Banker's Discount

1. Core Concepts & Theoretical Blueprint

Banker's Discount (BD) is the amount a banker deducts when discounting a bill BEFORE its due date — calculated as Simple Interest on the FULL FACE VALUE (the amount due) for the unexpired time, in contrast to True Discount (TD), which is Simple Interest on the smaller PRESENT WORTH. This structural difference between BD and TD is the entire foundation of the chapter.

Absolute Core Definitions:

• Face Value / Amount (A): the sum stated on the bill, payable at the due date.
• Banker's Discount (BD): Simple Interest on A (the full face value) for the unexpired time — this is what the banker actually deducts in practice.
• True Discount (TD): Simple Interest on the Present Worth (PW) for the unexpired time — the mathematically "fair" discount (as covered in the True Discount chapter).
• Banker's Gain (BG): the banker's profit from using BD instead of the fair TD: $BG=BD-TD$.

Absolute Core Formulas:

Key Derived Relationships (heavily tested):

Present Worth in terms of BD and BG:

The Universal Trap: Four persistent traps:

1. Computing BD as interest on Present Worth instead of the full Face Value — this is precisely the definition of TD, not BD; swapping these two is the single defining error of the chapter (and its mirror-image error in the True Discount chapter).
2. Forgetting that BG is always POSITIVE (BD > TD always, for positive R, T) — since BD is interest on the LARGER amount (A) while TD is interest on the smaller amount (PW), BD must always exceed TD; a computed negative BG signals an error in the setup.
3. Misapplying the $TD^2=PW\times BG$ identity — this is an extremely useful CROSS-CHECK and solving tool, but requires knowing (or first deriving) all three quantities' relationships correctly; plugging in mismatched values (e.g., BD instead of BG) breaks the identity.
4. Confusing "sum due" (Amount/Face Value) with "sum received after discounting" — the amount a bill-holder actually RECEIVES when they discount a bill with a banker is $A-BD$ (not $A-TD$, since the banker deducts BD, not the fairer TD).

2. Exhaustive Question Typology

                          BANKER'S DISCOUNT
                                  |
    -----------------------------------------------------------------------
    |            |              |               |               |          |
Type 1:       Type 2:        Type 3:        Type 4:         Type 5:     Type 6:
Basic BD      Basic TD       Find BG        Find Face       Find TD     Find Rate
Computation   Computation    Given BD       Value (A)       Given BD    or Time
(Given A,     (Given A,      and TD         Given BD, R,T   and BG      Given BD
R, T)         R, T,                                         (Using      and A
              Cross-Ref)                                    BG = TD×
                                                              RT/100)
    |            |              |
Type 7:       Type 8:        Type 9:
TD² = PW×BG   Find PW        Comparison/
Application   Given BD and   Ratio
(Cross-       BG (or BD                     Problems (BD
Check/         and R,T)                      vs TD vs SI)
Solve)

Type 1 — Basic BD computation:

• Core Scenario: "Find the Banker's Discount on a bill of ₹2,550 due 4 months hence at 6% per annum."
• Governing Equation: $BD=\dfrac{A\times R\times T}{100}$

Type 2 — Basic TD computation (cross-referenced from the True Discount chapter):

• Core Scenario: "Find the True Discount on the same bill (₹2,550 due 4 months hence at 6%)."
• Governing Equation: $TD=\dfrac{A\times R\times T}{100+RT}$

Type 3 — Find BG given BD and TD:

• Core Scenario: "The BD on a bill is ₹300 and the TD is ₹260. Find the Banker's Gain."
• Governing Equation: $BG=BD-TD$

Type 4 — Find Face Value (A) given BD, R, T:

• Core Scenario: "The Banker's Discount on a bill due 6 months hence at 8% per annum is ₹240. Find the face value of the bill."
• Governing Equation: Rearrange $BD=\dfrac{A\times RT}{100}\Rightarrow A=\dfrac{BD\times100}{RT}$

Type 5 — Find TD given BD and BG:

• Core Scenario: "If the BD on a certain sum is ₹420 and the BG is ₹20, find the TD."
• Governing Equation: $TD=BD-BG$

Type 6 — Find rate or time given BD and A:

• Core Scenario: "The BD on a bill of ₹5,000 due 9 months hence is ₹225. Find the rate of interest."
• Governing Equation: Rearrange $BD=\dfrac{A\times RT}{100}$ to solve for R (or T).

Type 7 — $TD^2=PW\times BG$ application (cross-check/solve):

• Core Scenario: "The TD on a bill is ₹40, and the BG is ₹2. Find the Present Worth."
• Governing Equation: $PW=\dfrac{TD^2}{BG}$

Type 8 — Find PW given BD and BG (or BD and R, T):

• Core Scenario: "The BD on a certain sum due some time hence is ₹72, and the BG is ₹6. Find the Present Worth and the sum due."
• Governing Equation: First find $TD=BD-BG$, then $PW=\dfrac{TD^2}{BG}$, then $A=PW+TD$.

Type 9 — Comparison/ratio problems (BD vs TD vs SI):

• Core Scenario: "For a given sum, rate, and time, compare the values of BD, TD, and ordinary SI, and rank them."
• Governing Equation: Since $BD=$ SI on A, and A>PW: BD>TD always. Compare directly using their respective formulas.

3. Type-wise Practice MCQs with Full Solutions

Type 1 — Basic BD Computation

MCQ 1. Find the Banker's Discount on a bill of ₹2,550 due 4 months hence at 6% per annum. (A) ₹51 (B) ₹48 (C) ₹55 (D) ₹45

Correct Answer: (A) Solution: $T=4/12=1/3$. $BD=\dfrac{2550\times6\times1/3}{100}=\dfrac{5100}{100}=51$.

MCQ 2. Find the Banker's Discount on a bill of ₹6,000 due 8 months hence at 9% per annum. (A) ₹360 (B) ₹340 (C) ₹380 (D) ₹350

Correct Answer: (A) Solution: $T=8/12=2/3$. $BD=\dfrac{6000\times9\times2/3}{100}=\dfrac{36000}{100}=360$.

MCQ 3. Find the Banker's Discount on a bill of ₹4,000 due 1 year hence at 5% per annum. (A) ₹200 (B) ₹190 (C) ₹210 (D) ₹180

Correct Answer: (A) Solution: $BD=\dfrac{4000\times5\times1}{100}=200$.

Type 2 — Basic TD Computation (Cross-Referenced)

MCQ 1. Find the True Discount on a bill of ₹2,550 due 4 months hence at 6% per annum. (A) ₹50 (B) ₹48 (C) ₹52 (D) ₹45

Correct Answer: (A) Solution: $T=1/3$; $RT=2$. $TD=\dfrac{2550\times2}{100+2}=\dfrac{5100}{102}=50$.

MCQ 2. Find the True Discount on a bill of ₹6,000 due 8 months hence at 9% per annum. (A) ₹339.62 (approx) (B) ₹340 (C) ₹335 (D) ₹330

Correct Answer: (A) Solution: $T=2/3$; $RT=6$. $TD=\dfrac{6000\times6}{106}=\dfrac{36000}{106}=339.62$.

MCQ 3. Find the True Discount on a bill of ₹4,000 due 1 year hence at 5% per annum. (A) ₹190.48 (B) ₹195 (C) ₹185 (D) ₹200

Correct Answer: (A) Solution: $RT=5$. $TD=\dfrac{4000\times5}{105}=\dfrac{20000}{105}=190.48$.

Type 3 — Find BG Given BD and TD

MCQ 1. The BD on a bill is ₹300 and the TD is ₹260. Find the Banker's Gain. (A) ₹40 (B) ₹35 (C) ₹45 (D) ₹30

Correct Answer: (A) Solution: $BG=BD-TD=300-260=40$.

MCQ 2. For a bill with BD=₹51 and TD=₹50 (same bill), find the Banker's Gain. (A) ₹1 (B) ₹2 (C) ₹1.5 (D) ₹0.5

Correct Answer: (A) Solution: $BG=51-50=1$.

MCQ 3. The BD and TD on a certain sum for the same time are ₹420 and ₹360 respectively. Find the Banker's Gain. (A) ₹60 (B) ₹55 (C) ₹65 (D) ₹50

Correct Answer: (A) Solution: $BG=420-360=60$.

Type 4 — Find Face Value (A) Given BD, R, T

MCQ 1. The Banker's Discount on a bill due 6 months hence at 8% per annum is ₹240. Find the face value of the bill. (A) ₹6000 (B) ₹5800 (C) ₹6200 (D) ₹5500

Correct Answer: (A) Solution: $T=0.5$; $RT=4$. $A=\dfrac{BD\times100}{RT}=\dfrac{240\times100}{4}=6000$.

MCQ 2. The BD on a certain bill due 3 months hence at 12% per annum is ₹90. Find the face value. (A) ₹3000 (B) ₹2800 (C) ₹3200 (D) ₹2900

Correct Answer: (A) Solution: $T=0.25$; $RT=3$. $A=\dfrac{90\times100}{3}=3000$.

MCQ 3. The BD on a bill due 9 months hence at 6% per annum is ₹135. Find the face value. (A) ₹3000 (B) ₹2900 (C) ₹3100 (D) ₹2800

Correct Answer: (A) Solution: $T=0.75$; $RT=4.5$. $A=\dfrac{135\times100}{4.5}=3000$.

Type 5 — Find TD Given BD and BG

MCQ 1. If the BD on a certain sum is ₹420 and the BG is ₹20, find the TD. (A) ₹400 (B) ₹410 (C) ₹390 (D) ₹380

Correct Answer: (A) Solution: $TD=BD-BG=420-20=400$.

MCQ 2. The BD on a sum is ₹360, and the BG is ₹15. Find the TD. (A) ₹345 (B) ₹350 (C) ₹340 (D) ₹355

Correct Answer: (A) Solution: $TD=360-15=345$.

MCQ 3. If the BD is ₹800 and the BG is ₹64, find the TD. (A) ₹736 (B) ₹740 (C) ₹730 (D) ₹745

Correct Answer: (A) Solution: $TD=800-64=736$.

Type 6 — Find Rate or Time Given BD and A

MCQ 1. The BD on a bill of ₹5,000 due 9 months hence is ₹225. Find the rate of interest. (A) 6% (B) 5% (C) 7% (D) 8%

Correct Answer: (A) Solution: $T=0.75$. $225=\dfrac{5000\times R\times0.75}{100}\Rightarrow225=37.5R\Rightarrow R=6\%$.

MCQ 2. The BD on a bill of ₹8,000 at 10% per annum is ₹400. Find the time (in months). (A) 6 months (B) 5 months (C) 7 months (D) 8 months

Correct Answer: (A) Solution: $400=\dfrac{8000\times10\times T}{100}\Rightarrow400=800T\Rightarrow T=0.5$ year $=6$ months.

MCQ 3. The BD on a bill of ₹12,000 due in a certain time at 9% per annum is ₹540. Find the time in months. (A) 6 months (B) 5 months (C) 7 months (D) 4 months

Correct Answer: (A) Solution: $540=\dfrac{12000\times9\times T}{100}\Rightarrow540=1080T\Rightarrow T=0.5$ year $=6$ months.

Type 7 — $TD^2=PW\times BG$ Application

MCQ 1. The TD on a bill is ₹40, and the BG is ₹2. Find the Present Worth. (A) ₹800 (B) ₹750 (C) ₹850 (D) ₹780

Correct Answer: (A) Solution: $PW=\dfrac{TD^2}{BG}=\dfrac{1600}{2}=800$.

MCQ 2. The TD on a sum is ₹60, and the BG is ₹3.60. Find the Present Worth. (A) ₹1000 (B) ₹950 (C) ₹1050 (D) ₹900

Correct Answer: (A) Solution: $PW=\dfrac{60^2}{3.6}=\dfrac{3600}{3.6}=1000$.

MCQ 3. The TD on a certain sum is ₹96, and the BG is ₹9.60. Find the Present Worth, and hence the sum due (Face Value). (A) PW=₹960, Sum due=₹1056 (B) PW=₹900,Sum due=₹1000 (C) PW=₹950,Sum due=₹1046 (D) PW=₹980,Sum due=₹1076

Correct Answer: (A) Solution: $PW=\dfrac{96^2}{9.6}=\dfrac{9216}{9.6}=960$. Sum due $=PW+TD=960+96=1056$.

Type 8 — Find PW Given BD and BG

MCQ 1. The BD on a certain sum due some time hence is ₹72, and the BG is ₹6. Find the Present Worth and the sum due. (A) PW=₹726, Sum due=₹792 (B) PW=₹800,Sum due=₹870 (C) PW=₹780,Sum due=₹850 (D) PW=₹810,Sum due=₹880

Correct Answer: (A) Solution: First find TD: $TD=BD-BG=72-6=66$. Then $PW=\dfrac{TD^2}{BG}=\dfrac{66^2}{6}=\dfrac{4356}{6}=726$. Sum due$=PW+TD=726+66=792$.

MCQ 2. The BD on a sum is ₹36, and the BG is ₹4. Find the sum due (Face Value). (A) ₹288 (B) ₹300 (C) ₹280 (D) ₹296

Correct Answer: (A) Solution: $TD=36-4=32$. $PW=\dfrac{32^2}{4}=\dfrac{1024}{4}=256$. Sum due$=256+32=288$.

MCQ 3. The BD on a bill due after a certain time is ₹18, and the BG is ₹1.80. Find the sum due. (A) ₹162 (B) ₹180 (C) ₹170 (D) ₹175

Correct Answer: (A) Solution: $TD=18-1.8=16.2$. $PW=\dfrac{16.2^2}{1.8}=\dfrac{262.44}{1.8}=145.8$. Sum due$=145.8+16.2=162$.

Type 9 — Comparison/Ratio Problems (BD vs TD vs SI)

MCQ 1. For a sum of ₹1,000 due 2 years hence at 10% per annum, compute and compare BD, TD, and ordinary SI (on the sum itself as principal). (A) SI=₹200, BD=₹200, TD≈₹166.67 (B) SI=₹200,BD=₹180,TD=₹160 (C) SI=₹180,BD=₹200,TD=₹150 (D) SI=₹190,BD=₹210,TD=₹170

Correct Answer: (A) Solution: $SI=\dfrac{1000\times10\times2}{100}=200$. $BD=\dfrac{1000\times10\times2}{100}=200$ (same formula/value as SI, since both are "interest on the face value/principal amount" — BD is ALWAYS numerically equal to SI computed on the sum due). $TD=\dfrac{1000\times20}{120}=166.67$.

MCQ 2. Verify that BD always equals SI on the face value, using a sum of ₹2,000 due 3 years hence at 5% per annum. (A) BD=SI=₹300 (B) BD=₹280,SI=₹300 (C) BD=₹320,SI=₹300 (D) BD=₹300,SI=₹280

Correct Answer: (A) Solution: $SI=\dfrac{2000\times5\times3}{100}=300$. $BD=\dfrac{2000\times5\times3}{100}=300$ — confirms $BD=SI$ always.

MCQ 3. For a sum of ₹1,500 due 4 years hence at 6% per annum, find BD, TD, and BG, and rank them. (A) BD=₹360, TD≈₹290.32, BG≈₹69.68; BD>TD, confirming BG>0 (B) BD=TD=₹360 (C) TD>BD (D) BD=₹300,TD=₹350

Correct Answer: (A) Solution: $BD=\dfrac{1500\times6\times4}{100}=360$. $RT=24$. $TD=\dfrac{1500\times24}{124}=290.32$. $BG=360-290.32=69.68$. Confirms BD>TD, with positive BG, consistent with the chapter's core principle.

4. High-Yield Speed Tricks & Shortcut Mental Models

Shortcut 1 — Recognize BD ≡ SI on Face Value (Skip Re-Deriving)

• Application: Every Type 1 problem, and any time BD needs to be cross-checked against SI.
• Mental Model: BD is not a "new" formula to memorize separately — it is EXACTLY the Simple Interest formula, applied to the face value A instead of a principal P. Whenever a question gives BD alongside A, R, T, treat it as an ordinary SI problem from the Simple Interest chapter, reusing all of that chapter's rearrangement techniques directly (solving for A, R, or T using the standard SI algebra).

Shortcut 2 — The $TD^2=PW\times BG$ Identity as a One-Line Solver

• Application: Every Type 7/8 problem where TD and BG (or enough information to find both) are known, and PW or the sum due is required.
• Mental Model: Never set up a full system of BD/TD/PW/A equations from scratch. The moment TD and BG are both known (or easily derived via $TD=BD-BG$), apply $PW=\dfrac{TD^2}{BG}$ directly as a single-step formula, then find the sum due via $A=PW+TD$ — this two-step chain is dramatically faster than re-deriving PW from the rate and time.

5. Deep-Dive: Most Frequently Asked Questions

Problem 1 (SSC/RRB Standard): The Banker's Discount on a sum of money due 3 years hence at 5% per annum is ₹360. Find the True Discount on the same sum for the same time and rate.

Traditional Method (Slow): First find the face value A from BD: $360=\dfrac{A\times5\times3}{100}\Rightarrow360=0.15A\Rightarrow A=2400$. Now find TD using A: $RT=15$. $TD=\dfrac{2400\times15}{115}=\dfrac{36000}{115}=313.04$. (Requires solving for A first, then plugging into the separate TD formula — ~35-40 seconds.)

Exam Shortcut (Fast): Recognize the direct BD-to-TD relationship without solving for A explicitly: since $BD=\frac{A\cdot RT}{100}$ means $A=\frac{100\cdot BD}{RT}$, substituting into $TD=\frac{A\cdot RT}{100+RT}$ gives the direct formula $TD=\dfrac{BD\times100}{100+RT}$ (the RT cancels): $TD=\dfrac{360\times100}{100+15}=\dfrac{36000}{115}=313.04$. Answer: ≈₹313.04, reached via a DIRECT BD-to-TD conversion formula (skipping the intermediate step of solving for the face value A) — under 15 seconds once this direct relationship is memorized as a standalone tool.

Problem 2 (UPSC/Banking Advanced): A bill of a certain face value is due after a certain time. The Banker's Discount on the bill is ₹210, and the True Discount is ₹200. Find the face value of the bill, the rate of interest, and the time, given that the time is 6 months.

Step-by-Step Breakdown:

1. First find the Banker's Gain: $BG=BD-TD=210-200=10$.
2. Use the identity $TD^2=PW\times BG$ to find the Present Worth: $PW=\dfrac{TD^2}{BG}=\dfrac{200^2}{10}=\dfrac{40000}{10}=4000$.
3. Find the face value (sum due): $A=PW+TD=4000+200=4200$.
4. Now find the rate of interest using the BD formula, since BD is SI on the face value A, with T=0.5 years (6 months) given: $BD=\dfrac{A\times R\times T}{100}\Rightarrow210=\dfrac{4200\times R\times0.5}{100}\Rightarrow210=21R\Rightarrow R=10\%$.
5. Verify using TD: $RT=10\times0.5=5$. $TD=\dfrac{4200\times5}{105}=\dfrac{21000}{105}=200$ ✓ (matches the given TD, confirming consistency).
6. Answer: Face value = ₹4,200, Rate = 10% per annum, Time = 6 months (as given). This demonstrates the complete advanced-level workflow for Banker's Discount problems with multiple unknowns: use BG (from BD−TD) and TD together via the $TD^2=PW\times BG$ identity to find PW and hence the face value FIRST, then use the now-known face value together with the ORIGINAL BD value to solve for the rate (or time, if that were the unknown instead) — this sequential "solve PW/A first, then rate/time" workflow is the standard technique for any Banker's Discount problem giving BD and TD (or BD and BG) but leaving rate or time unknown.

6. Chapter Checklist for Students

• I always compute BD as Simple Interest on the FULL FACE VALUE (A), and TD as Simple Interest on the PRESENT WORTH (PW) — never swapping these two.
• I recognize that BD is numerically identical to ordinary Simple Interest computed on the sum due, and reuse SI-chapter techniques directly for BD problems.
• I verify that BG is always positive (BD > TD) as a standard sanity check on my calculations.
• I apply $PW=\dfrac{TD^2}{BG}$ as a direct one-line solver whenever TD and BG are both known or easily derived.
• I follow the "solve for PW/Face Value first (using TD and BG), then solve for rate or time" sequential workflow for multi-unknown Banker's Discount problems.