Question 1

Compute Banker’s Discount directly: Find the banker's discount on a bill of ₹ 12,750 due four months hence when the rate is 6% per annum.

Accepted Answer

Introduction / Context: Banker's Discount is the simple-interest charge on the face value for the time remaining: BD = F * r * t, where t is in years and r is per annum. Given Data / Assumptions: F = ₹ 12,750. r = 6% per annum. t = 4 months = 4/12 year = 1/3 year. Concept / Approach: Apply BD = F * r * t directly. Step-by-Step Solution: BD = 12,750 * 0.06 * (1/3) = 12,750 * 0.02 = ₹ 255. Verification / Alternative check: Monthly interest at 6% p.a. is 0.5% per month; for 4 months ≈ 2% ⇒ 2% of 12,750 is 255, same result. Why Other Options Are Wrong: 250 and 240 assume approximate rates; 120 and 300 correspond to different times/rates. Common Pitfalls: Using true discount formula instead of BD; mis-converting months to years. Final Answer: ₹ 255

Question 2

True Discount given (16 months) → Banker’s Gain: The true discount on a bill of ₹ 7440 due 16 months hence is ₹ 240. Find the banker's gain.

Accepted Answer

Introduction / Context: Once TD is given, we can solve for x = r * t using TD = F * x / (1 + x). Banker's Discount is BD = F * x, and Banker's Gain is BG = BD − TD. Given Data / Assumptions: F = ₹ 7440. t = 16 months = 16/12 years = 4/3 years. TD = ₹ 240. Concept / Approach: Let x = r * t. Solve for x from the TD equation, then compute BD and BG. Step-by-Step Solution: TD = 7440 x / (1 + x) = 240 ⇒ 7440 x = 240 + 240 x ⇒ 7200 x = 240 ⇒ x = 1/30.BD = 7440 * (1/30) = ₹ 248.BG = BD − TD = 248 − 240 = ₹ 8. Verification / Alternative check: P = F − TD = 7440 − 240 = 7200; also P = F / (1 + x) = 7440 / (31/30) = 7200, consistent. Why Other Options Are Wrong: 10, 6, 4, 12 do not match BG once x = 1/30 is established. Common Pitfalls: Arithmetic slips when isolating x; forgetting BG = BD − TD. Final Answer: ₹ 8

Question 3

Banker’s Gain known (8 years @ 24%) → Present Worth: The banker's gain of a sum due 8 years hence at 24% per annum is ₹ 144. Find the present worth of that bill.

Accepted Answer

Introduction / Context: Given BG and the product x = r * t, we can compute F using BG = F * x^2 / (1 + x), then present worth P = F / (1 + x). Here r is large and t is long, so x > 1, but the simple-interest relationships still apply. Given Data / Assumptions: r = 24% per annum; t = 8 years ⇒ x = r * t = 0.24 * 8 = 1.92. BG = ₹ 144. Concept / Approach: Compute F from the BG identity, then divide by (1 + x) to get P. Step-by-Step Solution: BG = F * x^2 / (1 + x) ⇒ F = BG * (1 + x) / x^2 = 144 * 2.92 / (1.92^2).1.92^2 = 3.6864; hence F ≈ 144 * 2.92 / 3.6864 ≈ 144 * 0.7920 ≈ ₹ 114.05.P = F / (1 + x) = 114.05 / 2.92 ≈ ₹ 39.06. Verification / Alternative check: BD = F * x ≈ 114.05 * 1.92 ≈ 219.0; TD = F * x / (1 + x) ≈ 219.0 / 2.92 ≈ 75.0; BG = BD − TD ≈ 144, matching the given. Why Other Options Are Wrong: 45, 38.06, 50, 36 are off once the precise BG relation is applied. Common Pitfalls: Using compound-interest discounting; rounding too early; miscomputing x = r * t. Final Answer: ₹ 39.06

Question 4

Banker’s Discount (BD) and Banker’s Gain (BG) on a certain bill are ₹196 and ₹28, respectively. Using standard BD–TD–BG relations for simple interest bills, determine the face value (amount due at maturity) of the bill.

Accepted Answer

Introduction / Context: This question uses the classic relationships among Banker’s Discount (BD), True Discount (TD), Banker’s Gain (BG), and the face value F of a bill under simple interest conventions. We will extract TD and then back-compute F. Given Data / Assumptions: BD = ₹196. BG = ₹28. All discounts follow simple interest bill-discounting definitions. Concept / Approach: Key identities (with x = r * t): BD = F * x, TD = F * x / (1 + x), BG = BD − TD. Also BD/TD = 1 + x and BG = x * TD. If BD and BG are known, TD = BD − BG immediately. Then 1 + x = BD/TD ⇒ x obtained; next F = BD / x. Step-by-Step Solution: TD = BD − BG = 196 − 28 = ₹168.BD/TD = 196/168 = 7/6 ⇒ 1 + x = 7/6 ⇒ x = 1/6.BD = F * x ⇒ 196 = F * (1/6) ⇒ F = 196 * 6 = ₹1176. Verification / Alternative check: Compute TD from F and x: TD = F*x/(1+x) = 1176*(1/6)/(7/6) = 1176*(1/7) = ₹168; BG = 196 − 168 = ₹28, consistent. Why Other Options Are Wrong: ₹1200, ₹1376, and ₹1400 do not satisfy BD/TD = 7/6 while keeping BD = ₹196. Common Pitfalls: Confusing BD with TD or forgetting that BG = BD − TD; mixing rate and time unnecessarily when x can be inferred from BD/TD. Final Answer: ₹ 1176

Question 5

The present worth (PW) of a sum due sometime hence is ₹576 and the banker’s gain (BG) is ₹1. Find the true discount (TD).

Accepted Answer

Introduction / Context: For simple-interest bill discounting, PW (present worth), TD (true discount), BD (banker’s discount), and BG (banker’s gain) are tightly related via the parameter x = r * t. With PW and BG given, TD can be computed directly using compact identities. Given Data / Assumptions: PW = ₹576. BG = ₹1. All quantities based on simple interest conventions. Concept / Approach: Let x = r * t. Then PW = F / (1 + x), TD = F * x / (1 + x) and BG = F * x^2 / (1 + x). Eliminating F gives TD = x * PW and BG = x^2 * PW. Hence x = sqrt(BG / PW) and then TD = x * PW = sqrt(BG * PW). Step-by-Step Solution: x = sqrt(BG / PW) = sqrt(1 / 576) = 1/24.TD = x * PW = (1/24) * 576 = ₹24. Verification / Alternative check: Compute BG from x and PW: BG = x^2 * PW = (1/24)^2 * 576 = (1/576) * 576 = ₹1, matches. Why Other Options Are Wrong: ₹16, ₹18, and ₹32 do not satisfy the identity TD = sqrt(BG * PW) with BG = 1 and PW = 576. Common Pitfalls: Attempting to find r or t separately; the problem only needs the combined factor via x. Final Answer: Rs. 24

Question 6

A bill is due 1 year hence at 5% simple interest. The banker’s gain (BG) is Re. 1. What is the true discount (TD)?

Accepted Answer

Introduction / Context: Here the period and rate are known, so x = r * t is known. A direct identity links banker’s gain (BG) and true discount (TD): BG = x * TD. We can compute TD from BG and x immediately. Given Data / Assumptions: t = 1 year, r = 5% p.a. BG = Re. 1. Simple-interest bill-discounting framework applies. Concept / Approach: Let x = r * t = 0.05 * 1 = 0.05. Using identities: TD = F * x / (1 + x) and BG = F * x^2 / (1 + x). Dividing gives BG = x * TD ⇒ TD = BG / x. Step-by-Step Solution: x = 0.05.TD = BG / x = 1 / 0.05 = ₹20. Verification / Alternative check: If TD = ₹20, BG should be x * TD = 0.05 * 20 = Re. 1, consistent. Why Other Options Are Wrong: ₹5, ₹15, and ₹25 violate BG = x * TD with x = 0.05. Common Pitfalls: Confusing TD with BD; note BD = (1 + x) * TD, not needed here because BG gives TD directly. Final Answer: ₹ 20

Question 7

A bill is due 3 years hence at 5% p.a. The banker’s gain (BG) is ₹90. Find the banker’s discount (BD).

Accepted Answer

Introduction / Context: We are given BG and the period and rate. From BG we first recover TD using x = r * t. Then BD follows from BD = (1 + x) * TD or directly via BD = F * x once F is known. The compact route is via TD and the (1 + x) multiplier. Given Data / Assumptions: r = 5% p.a., t = 3 years ⇒ x = 0.15. BG = ₹90. Concept / Approach: Key identities: BG = x * TD and BD = (1 + x) * TD. Hence TD = BG / x and then BD = (1 + x) * TD. Step-by-Step Solution: x = 0.05 * 3 = 0.15.TD = BG / x = 90 / 0.15 = ₹600.BD = (1 + x) * TD = 1.15 * 600 = ₹690. Verification / Alternative check: With BD = ₹690 and TD = ₹600, BG = BD − TD = ₹90, consistent. Why Other Options Are Wrong: ₹550, ₹600, and ₹650 do not maintain BD − TD = ₹90 with x = 0.15. Common Pitfalls: Using BD = x * TD (incorrect); the correct relation is BD = (1 + x) * TD and BG = x * TD. Final Answer: Rs. 690

Question 8

The banker’s gain (BG) on a bill due 2 years hence at 5% p.a. is ₹8. Find the present worth (PW) of the bill.

Accepted Answer

Introduction / Context: Given banker’s gain and (rate × time), there is a direct formula linking BG and PW (present worth): BG = x^2 * PW with x = r * t. This avoids computing face value first. Given Data / Assumptions: r = 5% p.a., t = 2 years. BG = ₹8. Concept / Approach: Let x = r * t = 0.05 * 2 = 0.10. Identities: PW = F / (1 + x); TD = x * PW; BG = x * TD = x^2 * PW. Hence PW = BG / x^2. Step-by-Step Solution: x = 0.10 ⇒ x^2 = 0.01.PW = BG / x^2 = 8 / 0.01 = ₹800. Verification / Alternative check: If PW = 800 and x = 0.1, then TD = x * PW = 80; BG = x * TD = 0.1 * 80 = 8, as given. Why Other Options Are Wrong: ₹650, ₹750, and ₹850 do not satisfy BG = x^2 * PW with x = 0.1. Common Pitfalls: Trying to compute face value F or rate separately; the compact identity BG = x^2 * PW is sufficient. Final Answer: Rs. 800

Question 9

Find the true discount (TD) on a bill of ₹2520 due 6 months hence at 10% per annum (simple interest).

Accepted Answer

Introduction / Context: We directly compute the true discount for a known face value F, using the simple-interest discounting factor x = r * t. TD discounts the face value to its present worth by accounting for time and rate until maturity. Given Data / Assumptions: F = ₹2520. t = 6 months = 0.5 year; r = 10% p.a. Concept / Approach: Let x = r * t = 0.10 * 0.5 = 0.05. Identities: PW = F / (1 + x); TD = F − PW = F * x / (1 + x). Substitute F and x. Step-by-Step Solution: TD = (F * x) / (1 + x) = (2520 * 0.05) / 1.05 = 126 / 1.05 = ₹120. Verification / Alternative check: PW = 2520 − 120 = ₹2400. Recheck: PW = F / (1 + x) = 2520 / 1.05 = ₹2400, consistent. Why Other Options Are Wrong: ₹80, ₹140, and ₹180 do not match the exact computation using x = 0.05. Common Pitfalls: Applying simple interest on PW instead of F when forming TD; forgetting to divide by (1 + x). Final Answer: Rs. 120

Question 10

The true discount (TD) on a bill of ₹2160 is ₹360. Compute the banker’s discount (BD).

Accepted Answer

Introduction / Context: True discount and face value are known; we must find BD. Use the identity TD = F * x / (1 + x) to find x, then use BD = F * x or BD = (1 + x) * TD. This avoids needing r or t individually. Given Data / Assumptions: F = ₹2160. TD = ₹360. Concept / Approach: Let x = r * t. Then 360 = 2160 * x / (1 + x). Solve for x, then BD = 2160 * x. Step-by-Step Solution: 360(1 + x) = 2160x ⇒ 360 = 2160x − 360x = 1800x ⇒ x = 0.2.BD = F * x = 2160 * 0.2 = ₹432. Verification / Alternative check: Also BD = (1 + x) * TD = 1.2 * 360 = ₹432, same result. Why Other Options Are Wrong: Values like 412, 422, 442 do not satisfy both TD and the derived x. Common Pitfalls: Using BD = TD / (1 + x) (incorrect direction); BD is larger than TD for x > 0. Final Answer: Rs. 432

Question 11

The true discount (TD) on a certain sum due 6 months hence at 15% p.a. is ₹240. What is the banker’s discount (BD) on the same sum for the same time and rate?

Accepted Answer

Introduction / Context: Given TD and (rate × time), BD is just a simple multiplier away. The ratio BD/TD equals 1 + x where x = r * t. We leverage this direct relation for a quick computation. Given Data / Assumptions: TD = ₹240. t = 6 months = 0.5 year; r = 15% p.a. ⇒ x = 0.075. Concept / Approach: Identity: BD = (1 + x) * TD. No need to find the face value explicitly. Compute x, multiply TD by (1 + x). Step-by-Step Solution: x = 0.15 * 0.5 = 0.075.BD = (1 + x) * TD = 1.075 * 240 = ₹258. Verification / Alternative check: If TD = 240 and x = 0.075, then BD − TD = x * TD = 18 ⇒ BD = 258, consistent. Why Other Options Are Wrong: ₹228, ₹248, and ₹278 do not equal 1.075 * 240. Common Pitfalls: Forgetting to convert months to years before computing x; mixing up BD and BG. Final Answer: Rs. 258

Question 12

A bill is due 4 months hence at 15% p.a. The banker’s discount (BD) is ₹420. Find the true discount (TD).

Accepted Answer

Introduction / Context: When BD and x = r * t are known, TD follows from TD = BD / (1 + x). This is because BD = (1 + x) * TD under simple-interest bill discounting. Given Data / Assumptions: BD = ₹420. t = 4 months = 1/3 year; r = 15% p.a. Concept / Approach: Compute x = r * t; then TD = BD / (1 + x). No need to compute the face value explicitly. Step-by-Step Solution: x = 0.15 * (1/3) = 0.05.TD = BD / (1 + x) = 420 / 1.05 = ₹400. Verification / Alternative check: BG = BD − TD = 420 − 400 = ₹20. Also BG = x * TD = 0.05 * 400 = ₹20, matches. Why Other Options Are Wrong: ₹380, ₹390, and ₹410 are off the exact ratio division by 1.05. Common Pitfalls: Using months directly without converting to years; confusing TD with PW. Final Answer: Rs. 400

Question 13

The present worth (PW) of a sum due sometime hence is ₹5760 and the banker’s gain (BG) is ₹10. Compute the true discount (TD).

Accepted Answer

Introduction / Context: Given PW and BG, the true discount can be obtained without knowing the face value or the rate/time individually, by employing identities in x = r * t. Given Data / Assumptions: PW = ₹5760. BG = ₹10. Concept / Approach: Key relations: TD = x * PW and BG = x^2 * PW. Hence x = sqrt(BG / PW), and TD = x * PW = sqrt(BG * PW). This is the fastest computation path here. Step-by-Step Solution: x = sqrt(10 / 5760).TD = sqrt(BG * PW) = sqrt(10 * 5760) = sqrt(57600) = ₹240. Verification / Alternative check: BG = x * TD = (TD / PW) * TD = TD^2 / PW = 240^2 / 5760 = 57600 / 5760 = ₹10, checks out. Why Other Options Are Wrong: ₹120, ₹420, and ₹480 do not satisfy TD = sqrt(BG * PW) with the given values. Common Pitfalls: Trying to determine rate or time separately; it is unnecessary here. Final Answer: Rs. 240

Question 14

On a certain sum, the banker’s discount (BD) is ₹36 and the true discount (TD) is ₹30 for the same time and rate. What is the sum due (face value F)?

Accepted Answer

Introduction / Context: Given BD and TD only, we can extract x = r * t using BD/TD, then use BD = F * x to compute the face value F directly. This avoids explicit rate/time values. Given Data / Assumptions: BD = ₹36. TD = ₹30. Concept / Approach: Identity: BD/TD = 1 + x ⇒ x = BD/TD − 1 = 36/30 − 1 = 1/5. Then BD = F * x ⇒ F = BD / x. Step-by-Step Solution: x = (36/30) − 1 = 6/5 − 1 = 1/5 = 0.2.F = BD / x = 36 / 0.2 = ₹180. Verification / Alternative check: TD = F * x / (1 + x) = 180 * 0.2 / 1.2 = 36 / 1.2 = ₹30, matches. Why Other Options Are Wrong: ₹120, ₹200, ₹220 do not satisfy both BD and TD under the same x. Common Pitfalls: Using TD = F * x (incorrect); TD is discounted by 1 + x in the denominator. Final Answer: Rs. 180

Question 15

The present worth (PW) of a certain sum due sometime hence is ₹3400 and the true discount (TD) is ₹340. Find the banker’s gain (BG).

Accepted Answer

Introduction / Context: Given PW and TD, we can compute x = TD / PW because TD = x * PW. Banker’s gain then equals BG = x * TD = x^2 * PW. This provides BG directly without full details of rate and time. Given Data / Assumptions: PW = ₹3400. TD = ₹340. Concept / Approach: Compute x = TD / PW. Then BG = x * TD. Alternatively, BG = TD^2 / PW, which is the same identity rearranged. Step-by-Step Solution: x = 340 / 3400 = 0.1.BG = x * TD = 0.1 * 340 = ₹34. Verification / Alternative check: Using BG = TD^2 / PW = 340^2 / 3400 = 115600 / 3400 = ₹34; consistent. Why Other Options Are Wrong: ₹17, ₹18, and ₹21 are not equal to TD^2 / PW. Common Pitfalls: Confusing BG with BD − TD when BD is not given; here the formula avoids BD entirely. Final Answer: Rs. 34

Question 16

The banker’s gain (BG) on a certain sum due 2 years hence at 10% p.a. is ₹24. What is the present worth (PW)?

Accepted Answer

Introduction / Context: With BG and x known, PW follows from BG = x^2 * PW. This method is efficient because it does not require determining face value first or computing TD separately unless desired for checking. Given Data / Assumptions: r = 10% p.a., t = 2 years ⇒ x = 0.20. BG = ₹24. Concept / Approach: Use BG = x^2 * PW. Thus PW = BG / x^2. Straight substitution yields the answer immediately. Step-by-Step Solution: x = 0.1 * 2 = 0.2; x^2 = 0.04.PW = 24 / 0.04 = ₹600. Verification / Alternative check: TD = x * PW = 0.2 * 600 = ₹120; BG = x * TD = 0.2 * 120 = ₹24, correct. Why Other Options Are Wrong: ₹300, ₹400, ₹500 produce BG values different from ₹24 when multiplied by x^2 = 0.04. Common Pitfalls: Using BD or other intermediate steps unnecessarily; the direct identity suffices. Final Answer: Rs. 600

Question 17

Find the present worth (PW) of a bill of ₹1764 due 2 years hence at 5% compound interest per annum.

Accepted Answer

Introduction / Context: This item specifies compound interest, not simple discounting. The present worth under compound interest is obtained by dividing the future value (face value) by (1 + r)^t, where r is per annum rate and t is in years. Given Data / Assumptions: Face value at maturity F = ₹1764. r = 5% p.a.; t = 2 years. Concept / Approach: For compound interest: PW = F / (1 + r)^t. This differs from TD/BD formulas which belong to simple-interest bill discounting. Use exact powers for accuracy. Step-by-Step Solution: (1 + r)^t = 1.05^2 = 1.1025.PW = 1764 / 1.1025 = ₹1600. Verification / Alternative check: Forward check: 1600 × 1.1025 = 1764, matching the given future sum. Why Other Options Are Wrong: ₹1200, ₹1400, ₹1800 are not equal to 1764 / 1.1025. Common Pitfalls: Using simple-interest discounting formulas (TD/BD) instead of compound discounting when the problem explicitly states compound interest. Final Answer: Rs. 1600

Question 18

The true discount (TD) on a bill of ₹1860 due after 8 months is ₹60. Find the banker’s gain (BG).

Accepted Answer

Introduction / Context: We are given TD and face value F; the time (8 months) is mentioned but rate is not. However, we can find x = r * t directly from TD and F: TD = F * x / (1 + x). Having x, BG = x * TD. No need to split x into r and t separately. Given Data / Assumptions: F = ₹1860. TD = ₹60. Concept / Approach: Solve TD = F * x / (1 + x) for x, then compute BG = x * TD. This is efficient and general. Step-by-Step Solution: 60(1 + x) = 1860x ⇒ 60 = 1860x − 60x = 1800x ⇒ x = 1/30.BG = x * TD = (1/30) * 60 = ₹2. Verification / Alternative check: BD = (1 + x)*TD = (31/30)*60 = ₹62; BG = BD − TD = 62 − 60 = ₹2, consistent. Why Other Options Are Wrong: ₹1.5, ₹2.5, ₹4 do not equal x * TD with x = 1/30. Common Pitfalls: Attempting to find r explicitly; not required because BG depends on x only. Final Answer: Rs. 2

Question 19

The banker’s discount (BD) on a bill due 6 months hence at 6% p.a. is ₹37.08. Find the true discount (TD).

Accepted Answer

Introduction / Context: Given BD and x = r * t, TD can be recovered via TD = BD / (1 + x). This identity follows directly from the relation BD = (1 + x) * TD in simple-interest bill discounting problems. Given Data / Assumptions: BD = ₹37.08. t = 6 months = 0.5 year; r = 6% p.a. ⇒ x = 0.03. Concept / Approach: Compute TD = BD / (1 + x) = 37.08 / 1.03. Evaluate with care to avoid rounding errors. Step-by-Step Solution: TD = 37.08 / 1.03 = ₹36.00. Verification / Alternative check: BG = BD − TD = 37.08 − 36 = ₹1.08. Also BG = x * TD = 0.03 * 36 = ₹1.08, consistent. Why Other Options Are Wrong: ₹32 and ₹38 do not satisfy the exact division by 1.03; “None of these” is unnecessary since ₹36 fits perfectly. Common Pitfalls: Using months directly without conversion; rounding too early. Final Answer: Rs. 36

Question 20

For a sum due 2 years later, the banker’s discount (BD) is 11/10 of the true discount (TD). Find the annual rate percent (simple interest).

Accepted Answer

Introduction / Context: Relating BD and TD via their ratio immediately gives x = r * t, because BD/TD = 1 + x. With time known, we solve for r directly. This avoids needing monetary amounts. Given Data / Assumptions: BD = (11/10) * TD. t = 2 years. Concept / Approach: From BD/TD = 1 + x, we have 11/10 = 1 + x ⇒ x = 1/10 = 0.1. Since x = r * t, r = x / t = 0.1 / 2. Step-by-Step Solution: 1 + x = 11/10 ⇒ x = 1/10.t = 2 ⇒ r = x / t = (1/10) / 2 = 1/20 = 0.05 = 5% p.a. Verification / Alternative check: Check: BD/TD should be 1 + x = 1.1 = 11/10, matches the condition. Why Other Options Are Wrong: 10% and 11% imply x = 0.2 or 0.22 over 2 years, not consistent with BD/TD = 11/10. “51/3” is not a valid percent value here. Common Pitfalls: Forgetting BD/TD = 1 + x or dividing by years incorrectly when computing r. Final Answer: 5%

Banker's Discount Questions