Hello students, I am Rahul Sir from OdTutor, and today we are going to thoroughly understand a chapter that almost every competitive exam aspirant either partially prepares or completely avoids — True Discount. And I completely understand why. When students first encounter this topic, the terminology feels oddly similar to Simple Interest, the formulas look confusingly overlapping, and the questions seem to be asking something that isn’t entirely clear. The result is that most students make a half-hearted attempt at memorizing a formula or two, get confused when the question is even slightly differently worded, and end up skipping these questions entirely in the exam hall.
Let me tell you something that I tell every single batch of students I teach at OdTutor: True Discount is not a difficult chapter. It is a misunderstood chapter. And there is a very big difference between those two things. A difficult chapter requires exceptional mathematical ability. A misunderstood chapter simply requires someone to explain it clearly, once, in the right way. Once you genuinely understand what True Discount means — not just the formula, but the actual real-world situation it describes — everything else falls into place almost effortlessly.
The relationship between True Discount, Present Worth, and the Amount due is logical and intuitive. The formulas are few, the question types are limited, and the chapter is short enough to master completely within one focused week of preparation. In IBPS PO, IBPS Clerk, SBI PO, SBI Clerk, SSC, and Railway exams, True Discount questions are among the quickest to solve for a prepared student — typically under 45 seconds — which makes this chapter a tremendous scoring opportunity that you absolutely cannot afford to leave on the table.
In this article, I am going to teach you True Discount exactly the way I teach it in my live classes at OdTutor — starting from the real-world concept, building up through every formula logically, and walking you through every major question type with fully solved examples. Read carefully, practice every example, and by the end of this article you will approach True Discount questions with complete clarity and genuine confidence.
Let’s begin.
1. Understanding True Discount — The Real-World Concept First
Before any formula, before any shortcut, I want you to understand the actual situation that True Discount describes. This is the one investment I always insist on in my classes, and it pays back enormously when students sit down to solve questions.
Imagine your friend owes you ₹1,100 but this amount is due one year from now — not today. He will pay you ₹1,100 exactly one year later. Now you need money today, so you go to someone and say: “I have a document that says I will receive ₹1,100 one year from now. How much will you give me for it today?” The person calculates that if he gives you some amount today and charges interest at, say, 10% per annum, that amount should grow to ₹1,100 in one year. He works backward from ₹1,100 to find what amount, at 10%, becomes ₹1,100 after one year. That amount is called the Present Worth (PW).
In this case: PW × (1 + 10/100) = 1100, so PW = 1100/1.1 = ₹1,000.
So you receive ₹1,000 today, and the person waits one year to collect ₹1,100.
The difference between what is due in the future (₹1,100) and what is paid today (₹1,000) is called the True Discount (TD). Here, TD = 1100 − 1000 = ₹100.
The future amount due — ₹1,100 in this case — is called the Amount (A) or the Bill Value.
So the three key quantities are:
- Amount (A): The total sum due at a future date. Also called the Bill Value or the Sum Due.
- Present Worth (PW): The actual current value of that future amount — what someone would pay today to settle the debt.
- True Discount (TD): The difference between the Amount and the Present Worth. TD = A − PW.
One line summary I give every student: True Discount is the interest on the Present Worth, not on the Amount.
This single distinction — interest on PW, not on A — is what separates True Discount from the concept of Banker’s Discount, which we will revisit later. Internalize this now, and you will never confuse the two again.
2. The Core Formulas of True Discount
Now that the concept is crystal clear, let’s derive and list every formula you need. I want you to see where each formula comes from, because that understanding lets you reconstruct any formula you forget rather than panicking during the exam.
We know: TD = A − PW … (1)
We also know that TD is the Simple Interest on PW for the given time at the given rate.
So: TD = (PW × R × T) / 100 … (2)
From (1): PW = A − TD
Substituting in (2): TD = ((A − TD) × R × T) / 100
Solving for TD:
100 × TD = A × R × T − TD × R × T
TD(100 + RT) = A × R × T
TD = (A × R × T) / (100 + R × T) … (3)
This is the master formula for True Discount. And from this, we can derive Present Worth:
Since PW = A − TD:
PW = A × 100 / (100 + R × T) … (4)
And just as TD is the SI on PW, we have an extremely useful relationship:
TD = SI on PW
Also: SI on A > TD (always, because SI is calculated on A which is larger than PW)
One more important relationship that IBPS exams love to test:
TD = (SI × PW) / A
Or equivalently: SI − TD = SI × TD / PW
And: PW = TD² / (SI − TD) … when SI and TD are given
Let me also state the relationship between SI and TD clearly:
If SI is the simple interest on Amount A for the same rate and time:
TD / SI = PW / A
These relationships might look like a lot right now, but each one comes directly from the basic definitions. Once you practice enough questions using the master formula (Formula 3), the others will feel natural and you’ll know exactly when to use each one.
3. Solved Examples — Basic True Discount Calculations
Let’s immediately apply the formulas to the most straightforward question types that appear in IBPS exams. I will walk through each solution step by step so you can see the thought process, not just the answer.
Question 1: Find the True Discount on ₹1,650 due 2 years hence at 5% per annum simple interest.
Solution:
A = ₹1,650, R = 5%, T = 2 years
Using the master formula:
TD = (A × R × T) / (100 + R × T)
TD = (1650 × 5 × 2) / (100 + 5 × 2)
TD = 16500 / 110
TD = ₹150
And Present Worth = A − TD = 1650 − 150 = ₹1,500
Verification: SI on ₹1,500 at 5% for 2 years = (1500 × 5 × 2)/100 = ₹150 ✓ (TD equals SI on PW, confirmed.)
Question 2: What is the Present Worth of ₹2,200 due 4 years hence at 10% per annum?
Solution:
A = ₹2,200, R = 10%, T = 4 years
PW = (A × 100) / (100 + R × T)
PW = (2200 × 100) / (100 + 10 × 4)
PW = 220000 / 140
PW = ₹1,571.43
Question 3: The True Discount on a bill due 18 months hence at 8% per annum is ₹180. Find the Amount of the bill.
Solution:
TD = ₹180, R = 8%, T = 18 months = 3/2 years
From TD = (A × R × T) / (100 + R × T):
180 = (A × 8 × 3/2) / (100 + 8 × 3/2)
180 = (12A) / (100 + 12)
180 = 12A / 112
A = (180 × 112) / 12
A = 20160 / 12
A = ₹1,680
Notice how in Question 3, the formula is used in reverse — TD is given and A is unknown. This reverse application is a very common IBPS pattern that catches students who only practiced the forward direction. Always be comfortable using the master formula to find any one of the three quantities (A, TD, PW) when the other two are given or derivable.
4. Finding Present Worth — A Key Question Pattern
Finding the Present Worth is one of the most frequently tested calculations in True Discount problems. Let me show you multiple variations of this question type, including some that IBPS presents in a slightly disguised form.
Question 1: What sum of money should be paid now to settle a debt of ₹3,380 due 2 years hence at 13% per annum?
Solution:
The “sum to be paid now” is simply the Present Worth.
A = ₹3,380, R = 13%, T = 2 years
PW = (A × 100) / (100 + R × T)
PW = (3380 × 100) / (100 + 13 × 2)
PW = 338000 / 126
PW = ₹2,682.54
Question 2: A man wants to sell his scooter. There are two offers — one at ₹12,000 cash and another at ₹12,880 on credit, payable 8 months hence. If the rate of interest is 18% per annum, which offer is better?
Solution:
Find the Present Worth of the credit offer:
A = ₹12,880, R = 18%, T = 8/12 = 2/3 years
PW = (12880 × 100) / (100 + 18 × 2/3)
PW = (12880 × 100) / (100 + 12)
PW = 1288000 / 112
PW = ₹11,500
Since ₹12,000 (cash) > ₹11,500 (present worth of credit offer), the cash offer of ₹12,000 is better.
This type of question — comparing a cash offer with a credit offer by finding the present worth of the credit amount — is a beautifully practical question that IBPS loves to include. Students who understand the real-world concept from section 1 solve this naturally; those who only memorized the formula often get confused about which value to compare with which.
Question 3: The Present Worth of a sum due some time hence is ₹576 and the True Discount is ₹144. Find the sum due.
Solution:
Sum due (A) = PW + TD = 576 + 144 = ₹720
Simple but elegant. Always remember: Amount = Present Worth + True Discount. This is the most direct application of the fundamental definition and often appears as a one-step question embedded within a larger problem.
5. The Relationship Between SI and TD — A Crucial Exam Topic
One of the most commonly tested concepts in IBPS Stocks and Shares True Discount problems is the relationship between Simple Interest calculated on the Amount and the True Discount. Understanding this relationship deeply gives you access to a powerful set of shortcuts.
The Key Relationships:
Let SI = Simple Interest on Amount A for time T at rate R.
Let TD = True Discount on the same Amount A for the same time and rate.
Then:
SI > TD always (because SI is on the larger amount A, while TD is interest on the smaller PW)
TD = SI × PW / A
PW = TD² / (SI − TD)
TD = √(SI × PW) … when SI and PW are known
SI − TD = Interest on (SI − PW) … this is a subtle but testable relationship
Question 1: The Simple Interest on a sum for 2 years at 10% is ₹400. Find the True Discount on the same sum for the same time and rate.
Solution:
SI = ₹400, R = 10%, T = 2 years
First find Amount A using SI formula:
SI = (A × R × T) / 100… wait, here SI is on Amount A? No — let me re-clarify.
In True Discount problems, SI is calculated on the Amount (A), not on PW.
SI = (A × R × T) / 100 = 400
A = (400 × 100) / (10 × 2) = ₹2,000
Now TD = (A × R × T) / (100 + R × T) = (2000 × 10 × 2) / (100 + 20) = 40000 / 120 = ₹333.33
Question 2: The True Discount on a certain sum is ₹150 and the Simple Interest on the same sum for the same time and rate is ₹180. Find the sum.
Solution:
TD = ₹150, SI = ₹180
Using the relationship: PW = TD² / (SI − TD)
PW = (150)² / (180 − 150) = 22500 / 30 = ₹750
Sum due (A) = PW + TD = 750 + 150 = ₹900
Verification: Check that SI on A = 180 and TD = 150.
SI − TD = 180 − 150 = 30, and SI × TD / A = 180 × 150 / 900 = 30. ✓
This SI and TD relationship question type appears regularly in IBPS PO Mains. The formula PW = TD² / (SI − TD) is a particularly elegant shortcut that bypasses the need to know the rate and time individually. Whenever a question gives you both SI and TD and asks for the Amount or Present Worth, reach for this formula immediately.
6. True Discount on Bills — Practical Application Questions
A significant portion of True Discount questions in competitive exams are framed around the concept of bills — commercial documents promising future payment. These questions use the same formulas but are worded in a business context that can initially confuse students who aren’t familiar with the setting.
Let me explain the context first: In trade and commerce, a seller often delivers goods and gives the buyer a bill that promises payment after a fixed period — say, 3 months or 6 months. This document is called a bill of exchange. The amount written on the bill (to be paid at the end of the period) is the Amount (A). If the holder needs immediate cash, they can sell the bill to a bank or money-lender at its Present Worth, and the bank collects the full Amount later.
Key Terms in Bill-Based Questions:
Face Value of Bill: The Amount (A) written on the bill — the total sum payable on the due date.
Legal Due Date: The actual date on which payment is due, usually the date mentioned on the bill.
Unexpired Time: The time remaining from today until the legal due date. This is the ‘T’ used in all formulas.
Question 1: A bill for ₹5,300 is due in 2 years. What is the Present Worth of this bill if the rate of interest is 6% per annum?
Solution:
A = ₹5,300, R = 6%, T = 2 years
PW = (A × 100) / (100 + R × T) = (5300 × 100) / (100 + 12) = 530000 / 112 = ₹4,732.14
Question 2: What is the True Discount on a bill of ₹3,720 due 9 months hence at 16% per annum?
Solution:
A = ₹3,720, R = 16%, T = 9/12 = 3/4 years
TD = (A × R × T) / (100 + R × T)
TD = (3720 × 16 × 3/4) / (100 + 16 × 3/4)
TD = (3720 × 12) / (100 + 12)
TD = 44640 / 112
TD = ₹398.57
Question 3: A person buys a bill with a face value of ₹7,260 due in 15 months for ₹6,600 cash. What rate of interest is implied?
Solution:
A = ₹7,260, PW = ₹6,600, T = 15/12 = 5/4 years
TD = A − PW = 7260 − 6600 = ₹660
Using TD = (PW × R × T) / 100:
660 = (6600 × R × 5/4) / 100
660 = (6600 × 5 × R) / 400
660 = 82.5R
R = 660 / 82.5 = 8% per annum
Bill-based questions require no new formulas — they simply use the same True Discount formulas in a commercial setting. The only additional skill needed is correctly identifying which value is A, which is PW, and what the unexpired time T is. Practice reading these questions carefully and identifying these three elements before applying any formula.
7. True Discount vs Banker’s Discount — The Critical Distinction
At this point in my live classes, I always pause and spend dedicated time on the distinction between True Discount and Banker’s Discount, because IBPS PO exams frequently test both concepts in the same paper, and students who confuse the two lose marks on questions they should easily get right.
True Discount (TD): As we have thoroughly established, TD is the discount based on the Present Worth. It is the interest calculated on PW for the unexpired time.
TD = (A × R × T) / (100 + R × T)
Banker’s Discount (BD): This is the discount calculated by a bank or money-lender, but here the interest is charged on the Amount (A) — the face value of the bill — rather than on the Present Worth. Banks use this approach because it is simpler and more profitable for them.
BD = (A × R × T) / 100 (Simple Interest on A)
The Fundamental Difference:
TD uses SI on PW.
BD uses SI on A.
Since A > PW, we always have: BD > TD
Banker’s Gain (BG): The extra amount a banker earns over the True Discount.
BG = BD − TD
And here is a beautiful relationship: BG = TD² / PW (since BG is the SI on TD)
Also: BG = (BD × TD) / A … derived from above
Quick Example:
A = ₹1,300, R = 10%, T = 2 years
TD = (1300 × 10 × 2) / (100 + 20) = 26000 / 120 = ₹216.67
BD = (1300 × 10 × 2) / 100 = 26000 / 100 = ₹260
BG = BD − TD = 260 − 216.67 = ₹43.33
Verify: BG = TD²/PW = (216.67)²/(1300 − 216.67) = 46945.89 / 1083.33 ≈ ₹43.33 ✓
I always tell students: think of Banker’s Discount as the “greedy” version of True Discount — the bank charges interest on the full face value rather than the smaller present worth, earning itself a little extra. That extra earning is the Banker’s Gain. Remembering this real-world logic prevents formula confusion under exam pressure.
8. Advanced Question Types — Combined and Reverse Problems
As you progress toward IBPS PO Mains and SBI PO level preparation, True Discount questions become more multi-layered. Here I will cover the advanced patterns that trip up even well-prepared students.
Type 1 — Finding Rate When PW and TD Are Given:
Question: The Present Worth of a sum is ₹800 and the True Discount is ₹200 for 2 years. Find the rate of interest.
Solution:
PW = ₹800, TD = ₹200, T = 2 years
Using TD = (PW × R × T) / 100:
200 = (800 × R × 2) / 100
200 = 16R
R = 200/16 = 12.5% per annum
Type 2 — Finding Time When A, TD, and R Are Given:
Question: The True Discount on ₹2,160 is ₹360 at 10% per annum. Find the time.
Solution:
A = ₹2,160, TD = ₹360, R = 10%
PW = A − TD = 2160 − 360 = ₹1,800
Using TD = (PW × R × T) / 100:
360 = (1800 × 10 × T) / 100
360 = 180T
T = 360/180 = 2 years
Type 3 — Two Sums with Same TD at Different Rates and Times:
Question: The True Discount on a certain sum at 5% for 3 years is ₹300. What would the discount be at 4% for 5 years on the same sum?
Solution:
Using TD = (A × R × T) / (100 + R × T):
300 = (A × 5 × 3) / (100 + 15) = 15A / 115
A = (300 × 115) / 15 = ₹2,300
Now TD at 4% for 5 years:
TD = (2300 × 4 × 5) / (100 + 4 × 5) = 46000 / 120 = ₹383.33
Type 4 — Sum Due Payable in Parts:
Question: A man owes ₹1,573 in two equal installments payable 2 years and 4 years hence respectively at 7% per annum. Find the value of each installment.
Solution:
Let each installment = ₹x
PW of first installment (due in 2 years) = (x × 100) / (100 + 7 × 2) = 100x / 114
PW of second installment (due in 4 years) = (x × 100) / (100 + 7 × 4) = 100x / 128
Sum of present worths = total debt:
100x/114 + 100x/128 = 1573
x(100/114 + 100/128) = 1573
x × 100 × (128 + 114) / (114 × 128) = 1573
x × 100 × 242 / 14592 = 1573
x × 24200 / 14592 = 1573
x = (1573 × 14592) / 24200
x = 22,953,456 / 24200
x ≈ ₹948
Installment-based problems require setting up the equation of Present Worths carefully. Always translate “amount due in T years” into present worth and sum all present worths to equal the total debt. This is the universal approach for any installment variant.
9. Common Mistakes Students Make in True Discount
Years of teaching this chapter at OdTutor have shown me the same mistakes appearing again and again. Here is your complete guide to what goes wrong and how to ensure it never happens to you.
Mistake 1 — Calculating TD as SI on Amount instead of SI on PW. This is the single most common and costly mistake in this entire chapter. True Discount is always the Simple Interest on Present Worth, not on the Amount. If you calculate TD as (A × R × T)/100, you are calculating Banker’s Discount, not True Discount. This error gives a wrong answer while making you feel confident you’ve done it right — a particularly dangerous type of mistake.
Mistake 2 — Confusing True Discount with Banker’s Discount. These are two different things. TD uses SI on PW; BD uses SI on A. BD is always larger than TD. Whenever a question says “true discount,” use the formula TD = (A × R × T)/(100 + RT). Whenever it says “banker’s discount,” use BD = (A × R × T)/100. Never mix them.
Mistake 3 — Not converting time into years before applying the formula. If time is given in months, always convert to years first. 9 months = 3/4 year, 18 months = 3/2 years. Using months directly in a formula that expects years produces completely wrong answers.
Mistake 4 — Using Amount in the SI-based TD formula instead of PW. In the formula TD = (PW × R × T)/100, PW must be used, not A. Students sometimes write A here by mistake, especially under time pressure. Slow down when writing the formula and double-check which quantity you’re plugging in.
Mistake 5 — Not recognizing that A = PW + TD. This is the most fundamental relationship and also the easiest shortcut for many questions. When any two of the three values (A, PW, TD) are given, the third follows immediately from this relationship. Students who overlook this end up doing unnecessary formula calculations.
Mistake 6 — Mishandling the PW = TD² / (SI − TD) formula. This formula requires SI to be the simple interest on Amount A, not on PW. Students sometimes confuse which quantity the SI is calculated on, leading to wrong values for PW. Clarify this every time before applying the formula.
Mistake 7 — Skipping True Discount in preparation because “it rarely appears.” True Discount and Banker’s Discount together form a small but consistently present chapter in IBPS PO and SBI PO exams. More importantly, the questions are fast to solve when you’re prepared. Skipping guaranteed fast marks is never a smart strategy in a competitive exam environment.
10. Practice Strategy for Mastering True Discount Before the Exam
Let me close this article with the structured preparation plan I give my OdTutor students for completely mastering True Discount in a focused, efficient way. This roadmap is designed to take you from zero to fully exam-ready within eight to ten days.
Days 1–2 — Conceptual Foundation: Do not touch any formula on Day 1. Instead, read section 1 of this article and genuinely understand the real-world story behind True Discount — the future debt, the present payment, the difference between them. Draw a simple timeline for yourself: mark the present date on the left, the due date on the right, write A on the right, PW on the left, and TD as the gap between them. This visual anchor will make every formula feel logical rather than arbitrary. On Day 2, read section 2 carefully and derive each formula from the basic definition at least once by hand. Understanding where formulas come from is what lets you reconstruct them under pressure.
Day 3 — Formula Consolidation: Write all the core formulas — the master TD formula, the PW formula, the SI-TD relationship, and the BG formula — from memory on a blank sheet. Fill in any gaps by referring back to section 2. Repeat until you can write all formulas from scratch without looking. At this stage, you should also be clear on the difference between TD and BD as covered in section 7.
Days 4–5 — Basic Question Practice: Solve 25 to 30 straightforward questions covering finding TD, finding PW, and finding Amount — similar to the examples in sections 3 and 4. Focus entirely on accuracy here, not speed. Check every answer using the verification method: confirm that TD equals SI on PW. This cross-check habit catches errors before they become patterns.
Days 6–7 — SI and TD Relationship Questions: Work through the question types from section 5, particularly the formula PW = TD²/(SI − TD). These questions appear frequently in IBPS PO exams and require a slightly deeper understanding. Spend extra time here if needed, as this is the concept that separates average preparation from thorough preparation in this chapter.
Day 8 — Bill Problems and Advanced Question Types: Study section 6 carefully — practice reading bill-based problems and correctly identifying A, PW, and unexpired time T before applying formulas. Then move to section 8 and solve all four advanced question types: finding rate, finding time, two-rate comparison, and installment problems. Solve at least two examples of each type.
Days 9–10 onwards — Timed Mixed Practice and Mock Integration: Set a 45-second timer per question and solve mixed True Discount question sets that include all types in random order. The goal here is not just accuracy but the ability to identify the question type within the first reading and select the correct formula immediately. Include two to three True Discount or Banker’s Discount questions in your daily mock test routine so the chapter stays fresh alongside your preparation for other quantitative aptitude topics.
Throughout — Error Log and Weekly Review: Every wrong answer deserves a written diagnosis. Was it a wrong formula, a time-unit error, a confusion between TD and BD, or a misread question? Categorize every error and review the log at the end of each week. This disciplined review process is what consistently converts understanding into exam-day reliability, and it is the single habit that separates my top-performing students from those who plateau at an average score.
True Discount is a chapter that rewards honest, concept-first learning over formula-heavy cramming. The number of formulas is small, the question variety is limited, and the chapter is short — all of which means that a student who invests eight to ten focused days here essentially guarantees themselves a reliable, fast-scoring advantage in every exam that includes this topic. At OdTutor, I have seen this transformation happen with students of every background and ability level. It is not a matter of talent. It is entirely a matter of structured, patient, concept-driven preparation — and that is completely within your control.
How Teachers from OdTutor Can Help
At OdTutor, our trainers understand that True Discount feels abstract to many students simply because it is rarely taught with enough real-world context, and that is precisely where we begin. Rahul Sir and the OdTutor teaching team use timeline-based visual explanations, concept-first live sessions, and dedicated workshops on the critical distinction between True Discount and Banker’s Discount — all built around the exact question types and difficulty levels tested in IBPS PO, IBPS Clerk, and SBI PO exams. With personalized doubt-clearing sessions, topic-wise practice sheets covering every question pattern from basic to advanced, and full-length mock tests with detailed performance analysis, OdTutor ensures every student builds complete mastery of this chapter — transforming one of the most commonly skipped topics into a dependable, consistently fast-scoring strength on the actual exam day.