Profit and Loss — Study Notes for UP Police Constable

Overview

Profit and Loss is a high-frequency topic in the Numerical & Mental Ability section of UP Police Constable exams, typically yielding 3–5 questions. This topic tests your ability to calculate commercial transactions, understand pricing strategies, and solve real-world buying-selling scenarios under time pressure.

The core challenge is distinguishing between Cost Price (CP), Selling Price (SP), and Marked Price (MP), then applying formulas accurately. Questions often involve successive discounts, false weights, and combination scenarios that require multi-step calculations. Mastering this topic requires memorizing key formulas and practicing quick mental calculations, as the exam rewards speed alongside accuracy.

Understanding percentage conversions is crucial — most profit/loss problems boil down to percentage increase or decrease calculations. Students who can instantly recall formulas and avoid sign errors (profit vs loss) typically score full marks in this section.

Key Concepts

• **Cost Price (CP)** is the amount paid to acquire or produce an item. All profit/loss calculations reference this baseline.

• **Selling Price (SP)** is the amount received when selling the item. Profit occurs when SP > CP; loss occurs when SP < CP.

• **Marked Price (MP)** is the listed or tagged price before any discount. Discount is always calculated on MP, not CP.

• **Profit** is calculated as SP − CP, while **Loss** is CP − SP. Profit% and Loss% are always calculated on CP unless stated otherwise.

• **Discount** reduces the MP to arrive at the final SP. A single discount differs from successive discounts — two 10% discounts do NOT equal 20%.

• **Break-even** occurs when SP = CP, meaning neither profit nor loss. This concept appears in problems involving partial sales.

• **False weight/measure** creates an implicit profit by giving less quantity while charging for more — a common trick question type.

• In **partnership/mix problems**, profit distribution follows the ratio of investments or quantities, connecting profit-loss with ratio concepts.

Formulas / Key Facts

**Basic Formulas:**

• Profit = SP − CP
• Loss = CP − SP
• Profit% = (Profit / CP) × 100 = [(SP − CP) / CP] × 100
• Loss% = (Loss / CP) × 100 = [(CP − SP) / CP] × 100
• SP = CP × (100 + Profit%) / 100 = CP × (100 − Loss%) / 100
• CP = SP × 100 / (100 + Profit%) = SP × 100 / (100 − Loss%)

**Discount Formulas:**

• Discount = MP − SP
• Discount% = (Discount / MP) × 100
• SP = MP × (100 − Discount%) / 100
• MP = SP × 100 / (100 − Discount%)

**Successive Discount:**

• For two successive discounts d₁% and d₂%, net discount% = d₁ + d₂ − (d₁ × d₂)/100
• For three discounts d₁%, d₂%, d₃%, apply the formula twice or use: Final SP = MP × (100−d₁)/100 × (100−d₂)/100 × (100−d₃)/100

**Profit with MP and Discount:**

• When MP is given with discount and profit: SP = MP × (100−Discount%)/100 and also SP = CP × (100+Profit%)/100

**Quick Conversion:**

• 25% profit means SP = 1.25 × CP
• 20% loss means SP = 0.80 × CP
• 10% discount on MP means SP = 0.90 × MP

Worked Examples

**Example 1: Basic Profit Percent** A shopkeeper buys an article for ₹500 and sells it for ₹650. Find the profit percent.

*Solution:*

• CP = ₹500, SP = ₹650
• Profit = SP − CP = 650 − 500 = ₹150
• Profit% = (Profit / CP) × 100 = (150 / 500) × 100 = 30%

**Answer: 30% profit**

**Example 2: Finding CP from Loss** An article sold for ₹840 results in a 16% loss. Find the CP.

*Solution:*

• SP = ₹840, Loss% = 16%
• CP = SP × 100 / (100 − Loss%) = 840 × 100 / (100 − 16) = 840 × 100 / 84
• CP = 84000 / 84 = ₹1000

**Answer: CP = ₹1000**

**Example 3: Successive Discount** A shirt marked at ₹2000 is sold with two successive discounts of 20% and 10%. Find the final selling price.

*Solution:* Method 1 (Formula):

• Net discount% = 20 + 10 − (20 × 10)/100 = 30 − 2 = 28%
• SP = MP × (100 − 28)/100 = 2000 × 72/100 = ₹1440

Method 2 (Step-by-step):

• After 1st discount: 2000 × 80/100 = ₹1600
• After 2nd discount: 1600 × 90/100 = ₹1440

**Answer: ₹1440**

**Example 4: MP, Discount and Profit Combined** A trader marks goods 40% above CP and allows 20% discount. Find his profit percent.

*Solution:*

• Let CP = ₹100 (assume for convenience)
• MP = 100 + 40% of 100 = ₹140
• SP = 140 − 20% of 140 = 140 × 80/100 = ₹112
• Profit = 112 − 100 = ₹12
• Profit% = 12%

**Answer: 12% profit**

Common Mistakes

**Mistake 1: Calculating discount on CP instead of MP** — Students often apply discount% to CP, but discount ALWAYS applies to the marked price. *Fix:* Remember the hierarchy: CP → add markup → MP → apply discount → SP.

**Mistake 2: Adding successive discounts directly** — Assuming 10% + 10% = 20% discount is incorrect because the second discount applies to an already reduced price. *Fix:* Use the successive discount formula or multiply step-by-step: (0.9) × (0.9) = 0.81, which is 19% discount, not 20%.

**Mistake 3: Confusing profit% base** — Calculating profit% on SP instead of CP leads to wrong answers. The question may trick you by asking "percent of selling price", but standard profit% is on CP. *Fix:* Always read whether the question asks for profit on CP or SP explicitly.

**Mistake 4: Sign errors in loss problems** — Writing CP = SP × (100 + Loss%) instead of (100 − Loss%). *Fix:* Internalize: profit increases CP, loss decreases CP. In formulas, profit is (+), loss is (−).

**Mistake 5: Not converting fractions/ratios to percentages** — Questions state "sold at 3/4 of CP" and students struggle to apply formulas. *Fix:* Convert immediately: SP = (3/4) × CP means 25% loss, then use standard formulas.

Quick Reference

• **Profit% = [(SP − CP) / CP] × 100**; Loss% uses same pattern with CP − SP.
• **Successive discounts d₁% and d₂%**: Net = d₁ + d₂ − (d₁ × d₂)/100
• **CP from SP with profit**: CP = SP × 100 / (100 + Profit%)
• **CP from SP with loss**: CP = SP × 100 / (100 − Loss%)
• Discount is calculated on **MP**, profit/loss on **CP**.
• Two 10% successive discounts = 19% net discount (not 20%).