True Discount and Banker’s Discount

Suppose you must pay ₹12,100 after two years and the rate is 10% compounded annually. If you have the money today you need not pay the full ₹12,100 — paying ₹10,000 now clears the debt, because ₹10,000 grows to ₹12,100 in two years. That ₹10,000 is the present worth (PW); ₹12,100 is the sum due (amount); and the ₹2,100 relief is the true discount (TD). TD = sum due − PW, and it is really the interest on the PW.

A = P W + T D, T D = \frac{P W \times R \times T}{100}, P W = \frac{A \times 100}{100 + R T}, T D = \frac{A \times R T}{100 + R T}

Core formulas at simple interest

1

True discount

PW · A · TD

The relations between present worth, true discount and sum due.

Finding PW, TD and A

Divide the debt

Present worth and true discount at simple and compound interest.

To find	Formula
PW	$\frac{A \times 100}{100 + R T}$
TD	$A - P W = \frac{P W R T}{100} = \frac{A R T}{100 + R T}$
A	$P W + T D = \frac{T D ( 100 + R T )}{R T}$
PW at compound interest	$\frac{A}{( 1 + \frac{R}{100} ) ^{n}}$

Example 1 · Basic

A debt of ₹9,600 is due in 5 years at 12% simple interest per annum. Find the present worth and the true discount.

Solution, step by step

1 $P W = \frac{9600 \times 100}{100 + 12 \times 5} = \frac{960000}{160} = 6000$ .
2 $T D = 9600 - 6000 = 3600$ (check: $6000 \times \frac{12 \times 5}{100} = 3600$ ✓)

Answer PW ₹6,000; TD ₹3,600

Example 2 · Standard

A debt of ₹7,200 is due in 4 years at 5% simple interest per annum. Find the PW and the TD.

Solution, step by step

1 $P W = \frac{7200 \times 100}{100 + 5 \times 4} = \frac{720000}{120} = 6000$ .
2 $T D = 7200 - 6000 = 1200$ (check: $6000 \times \frac{5 \times 4}{100} = 1200$ ✓)

Answer PW ₹6,000; TD ₹1,200

Example 3 · Standard

What is the present worth of ₹13,310 due 3 years hence at 10% compound interest per annum?

Solution, step by step

1 $P W = \frac{13310}{( 1.1 ) ^{3}} = \frac{13310}{1.331}$ .
2 $= 10000$ and $T D = 13310 - 10000 = 3310$

Answer ₹10,000 (TD ₹3,310)

Example 4 · Standard

The true discount on a debt is ₹1,500 at 15% simple interest per annum for 4 years. What is the sum due?

Solution, step by step

1 $A : P W : T D = (100 + 60) : 100 : 60 = 160 : 100 : 60$ ⟹ 60 parts of TD are ₹1,500.
21 part $= 25$ ; $A = 160 \times 25 = 4000$ and $P W = 2500$ .

Answer ₹4,000

Example 5 · Advanced

A man pays ₹8,400 today to clear a sum due after 2 years. The rate is 10% compound interest per annum. Find the sum due and the true discount.

Solution, step by step

1 $A = P W \times (1.1)^{2} = 8400 \times 1.21 = 10164$ .
2 $T D = 10164 - 8400 = 1764$

Answer Due ₹10,164; TD ₹1,764

BasicTry it yourself first

A debt of ₹3,600 is due 2 years hence. If simple interest is 10% per annum what is its present worth?

BasicTry it yourself first

A debt of ₹6,200 is due 2 years hence. If simple interest is 12% per annum what is its present worth?

StandardTry it yourself first

A debt of ₹5,200 is due 5 years hence. If simple interest is 6% per annum what is its present worth?

StandardTry it yourself first

A debt of ₹9,100 is due 3 years hence. If simple interest is 10% per annum what is its present worth?

AdvancedTry it yourself first

A debt of ₹7,500 is due 2 years hence. If simple interest is 12.5% per annum what is its present worth?

BasicTry it yourself first

A debt of ₹6,500 is due 2 years hence at 15% simple interest per annum. What is the true discount if it is paid today?

StandardTry it yourself first

A debt of ₹4,960 is due 3 years hence at 8% simple interest per annum. What is the true discount if it is paid today?

StandardTry it yourself first

A debt of ₹3,960 is due 2 years hence at 16% simple interest per annum. What is the true discount if it is paid today?

AdvancedTry it yourself first

A debt of ₹4,600 is due 3 years hence at 5% simple interest per annum. What is the true discount if it is paid today?

BasicTry it yourself first

What is the true discount on a present worth of ₹6,000 at 12% per annum for 9 months?

StandardTry it yourself first

What is the true discount on a present worth of ₹8,000 at 9% per annum for 6 months?

StandardTry it yourself first

A debt due 2 years hence has a true discount of ₹1,200 at 12% per annum. What is the sum due?

AdvancedTry it yourself first

A debt due 3 years hence has a true discount of ₹2,100 at 10% per annum. What is the sum due?

StandardTry it yourself first

What is the present worth of ₹12,100 due 2 years hence at 10% compound interest per annum?

AdvancedTry it yourself first

What is the present worth of ₹15,625 due 3 years hence at 25% compound interest per annum?

AdvancedTry it yourself first

What is the present worth of ₹7,200 due 2 years hence at 20% compound interest per annum?

TD and simple interest

SI on the whole sum, TD only on the PW

On the same sum, time and rate the simple interest exceeds the TD.

A = \frac{S I \times T D}{S I - T D}, R = \frac{( S I - T D ) \times 100}{T D \times T}

From SI and TD

On the same sum $A$ , $S I = \frac{A R T}{100}$ is charged on the whole sum while $T D = \frac{P W R T}{100}$ only on the PW — so $S I > T D$ . The gap $S I - T D = \frac{( A - P W ) R T}{100} = \frac{T D \cdot R T}{100}$ is the interest on the TD for $T$ years.

Example 6 · Standard

On a sum due in 5 years the true discount is ₹300 and the simple interest ₹375. Find the sum and the rate.

Solution, step by step

1Sum $= \frac{375 \times 300}{375 - 300} = 1500$ .
2Rate $= \frac{( 375 - 300 ) \times 100}{300 \times 5} = 5%$

Answer ₹1,500 and 5%

Example 7 · Advanced

On a sum due in 2 years the true discount is ₹1,000 and the simple interest ₹1,200. Find the rate and the sum due.

Solution, step by step

1 $S I - T D = 200$ , the interest on the TD for 2 years ⟹ $R = \frac{200 \times 100}{1000 \times 2} = 10%$ .
2Sum due $= \frac{1200 \times 1000}{200} = 6000$ .

Answer 10% and ₹6,000

BasicTry it yourself first

The true discount on a sum due is ₹400 and the simple interest on the same sum for the same time and rate is ₹480. What is the sum due?

StandardTry it yourself first

The true discount on a sum due is ₹1,200 and the simple interest on the same sum for the same time and rate is ₹1,488. What is the sum due?

AdvancedTry it yourself first

The true discount on a sum due is ₹1,200 and the simple interest on the same sum for the same time and rate is ₹1,488. What is the sum due?

StandardTry it yourself first

For a sum due 3 years hence, the true discount is ₹1,200 and the simple interest is ₹1,488. What is the rate per annum?

AdvancedTry it yourself first

For a sum due 2 years hence, the true discount is ₹1,200 and the simple interest is ₹1,488. What is the rate per annum?

AdvancedTry it yourself first

For a sum due 3 years hence, the true discount is ₹600 and the simple interest is ₹690. What is the rate per annum?

2

Banker’s discount

Discounting a bill · BD · BG

A bank discounting a bill (hundi), the banker’s discount and the banker’s gain.

The bill and the banker’s discount

Interest on the face value for the unexpired time

What the bank deducts when a bill is cashed before its due date.

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Relations among BD, TD and BG

Given any two of the three

BG is interest on TD, so $A$ , PW and $RT$ can all be recovered.

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3

Traps and fast tricks

Slips · tricks

Where marks go in these two discounts.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

The 100 + RT denominator

Habits that shorten the formulas.

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4

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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5

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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True discount

Finding PW, TD and A

TD and simple interest

Banker’s discount

The bill and the banker’s discount

Relations among BD, TD and BG

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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