Discount — Study Notes for UP Police Constable

Overview

Discount questions are a staple of the Numerical Ability section in UP Police Constable exams, typically appearing in 3–5 questions per paper. This topic tests your understanding of how merchants offer price reductions and how financial institutions handle bills before their due date. You must master three core concepts: **marked price and trade discount** (straightforward percentage reduction), **true discount** (the difference between a future sum and its present value at simple interest), and **banker's discount** (simple interest on the face value of a bill). Successive discount problems — where multiple percentage reductions apply one after another — are particularly common and require careful calculation to avoid errors.

Unlike profit-loss problems that focus on cost price versus selling price, discount problems center on **marked price (MP)**, **selling price (SP)**, and various discount types. The exam favors practical scenarios: shopkeepers announcing "20% off," banks discounting bills 6 months early, or festival sales with "buy one get 40% off the second item." Strong command of percentage calculations and the relationship between present value, future value, time, and rate of interest will serve you well here. Expect both direct calculation questions and word problems requiring 2–3 steps.

Most importantly, distinguish clearly between true discount (TD) and banker's discount (BD) — confusing these definitions is the most common error students make. Practice converting time periods (months to years) and applying formulas accurately under time pressure.

Key Concepts

• **Marked Price (MP)** is the labeled or list price before any discount. Selling Price (SP) is what the customer actually pays after discount is applied. The relationship is: SP = MP − Discount.

• **Discount Percentage** is always calculated on the marked price unless otherwise specified. If a 25% discount is offered on MP of ₹800, discount amount = 25% of 800 = ₹200, so SP = ₹600.

• **True Discount (TD)** is the simple interest on the present value (PV) of a sum due at a future date. Formula: TD = (PW × R × T)/100, where PW is present worth. Also, TD = Amount − PW, and PW = (Amount × 100)/(100 + R × T).

• **Banker's Discount (BD)** is the simple interest on the face value (amount) of a bill for the unexpired time. Formula: BD = (Amount × R × T)/100. Banker's discount is always greater than true discount because it's calculated on the larger face value rather than present worth.

• **Relationship between BD and TD**: BD − TD = SI on TD. This means BD = TD + (TD × R × T)/100. Also, Amount = (BD × TD)/(BD − TD).

• **Successive Discounts** are multiple discounts applied one after another, not added together. If discounts are d₁% and d₂%, the single equivalent discount = d₁ + d₂ − (d₁ × d₂)/100. For three discounts d₁%, d₂%, d₃%, apply the formula twice sequentially.

• **Conversion tip**: When time is given in months, convert to years by dividing by 12. For example, 9 months = 9/12 = 3/4 year. Always maintain consistency in time units.

• In bills of exchange problems, the "unexpired time" or "period of discount" is the time remaining until the bill's due date. If a bill due 8 months from now is discounted today after 2 months, the unexpired time is 6 months.

Formulas / Key Facts

1. **Basic Discount**: Discount = MP − SP, or Discount = (Discount% × MP)/100 2. **Selling Price after discount**: SP = MP × (100 − Discount%)/100 3. **True Discount (TD)**: TD = (Amount × R × T)/(100 + R × T), where Amount is the sum due 4. **Present Worth (PW)**: PW = Amount − TD = (Amount × 100)/(100 + R × T) 5. **Banker's Discount (BD)**: BD = (Amount × R × T)/100 = SI on Amount 6. **Relation**: BD − TD = SI on TD for the given time and rate 7. **Amount from BD and TD**: Amount = (BD × TD)/(BD − TD) 8. **Sum from BD and TD**: Sum (Amount) = BD × TD/(BD − TD) when both BD and TD are known 9. **Successive Discount (two)**: Single Equivalent = d₁ + d₂ − (d₁ × d₂)/100 10. **Successive Discount (general)**: Apply discounts sequentially — if MP = 1000, first 20% off gives 800, then 10% off 800 gives 720. Final SP = MP × (1 − d₁/100) × (1 − d₂/100)

Worked Examples

**Example 1: Simple Discount** A shirt with marked price ₹1200 is sold at 15% discount. Find the selling price.

*Solution:* Discount = 15% of 1200 = (15/100) × 1200 = ₹180 SP = MP − Discount = 1200 − 180 = ₹1020 **Answer: ₹1020**

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**Example 2: Successive Discounts** A shopkeeper offers two successive discounts of 20% and 10% on an article marked at ₹500. What is the final selling price?

*Solution:* Method 1 (Step-by-step): First discount 20% on ₹500 = 100, so price after first discount = 400 Second discount 10% on ₹400 = 40, so final SP = 400 − 40 = ₹360

Method 2 (Formula): Single equivalent discount = 20 + 10 − (20 × 10)/100 = 30 − 2 = 28% SP = 500 × (100 − 28)/100 = 500 × 72/100 = ₹360 **Answer: ₹360**

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**Example 3: True Discount** The true discount on ₹1080 due 3 years hence at 10% per annum simple interest is what amount?

*Solution:* Amount = ₹1080, R = 10%, T = 3 years TD = (Amount × R × T)/(100 + R × T) TD = (1080 × 10 × 3)/(100 + 10 × 3) = 32400/130 = ₹249.23 (approximately ₹249) Also, PW = 1080 − 249 = ₹831 **Answer: ₹249 (approx)**

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**Example 4: Banker's Discount vs True Discount** The banker's discount on a bill due 6 months hence at 12% per annum is ₹180. Find the true discount and the amount of the bill.

*Solution:* BD = ₹180, R = 12% per annum, T = 6 months = 0.5 year BD = (Amount × R × T)/100 180 = (Amount × 12 × 0.5)/100 180 = Amount × 6/100 Amount = (180 × 100)/6 = ₹3000

Now find TD: TD = (Amount × R × T)/(100 + R × T) = (3000 × 12 × 0.5)/(100 + 6) = 18000/106 = ₹169.81 (approx ₹170) **Answer: Amount = ₹3000, TD = ₹170**

Common Mistakes

• **Adding successive discounts directly**: Students often add 20% + 10% = 30% and apply 30% discount on the original price. Wrong! Each discount applies to the reduced price after the previous discount. Always use the formula or apply step-by-step.

• **Confusing BD with TD**: Banker's discount is SI on the face value (amount), while true discount is SI on present value. BD is always larger than TD. If a question asks for "discount on a bill," it typically means banker's discount unless specified otherwise.

• **Forgetting time conversion**: When time is given in months, students often use it directly in formulas meant for years. Always convert months to years (divide by 12) before applying SI-based discount formulas.

• **Using wrong base for discount percentage**: Discount percentage is calculated on **marked price**, not selling price. If SP = ₹800 after 20% discount, MP is not 800/0.8 = 1000; rather MP = 800/(1 − 0.20) = ₹1000. Be clear which value is the base.

• **Misapplying the BD−TD relationship**: Students sometimes write BD + TD instead of BD − TD = SI on TD. Remember: BD is greater, so BD − TD gives the difference, which equals the interest on the true discount amount itself for the given period and rate.

Quick Reference

• **Discount = MP − SP**; Discount% always on MP unless stated otherwise.
• **Successive discounts**: Use d₁ + d₂ − (d₁d₂)/100, not d₁ + d₂.
• **True Discount (TD)** = (Amt × R × T)/(100 + RT); calculated on present value.
• **Banker's Discount (BD)** = (Amt × R × T)/100; calculated on face value; BD > TD always.
• **BD − TD = SI on TD**; Amount = (BD × TD)/(BD − TD).
• Convert months to years (÷12) before using time in formulas.