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Quantitative AptitudeBeginner5 min study

How to Solve Profit and Loss Questions with Formulas & Solved Examples

लाभ और हानि (Profit & Loss) के प्रश्न कैसे हल करें: सूत्र, शॉर्टकट ट्रिक्स और हल सहित उदाहरण

Profit and Loss (लाभ और हानि) is one of the most fundamental and high-scoring arithmetic topics in MP Police Constable, MP Sub-Inspector, MP Patwari, MPPSC CSAT, and SSC exams. It covers transactions where commodities are acquired at a Cost Price (CP / क्रय मूल्य) and sold at a Selling Price (SP / विक्रय मूल्य). The foundational rule in competitive exams is that Profit Percentage and Loss Percentage are ALWAYS calculated relative to the Cost Price (CP), unless explicitly specified otherwise by the question.

Key Rules & Core Formulas

Understand these foundational properties before solving questions

1Basic Profit & Loss Definitions

Profit = SP - CP (when SP > CP) | Loss = CP - SP (when CP > SP)

Profit occurs when Selling Price exceeds Cost Price. Loss occurs when Cost Price exceeds Selling Price.

Cost Price (CP) is the total investment, including any overhead expenses like transportation or repairs.

2Profit % and Loss % (Anchor Rule)

Profit % = (Profit / CP) × 100% | Loss % = (Loss / CP) × 100%

Profit or loss percentage is strictly calculated with Cost Price (CP) in the denominator. Never divide by SP unless the problem specifically specifies 'profit calculated on selling price'.

A 20% profit means SP = 120% of CP (1.2 × CP). A 20% loss means SP = 80% of CP (0.8 × CP).

3Direct Multipliers for SP and CP

SP = CP × (100 ± Gain/Loss)% / 100 | CP = SP × 100 / (100 ± Gain/Loss)%

Skip calculating the rupee profit or loss amount and compute the final selling price or cost price directly in one step.

Use '+' for profit and '-' for loss.

4Marked Price, Discount, and SP

Discount = Marked Price (MP) - SP | Discount % = (Discount / MP) × 100%

Discount (छूट/बट्टा) is ALWAYS calculated on the Marked Price (अंकित मूल्य / List Price), never on the Cost Price.

Selling Price under discount: SP = MP × (100 - Discount %) / 100.

5Successive Discounts Shortcut

Net Equivalent Discount % = [d₁ + d₂ - (d₁ × d₂) / 100]%

When two successive discounts d₁% and d₂% are offered, their combined single equivalent discount is less than d₁ + d₂.

Example: Two successive discounts of 20% and 10% equal a net single discount of 20 + 10 - 2 = 28%.

6Equal Selling Price with Same % Gain and Loss

Net Result = Always a Loss of (x / 10)² % = (x² / 100) %

When two articles are sold for the exact same selling price, one at a gain of x% and the other at a loss of x%, the merchant always incurs a net loss.

Never choose 'No Profit, No Loss' on such questions; it is a very common exam trap!
Standard Method

Step-by-Step Method to Solve

Follow these universal steps in order to solve any Quantitative Aptitude question accurately:

  1. 1

    Step 1: Identify and classify all given values

    Read the problem carefully to label the knowns: Cost Price (CP), Selling Price (SP), Marked Price (MP), and the given Profit%, Loss%, or Discount%.

    Pro-Tip: Check whether any overhead expenses (shipping, repairs) must be added to the purchase price to obtain true CP.
  2. 2

    Step 2: Convert percentages to fractional ratios

    Express given percentages as simple fractions for lightning-fast calculations (e.g., 20% = 1/5, 25% = 1/4, 16 2/3% = 1/6, 12.5% = 1/8).

    Pro-Tip: A 25% profit (1/4) means CP : SP = 4 : 5. A 20% loss (1/5) means CP : SP = 5 : 4.
  3. 3

    Step 3: Anchor each percentage to its correct baseline

    Remember the fundamental distinction: Profit/Loss percentage is anchored to CP, whereas Discount percentage is anchored to MP.

    Pro-Tip: Markup flows from CP → MP; Discount flows from MP → SP; Profit/Loss compares CP → SP.
  4. 4

    Step 4: Apply direct multipliers or unitary method

    Set up direct equations: SP = CP × (100 + P%)/100, or match the ratio units directly to the given rupee amount.

    Pro-Tip: When calculating CP from SP, divide by the multiplier: for 25% profit, CP = SP / 1.25 (or SP × 4/5).
  5. 5

    Step 5: Sanity check the answer direction

    Verify your result: if there was a profit, SP must be strictly greater than CP. If there was a loss, SP must be strictly less than CP.

    Pro-Tip: Use this check to immediately eliminate 2 out of 4 options in multiple-choice exams.

Worked Examples & Walkthroughs

Real questions solved step-by-step showing every calculation

1Example 1Method: Direct Multiplier & Percentage Method

Question

A shopkeeper purchases an electric kettle for ₹750 and sells it at a profit of 16%. Calculate the selling price.

Step-by-Step Solution:

Step 1: Identify given values:Cost Price (CP) = ₹750, Profit % = 16%.
Step 2: Calculate absolute profit:Profit = 16% of ₹750 = (16 / 100) × 750 = (16 × 15) / 2 = 8 × 15 = ₹120.
Step 3: Calculate Selling Price:Selling Price (SP) = CP + Profit = ₹750 + ₹120 = ₹870.
Step 4: Alternative direct multiplier:SP = CP × (100 + 16) / 100 = 750 × 1.16 = ₹870.
Final Result:
Selling Price (SP) = ₹870
Exam Shortcut / Verification: Fraction method: 16% = 4/25. If CP is 25 units, SP = 25 + 4 = 29 units. Since 25 units = ₹750 → 1 unit = ₹30 → SP = 29 × 30 = ₹870 in 10 seconds.
2Example 2Method: Two-Scenario Chain Method

Question

By selling a watch for ₹1,440, a shopkeeper incurs a loss of 10%. At what price should he sell it to gain 15%?

Step-by-Step Solution:

Step 1: Relate initial SP to CP:A 10% loss means initial SP (₹1,440) represents 90% of Cost Price (100% - 10%).
Step 2: Calculate Cost Price (CP):90% of CP = ₹1,440 → CP = (1,440 × 100) / 90 = 16 × 100 = ₹1,600.
Step 3: Calculate desired SP for 15% gain:Target Profit = 15%. Desired SP = 115% of CP = (115 / 100) × 1,600 = 115 × 16 = ₹1,840.
Final Result:
New Selling Price = ₹1,840
Exam Shortcut / Verification: Direct One-Line Ratio: SP₂ = SP₁ × (100 + Desired Gain%) / (100 - Initial Loss%) = 1,440 × (115 / 90) = 16 × 115 = ₹1,840!
3Example 3Method: Base-100 Assumption & Successive Formula Method

Question

A trader marks his goods 40% above the cost price and allows a discount of 20% on the marked price. What is his net profit percentage?

Step-by-Step Solution:

Step 1: Assume Cost Price:Let Cost Price (CP) = ₹100.
Step 2: Determine Marked Price (MP):Markup is 40% above CP: MP = 100 + 40 = ₹140.
Step 3: Calculate discount and Selling Price (SP):Discount is 20% of MP: Discount = 20% of 140 = ₹28. SP = MP - Discount = 140 - 28 = ₹112.
Step 4: Compute Net Profit %:Net Profit = SP - CP = 112 - 100 = ₹12 on CP of ₹100 → Profit % = 12%.
Final Result:
Net Profit = 12%
Exam Shortcut / Verification: Successive change formula: Net % = Markup - Discount - [(Markup × Discount) / 100] = 40 - 20 - [(40 × 20) / 100] = 20 - 8 = 12% profit.
4Example 4Method: Equal Selling Price Property

Question

A person sells two mobile phones for ₹9,900 each. On one he gains 10% and on the other he loses 10%. Find his overall gain or loss percentage and the net loss in rupees.

Step-by-Step Solution:

Step 1: Calculate CP of the first phone (10% profit):SP₁ = ₹9,900. Since Profit = 10%, CP₁ = 9,900 × (100 / 110) = ₹9,000.
Step 2: Calculate CP of the second phone (10% loss):SP₂ = ₹9,900. Since Loss = 10%, CP₂ = 9,900 × (100 / 90) = ₹11,000.
Step 3: Calculate total CP and total SP:Total CP = 9,000 + 11,000 = ₹20,000. Total SP = 9,900 + 9,900 = ₹19,800.
Step 4: Determine net loss and percentage:Total CP > Total SP, so Net Loss = 20,000 - 19,800 = ₹200. Net Loss % = (200 / 20,000) × 100% = 1%.
Final Result:
Overall 1% Loss (Net loss of ₹200)
Exam Shortcut / Verification: Exam Shortcut: When SP is identical and gain% = loss% = x%, there is always a loss of (x/10)² % = (10/10)² = 1% loss. Total SP (₹19,800) is 99% of total CP → 1% loss = ₹19,800 / 99 = ₹200.

Common Mistakes to Avoid

Avoid these frequent traps to protect yourself from negative marking

1

Calculating profit or loss percentage on the Selling Price instead of Cost Price

Why it happens: In problems where the Selling Price is stated first, students inadvertently place SP in the denominator.
Correct approach: Always place CP in the denominator: Profit % = (Profit / CP) × 100%. Only divide by SP if the question explicitly asks for 'profit on selling price'.
2

Calculating discount percentage on the Cost Price instead of Marked Price

Why it happens: Confusing Cost Price with Marked Price (List Price).
Correct approach: Remember: Profit/Loss is calculated on CP; Discount is always calculated on Marked Price (MP).
3

Assuming 'No Profit, No Loss' when two items are sold at equal SP with equal % gain and loss

Why it happens: Intuitively thinking that +10% and -10% cancel each other out.
Correct approach: When SPs are identical, the 10% loss is taken on a LARGER cost price than the 10% gain, resulting in a guaranteed net loss of (x/10)² %.
4

Misinterpreting 'Cost price of X articles = Selling price of Y articles'

Why it happens: Assuming X is the cost price and Y is the selling price.
Correct approach: Write down the algebraic relationship: X × CP = Y × SP → CP / SP = Y / X. Then compare Y and X to determine profit or loss.
Self-Assessment

Try It Yourself (Practice Questions)

Test your understanding on your own first, then reveal the step-by-step solution to check your work:

Practice #1

If the cost price of 15 tables is equal to the selling price of 20 tables, what is the profit or loss percentage?

Practice #2

A dishonest trader professes to sell his goods at cost price, but uses a false weight of 900 grams for a 1 kg (1,000 grams) weight. What is his profit percentage?

Practice #3

A shopkeeper marks an article at ₹500. If he offers two successive discounts of 10% and 20%, what is the final selling price of the article?

Quick Revision Summary

  • Profit = SP - CP (when SP > CP); Loss = CP - SP (when CP > SP).
  • Profit % and Loss % are always calculated with respect to Cost Price (CP).
  • Discount is always calculated on the Marked Price (MP): SP = MP × (100 - Discount %)/100.
  • Equal SP with same % profit and loss always results in an overall Net Loss of (x / 10)² %.
  • When successive discounts are given, use Net Discount = [d₁ + d₂ - (d₁ × d₂)/100]%.
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