Quick Concept Summary
Banker's Discount builds on True Discount. The key difference is what the interest is calculated on:
| Term | Interest calculated on | When |
|---|
| True Discount (TD) | Present Worth (PW) | Paying early — theoretically fair |
| Banker's Discount (BD) | Full Face Value (Amount) | What banks actually charge |
| Banker's Gain (BG) | BD − TD | Extra profit for the bank |
The core relationships:
Amount (Face Value) = PW + TD
BD = SI on Amount
TD = SI on PW
BG = BD − TD = SI on TD
Since Amount > PW: BD > TD > 0
Key Formulas
| Formula | Expression |
|---|
| Banker's Discount | BD = (A × R × T) / 100 |
| True Discount | TD = (A × R × T) / (100 + RT) |
| Present Worth | PW = (A × 100) / (100 + RT) |
| Banker's Gain | BG = BD − TD |
| BG alternate | BG = TD² / PW |
| BG alternate 2 | BG = BD × TD / A |
| TD from BD and BG | TD = BD − BG |
| PW from TD and BG | PW = TD² / BG |
| BD from BG and TD | BD = TD + BG |
| Amount from BD and BG | A = BD²/ BG × ... (use BD/TD = A/PW) |
The Ratio Relationship
BD : TD = Amount : PW = (100 + RT) : 100
Worked Examples
Example 1: Find BD, TD, and BG
Problem:
Find BD, TD, and BG on Rs 5,000 due 2 years hence at 10% per annum.
Given:
Amount A = Rs 5,000
R = 10%, T = 2 years
Step 1: Banker's Discount
BD = (A × R × T) / 100
BD = (5000 × 10 × 2) / 100
BD = Rs 1,000
Step 2: True Discount
TD = (A × R × T) / (100 + RT)
TD = (5000 × 10 × 2) / (100 + 20)
TD = 100000 / 120
TD = Rs 833.33
Step 3: Banker's Gain
BG = BD − TD = 1000 − 833.33 = Rs 166.67
Answer: BD = Rs 1,000 | TD = Rs 833.33 | BG = Rs 166.67
Example 2: Find the Amount from BD and Rate
Problem:
The banker's discount on a bill due 3 years hence at 12% per annum is Rs 360.
Find the true discount and the amount.
Given:
BD = Rs 360, R = 12%, T = 3 years
Step 1: Find Amount from BD formula
BD = (A × R × T) / 100
360 = (A × 12 × 3) / 100
360 = 36A / 100
A = 36000 / 36 = Rs 1,000
Step 2: Find TD
TD = (A × RT) / (100 + RT)
TD = (1000 × 36) / (100 + 36)
TD = 36000 / 136
TD = Rs 264.71
Step 3: BG = BD − TD = 360 − 264.71 = Rs 95.29
Answer: Amount = Rs 1,000 | TD ≈ Rs 264.71 | BG ≈ Rs 95.29
Example 3: Find TD Using BG and BD
Problem:
The banker's discount on a certain sum is Rs 72 and the banker's gain is Rs 12.
Find the true discount and the sum.
Given:
BD = Rs 72, BG = Rs 12
Step 1: True Discount
TD = BD − BG = 72 − 12 = Rs 60
Step 2: Find Present Worth
BG = TD² / PW
12 = 60² / PW
12 = 3600 / PW
PW = 3600 / 12 = Rs 300
Step 3: Amount
Amount = PW + TD = 300 + 60 = Rs 360
Answer: TD = Rs 60 | PW = Rs 300 | Amount = Rs 360
Example 4: Find Rate from BD and TD
Problem:
The banker's discount and true discount on a sum due in 1 year are Rs 84 and Rs 70
respectively. Find the rate of interest and the sum.
Given:
BD = Rs 84, TD = Rs 70, T = 1 year
Step 1: BG = BD − TD = 84 − 70 = Rs 14
Step 2: PW = TD² / BG = 70² / 14 = 4900 / 14 = Rs 350
Step 3: Amount = PW + TD = 350 + 70 = Rs 420
Step 4: Find Rate using TD formula
TD = (A × R × T) / (100 + RT)
70 = (420 × R × 1) / (100 + R)
70(100 + R) = 420R
7000 + 70R = 420R
7000 = 350R
R = 20% per annum
Verify BD: (420 × 20 × 1)/100 = 84 ✓
Answer: Rate = 20% | Sum = Rs 420
Example 5: BD Discounted at Different Rates
Problem:
The present worth of a bill due 6 months hence is Rs 1,500. If the banker's
discount is Rs 75, find the true discount and rate of interest.
Given:
PW = Rs 1,500, BD = Rs 75, T = 6 months = 1/2 year
Step 1: Amount = PW + TD → but we need to find TD first
From ratio: BD / TD = A / PW
Let TD = x. Then A = PW + x = 1500 + x
BD/TD = A/PW
75/x = (1500 + x)/1500
75 × 1500 = x(1500 + x)
112500 = 1500x + x²
x² + 1500x − 112500 = 0
(x + 1750)(x − ... ) — use quadratic formula or notice:
Actually easier: TD = BD × PW / A → circular.
Better: Use BG = BD − TD and BG = TD²/PW
Let TD = t
BD − t = t²/1500
75 − t = t²/1500
1500(75 − t) = t²
112500 − 1500t = t²
t² + 1500t − 112500 = 0
t = (−1500 + √(2250000 + 450000)) / 2
t = (−1500 + √2700000) / 2
t = (−1500 + 1643.17) / 2
t = 143.17 / 2 ≈ 71.58
Practical approach (cleaner):
Use BD/TD = A/PW and BG = TD²/PW
BG = BD²/(BD+TD) × (1/PW)...
Simpler given exam context:
Rate: BD = (PW+TD)×R×T/100 = A×R×T/100
Also SI on PW: TD = PW×R×T/100
75 (on A) and TD (on PW=1500)
Since TD = PW×R×T/100: TD = 1500×R×(1/2)/100 = 7.5R
Since BD = A×R×T/100 and A = 1500+TD = 1500+7.5R:
75 = (1500+7.5R)×R×(1/2)/100
75×200 = (1500+7.5R)×R
15000 = 1500R + 7.5R²
7.5R² + 1500R − 15000 = 0
R² + 200R − 2000 = 0
R = (−200 + √(40000+8000))/2 = (−200+√48000)/2 ≈ (−200+219.09)/2 ≈ 9.54%
Answer: TD ≈ Rs 71.58, R ≈ 9.54% (Note: this is an atypical problem; cleaner exam versions use whole numbers)
Example 6: Comparing BD and TD
Problem:
For a bill of Rs 1,300 due in 2 years at 10%, by how much does BD exceed TD?
Step 1: BD
BD = (1300 × 10 × 2) / 100 = Rs 260
Step 2: TD
TD = (1300 × 10 × 2) / (100 + 20) = 26000/120 = Rs 216.67
Step 3: BG = BD − TD = 260 − 216.67 = Rs 43.33
Answer: BD exceeds TD by Rs 43.33 (= Banker's Gain)
Shortcuts and Tricks
Trick 1: The Key Ratio
BD/TD = Face Value / PW = (100 + RT)/100
Use this to find any one value when others are known.
Trick 2: BG = SI on TD
Banker's Gain is the SI on the True Discount itself. BG = (TD × R × T) / 100
Trick 3: Given BD and BG, find TD
TD = BD − BG (simply subtract) Then PW = TD²/BG
Trick 4: Exam-Friendly Order
1. Find BG = BD − TD (if both given) 2. Find PW = TD²/BG 3. Find A = PW + TD
TNPSC / SSC Exam Pattern
| Question Type | Frequency | Difficulty |
|---|
| Find BD, TD, BG from Amount | Common | Medium |
| Find Amount from BD and Rate | Common | Medium |
| Find TD from BD and BG | Common | Medium |
| Find Rate given BD and TD | Occasional | Hard |
| Relationship questions (ratio) | Occasional | Medium |
Practice Questions
Q1. Find BD on Rs 2,000 due 1 year hence at 10%. Answer: BD = (2000×10×1)/100 = Rs 200
Q2. Find TD on Rs 2,000 due 1 year hence at 10%. Answer: TD = (2000×10×1)/(100+10) = 20000/110 = Rs 181.82
Q3. Find BG when BD = Rs 200 and TD = Rs 181.82. Answer: BG = 200 − 181.82 = Rs 18.18
Q4. BD = Rs 50, BG = Rs 5. Find TD. Answer: TD = BD − BG = 50 − 5 = Rs 45
Q5. BD = Rs 50, BG = Rs 5. Find PW and Amount. Answer: PW = TD²/BG = 45²/5 = 2025/5 = Rs 405. Amount = 405+45 = Rs 450.
Q6. BD on a bill is Rs 120, due 3 months hence at 8% p.a. Find the face value. Answer: BD = A×R×T/100 → 120 = A×8×(1/4)/100 = 2A/100 → A = Rs 6,000
Q7. The BG is 1/25 of TD. If TD = Rs 50, find BD. Answer: BG = 50/25 = Rs 2. BD = TD + BG = 50 + 2 = Rs 52
Q8. BD = Rs 40, TD = Rs 36. Find BG and PW. Answer: BG = 40−36 = Rs 4. PW = TD²/BG = 1296/4 = Rs 324
Q9. What is the BG when BD = Rs 100 and PW = Rs 400? Answer: TD = BD × PW/A → need A first: A = PW + TD → circular. Use: BG = BD²/(BD + PW×something)... Actually: BD/PW = (100+RT)/100, and TD = BD×100/(100+RT). BG = BD − TD. With BD=100, PW=400: A = PW+TD. From BD formula: 100 = A×RT/100 and PW = A×100/(100+RT)=400 → A/400 = (100+RT)/100. Let 100+RT = k: A = 400k/100 = 4k and BD = A×(k-100)/100×(100/k) ... Let me just use BG = TD²/PW. We need TD: TD² = BG×PW → need BG first. Use BG = BD×TD/A. This needs A. From PW=400 and BD=100: BG = (BD−TD). Set TD=t: (100−t) = t²/400 → 400(100−t)=t² → t²+400t−40000=0 → t=(−400+√(160000+160000))/2 = (−400+565.7)/2 ≈ 82.8. BG≈17.2
Q10. If BD exceeds TD by Rs 10 and TD = Rs 40, find the rate if T = 1 year. Answer: BG=10, TD=40. PW = TD²/BG = 1600/10 = Rs 160. A = 160+40 = Rs 200. Rate: TD = A×RT/(100+RT) → 40 = 200×R/(100+R) → 4000+40R = 200R → 160R = 4000 → R = 25%
