Bankers discount

ARITHMETIC HANDBOOK • BANKER'S DISCOUNT

PAGE 1 — BASIC CONCEPTS

1. What is a Banker's Discount?

Banker's Discount (BD) is the simple interest calculated on the amount due for the remaining period of a bill.

When a bill is discounted by a bank before its maturity date, the bank calculates interest on the face value or amount due. This interest is called Banker's Discount.

Important: Banker's Discount is calculated on the amount due (A), whereas True Discount is calculated on the present value (PV).

2. Important Terms

Term	Meaning
Amount Due (A)	Face value of the bill payable at maturity.
Present Value (PV)	Value of the bill today at the given rate and time.
Banker's Discount (BD)	Simple interest on the amount due for the unexpired period.
True Discount (TD)	Difference between amount due and present value.
Banker's Gain (BG)	Difference between Banker's Discount and True Discount.
Rate (R)	Annual rate of interest.
Time (T)	Unexpired time before maturity, generally in years.

3. Basic Formula

Banker's Discount = Amount Due × Rate × Time / 100

BD = ART / 100

Where:

A = Amount due
R = Rate percent per annum
T = Unexpired time in years

4. Time Conversion

Given Time	Convert Into Years
6 months	6/12 = 1/2 year
4 months	4/12 = 1/3 year
3 months	3/12 = 1/4 year
2 months	2/12 = 1/6 year
15 months	15/12 = 5/4 years

Example 1: Find the Banker's Discount on a bill of ₹8,000 due after 6 months at 10% per annum.

BD = ART/100
= 8000 × 10 × 1/2 ÷ 100
= ₹400

BANKER'S DISCOUNT • FORMULA HANDBOOK

PAGE 2 — IMPORTANT FORMULAS

5. Main Formula Set

BD = ART/100

A = 100 × BD / (RT)

R = 100 × BD / (AT)

T = 100 × BD / (AR)

6. Relation Between Banker's Discount and Present Value

Since Banker's Discount is simple interest on the amount due:

BD = A − PV_bank

Therefore, the amount received after deducting Banker's Discount is:

Amount Received = A − BD
= A(1 − RT/100)

Exam Alert: Do not confuse Banker's Discount with True Discount. Banker's Discount uses the amount due as the base, while True Discount uses the present value as the base.

7. True Discount and Banker's Discount

For a bill whose amount due is A, rate is R and time is T:

True Discount:
TD = ART / (100 + RT)

Banker's Discount:
BD = ART / 100

Banker's Gain:
BG = BD − TD

8. Important Ratios

BD : TD = (100 + RT) : 100

TD : PV = RT : 100

BD : BG = (100 + RT) : RT

BG = A(RT)² / [100(100 + RT)]

Example 2: Find the Banker's Discount on ₹12,000 due after 9 months at 8% per annum.

T = 9/12 = 3/4 year
BD = 12000 × 8 × 3/4 ÷ 100
= ₹720

Example 3: A bill of ₹15,000 is due after 4 months. If the Banker's Discount is ₹300, find the rate.

R = 100 × BD/(AT)
= 100 × 300 / [15000 × 4/12]
= 30000/5000
= 6%

BANKER'S DISCOUNT • SOLVED EXAMPLES

PAGE 3 — SOLVED PROBLEMS

9. Finding Banker's Discount

Example 4: A bill of ₹20,000 is due after 5 months. Find the Banker's Discount at 12% per annum.

T = 5/12 year
BD = 20000 × 12 × 5/12 ÷ 100
= ₹1,000

10. Finding Amount Due

Example 5: The Banker's Discount on a bill at 10% per annum for 6 months is ₹500. Find the amount due.

A = 100 × BD/(RT)
= 100 × 500/(10 × 1/2)
= 50000/5
= ₹10,000

11. Finding Time

Example 6: The Banker's Discount on ₹16,000 at 15% per annum is ₹600. Find the unexpired time.

T = 100 × BD/(AR)
= 100 × 600/(16000 × 15)
= 60000/240000
= 1/4 year
= 3 months

12. Banker's Gain

Banker's Gain is the excess of Banker's Discount over True Discount.

BG = BD − TD

Example 7: Find the Banker's Gain on a bill of ₹10,000 due after 6 months at 10% per annum.

``` Banker's Discount:
BD = 10000 × 10 × 1/2 ÷ 100 = ₹500

True Discount:
TD = ART/(100 + RT)
= 10000 × 10 × 1/2 / (100 + 5)
= 50000/105
≈ ₹476.19

Therefore,
BG = 500 − 476.19
= ₹23.81 ```

13. Shortcut for Banker's Gain

BG = A(RT)² / [100(100 + RT)]

Example 8: For ₹10,000, 10% rate and 6 months:

RT = 10 × 1/2 = 5
BG = 10000 × 25/[100 × 105]
= ₹23.81 approximately

Quick Tip: If BD is known and TD is required, first find the amount due or use the relationship:

BD : TD = (100 + RT) : 100

BANKER'S DISCOUNT • REVISION & PRACTICE

PAGE 4 — SHORTCUTS & PRACTICE SET

14. Quick Revision Formula Chart

Quantity	Formula
Banker's Discount	BD = ART/100
Amount Due	A = 100BD/(RT)
Rate	R = 100BD/(AT)
Time	T = 100BD/(AR)
True Discount	TD = ART/(100+RT)
Banker's Gain	BG = BD − TD
BD : TD	(100+RT) : 100
TD : PV	RT : 100
BD : BG	(100+RT) : RT
Banker's Gain	A(RT)²/[100(100+RT)]

15. Fast Calculation Tips

Tip 1: Always convert months into years before using BD = ART/100.

Tip 2: Remember the base:
Banker's Discount → Amount Due
True Discount → Present Value

Tip 3: For 6 months, use T = 1/2; for 3 months, use T = 1/4; for 9 months, use T = 3/4.

Tip 4: When both BD and TD are involved, remember:
Banker's Gain = BD − TD

16. Practice Questions

Find the Banker's Discount on ₹10,000 due after 6 months at 8% p.a.
A bill of ₹12,000 is due after 4 months. Find BD at 9% p.a.
The Banker's Discount on a bill for 1 year at 12% is ₹720. Find the amount due.
The Banker's Discount on ₹15,000 at 8% p.a. is ₹400. Find the time.
A bill of ₹20,000 is due after 3 months at 10% p.a. Find the Banker's Discount.
Find the Banker's Discount on ₹25,000 due after 8 months at 6% p.a.
Find the Banker's Gain on ₹10,000 due after 1 year at 10% p.a.
A bill of ₹16,000 is due after 9 months at 8% p.a. Find both BD and TD.
If BD is ₹600, rate is 10% and time is 6 months, find the amount due.
A bill of ₹24,000 is discounted at 12% p.a. for 5 months. Find the Banker's Discount.

17. Answer Key