True Discount

PAGE 1 — BASIC CONCEPTS

True Discount is an important topic in commercial mathematics. It is mainly used in problems involving bills, promissory notes, present value, future value and simple interest.

1. IMPORTANT TERMS

Term	Meaning
Present Value (PV)	The amount of money that is equivalent to a future sum at the present time, at the given simple-interest rate.
Amount / Sum Due (A)	The amount payable at the end of the specified period.
True Discount (TD)	The difference between the amount due and its present value.
Rate (R)	The annual simple-interest rate.
Time (T)	The remaining time until the amount becomes due.

True Discount = Amount Due − Present Value

TD = A − PV

2. PRESENT VALUE

If an amount A is due after T years and the simple interest rate is R%, then its present value is found by discounting the future amount back to the present.

PV = A × 100 ÷ (100 + RT)

Equivalently:

PV = A ÷ [1 + RT/100]

3. TRUE DISCOUNT FORMULA

Since:

TD = A − PV

We obtain:

TD = A × RT ÷ (100 + RT)

Example 1: Find the true discount on ₹1,000 due after 2 years at 10% per annum simple interest.

A = ₹1,000, R = 10%, T = 2

PV = 1000 × 100 ÷ (100 + 10 × 2)
= 100000 ÷ 120
= ₹833.33

TD = 1000 − 833.33
= ₹166.67

4. RELATION WITH SIMPLE INTEREST

The simple interest calculated on the present value for the remaining period is equal to the true discount.

TD = SI on Present Value

TD = PV × R × T ÷ 100

Example 2: If the present value is ₹800, the rate is 10% and the time is 2 years:

TD = 800 × 10 × 2 ÷ 100
= ₹160

Amount due = 800 + 160 = ₹960.

TIP: True discount is always calculated by finding the Present Value first, or by directly using the true-discount formula.

PAGE 2 — IMPORTANT FORMULAS

5. BASIC FORMULA TRIANGLE

Let:
A = Amount due
PV = Present Value
TD = True Discount
R = Rate per annum
T = Time in years

A = PV + TD

TD = A − PV

A = PV(1 + RT/100)

6. DIRECT FORMULAS

PV = 100A ÷ (100 + RT)

TD = ART ÷ (100 + RT)

TD = PV × RT ÷ 100

A = TD(100 + RT) ÷ RT

7. FINDING THE AMOUNT DUE

Example: The true discount on a sum due after 3 years at 8% per annum is ₹240. Find the amount due.

TD = ART/(100 + RT)

240 = A × 8 × 3/(100 + 8 × 3)
240 = 24A/124
A = 240 × 124/24
= ₹1,240

8. FINDING PRESENT VALUE FROM TRUE DISCOUNT

Present Value = Amount Due − True Discount

Example: A bill of ₹2,500 has a true discount of ₹250. Find its present value.

PV = 2500 − 250
= ₹2,250

9. FINDING RATE

From:

TD = PV × R × T ÷ 100

Therefore:

R = 100 × TD ÷ (PV × T)

Example: The true discount on a sum is ₹180 and its present value is ₹900 for 2 years. Find the rate.

R = 100 × 180 ÷ (900 × 2)
= 18000 ÷ 1800
= 10%

10. FINDING TIME

T = 100 × TD ÷ (PV × R)

Example: The present value of a bill is ₹1,000 and the true discount is ₹200 at 10% p.a. Find the time.

T = 100 × 200 ÷ (1000 × 10)
= 2 years

11. IMPORTANT RELATION

Amount : Present Value = (100 + RT) : 100

True Discount : Present Value = RT : 100

Shortcut: If RT is known, many true-discount problems can be solved directly without calculating simple interest separately.

PAGE 3 — BANKER'S DISCOUNT

12. BANKER'S DISCOUNT

Banker's Discount (BD) is the simple interest calculated on the face value or amount due for the unexpired period.

BD = A × R × T ÷ 100

Compare this with true discount, which is calculated using the present value.

TD = PV × R × T ÷ 100

Example: Find the Banker's Discount on ₹2,000 due after 3 years at 10% p.a.

BD = 2000 × 10 × 3 ÷ 100
= ₹600

13. BANKER'S DISCOUNT VS TRUE DISCOUNT

Particular	True Discount	Banker's Discount
Calculated on	Present Value	Amount Due
Formula	PV × RT/100	A × RT/100
Usually denoted	TD	BD

14. RELATION BETWEEN BD & TD

Since:

BD = A × RT/100
TD = PV × RT/100

And:

A = PV(100 + RT)/100

Therefore:

BD / TD = A / PV

BD / TD = (100 + RT) / 100

15. BANKER'S GAIN

Banker's Gain = Banker's Discount − True Discount

Thus:

BG = BD − TD

Example: A sum of ₹5,000 is due after 2 years at 10% p.a. Find BD, TD and Banker's Gain.

Step 1 — BD:
BD = 5000 × 10 × 2/100 = ₹1,000

Step 2 — Present Value:
PV = 5000 × 100/120
= ₹4,166.67

Step 3 — TD:
TD = 5000 − 4166.67
= ₹833.33

Step 4 — Banker's Gain:
BG = 1000 − 833.33
= ₹166.67

16. FORMULA FOR BANKER'S GAIN

BG = A(RT/100)² ÷ (1 + RT/100)

A commonly useful simplified form is:

BG = A(RT)² ÷ [100(100 + RT)]

Exam Shortcut: For BD and TD, first identify the base:
BD → Amount Due
TD → Present Value

PAGE 4 — QUICK REVISION

17. COMPLETE FORMULA SHEET

Concept	Formula
Amount	A = PV(1 + RT/100)
Present Value	PV = 100A/(100 + RT)
True Discount	TD = A − PV
True Discount — direct	TD = ART/(100 + RT)
True Discount from PV	TD = PV × RT/100
Banker's Discount	BD = ART/100
Banker's Gain	BG = BD − TD
Amount : PV	(100 + RT) : 100
TD : PV	RT : 100
BD : TD	(100 + RT) : 100

18. SOLVED MIXED EXAMPLES

Example 1: Find the true discount on ₹3,000 due after 4 years at 5% p.a.

TD = 3000 × 5 × 4/(100 + 5 × 4)
= 60000/120
= ₹500

Example 2: A bill of ₹2,400 is due after 2 years at 10% p.a. Find its present value.

PV = 2400 × 100/(100 + 20)
= 240000/120
= ₹2,000

Example 3: Find the Banker's Discount on ₹4,000 due after 3 years at 8% p.a.

BD = 4000 × 8 × 3/100
= ₹960

Example 4: A sum of ₹6,000 is due after 2 years at 10%. Find the Banker's Gain.

BD = 6000 × 10 × 2/100 = ₹1,200
TD = 6000 × 20/120 = ₹1,000
BG = 1200 − 1000
= ₹200

19. PRACTICE QUESTIONS

Find the true discount on ₹1,500 due after 2 years at 10% p.a.
Find the present value of ₹2,400 due after 3 years at 10% p.a.
Find the Banker's Discount on ₹5,000 due after 2 years at 8% p.a.
A bill of ₹3,000 is due after 2 years at 10%. Find its true discount.
A sum has a present value of ₹1,200 at 10% p.a. for 2 years. Find the true discount.
The true discount on a sum due after 2 years at 10% is ₹200. Find the amount due.
A sum of ₹4,000 is due after 3 years at 5%. Find the present value.
A bill of ₹6,000 is due after 2 years at 10%. Find the Banker's Gain.
The present value of a bill is ₹1,500 and the true discount is ₹300 for 2 years. Find the rate.
The present value is ₹2,000 and the true discount is ₹400 at 10% p.a. Find the time.

20. ANSWER KEY