 = (2/21) * (2/5) = 4 / 105 ≈ 0.038095.Rate % ≈ 3.8095% ≈ 4%. Verification / Alternative check: Plug r back: x = r * t ≈ 0.038095 * 2.5 ≈ 0.095238. Then BG/BD = x/(1 + x) ≈ 0.095238 / 1.095238 ≈ 0.086956 ≈ 2/23, consistent. Why Other Options Are Wrong: 5%, 6%, 8%, 3% give x values that do not satisfy BG = (2/23) * BD. Common Pitfalls: Using BD/BG instead of BG/BD; forgetting to divide by t to get r. Final Answer: 4%

Banker's Discount Questions