Profit and Loss Problems — Profit% on SP Cost ₹2 Lakh

Every business on the planet — from a street-side chai stall to Amazon — lives and breathes profit and loss. When a shopkeeper marks up a price, offers a discount, or sells goods in bulk, they're doing profit and loss calculations in their head. These aren't abstract math problems; they're the engine of commerce. That's exactly why aptitude tests and technical interviews use them: they reveal whether you can think clearly about numbers under pressure.

Profit and Loss Problems — The Real Math Behind Margin and Markup

Profit and loss aptitude problems test your ability to compute gain or loss as a percentage of cost price (CP) or selling price (SP). The core mechanic: Profit = SP - CP; Loss = CP - SP; Profit% = (Profit/CP) × 100; Loss% = (Loss/CP) × 100. A common twist is when profit is given as a percentage of SP — e.g., "profit% on SP" — which changes the denominator. For a ₹2 lakh cost item sold at 20% profit on SP, SP = CP / (1 - profit% on SP) = 200000 / 0.8 = ₹2.5 lakh; profit = ₹50,000.

The key property: profit% on CP always yields a higher profit amount than the same % on SP for the same CP. This is because CP is smaller than SP when there's profit. In practice, many e-commerce and retail systems use margin (profit% on SP) for pricing, while accounting uses markup (profit% on CP). Confusing the two leads to incorrect pricing, margin erosion, or overpricing. Always verify which base the percentage refers to.

Use these calculations when building pricing engines, discount logic, or financial reports. In real systems, a 10% margin error on a ₹2 lakh item means ₹20,000 lost per transaction. At scale, that's millions. Understanding the distinction prevents silent revenue leaks and ensures your pricing model matches business intent.

The Four Foundation Terms You Must Know Cold

Before you touch a single formula, you need to own these four terms. Confusing any two of them is the #1 reason candidates lose marks.

Cost Price (CP): The price at which you BUY something. This is your investment. If you bought a phone for ₹10,000, your CP is ₹10,000.

Selling Price (SP): The price at which you SELL something. If you sold that phone for ₹12,000, your SP is ₹12,000.

Profit: When SP > CP. You sold for MORE than you paid. Profit = SP − CP.

Loss: When CP > SP. You sold for LESS than you paid. Loss = CP − SP.

Here's the anchor image to burn into memory: think of CP as the floor. If your SP is above the floor, you're in profit territory. If it's below the floor, you're in a hole — a loss. Everything in profit and loss revolves around the relationship between these two numbers. Get this instinct right and every formula that follows will feel obvious, not memorised.

=== PROFIT AND LOSS — FOUNDATION FORMULAS ===

Given:
  Cost Price (CP)    = ₹10,000   ← what you paid to buy the phone
  Selling Price (SP) = ₹12,000   ← what you got when you sold it

Step 1 — Identify whether it's Profit or Loss:
  SP (12,000) > CP (10,000)  →  It's a PROFIT situation

Step 2 — Calculate Profit:
  Profit = SP − CP
         = 12,000 − 10,000
         = ₹2,000

Step 3 — Calculate Profit %:
  Profit % = (Profit / CP) × 100
           = (2,000 / 10,000) × 100
           = 20%

──────────────────────────────────────────
Now flip it — Loss scenario:
  CP = ₹10,000
  SP = ₹8,500

  CP (10,000) > SP (8,500)  →  It's a LOSS situation

  Loss = CP − SP
       = 10,000 − 8,500
       = ₹1,500

  Loss % = (Loss / CP) × 100
         = (1,500 / 10,000) × 100
         = 15%

══════════════════════════════════════════
KEY RULE: Profit % and Loss % are ALWAYS
calculated on the COST PRICE — not the
Selling Price. Forgetting this = wrong answer.
══════════════════════════════════════════

The Master Formula Sheet — Every Variation in One Place

Once you know CP and SP, everything else is derived. Here's every formula you'll ever need, explained with the reasoning behind each one.

Finding SP when CP and Profit% are given: SP = CP × (1 + Profit%/100) Think of it as: SP = CP + (Profit% of CP). You're adding the gain on top.

Finding SP when CP and Loss% are given: SP = CP × (1 − Loss%/100) You're subtracting the loss from what you paid.

Finding CP when SP and Profit% are given: CP = SP × 100 / (100 + Profit%) You're reverse-engineering the original price.

Finding CP when SP and Loss% are given: CP = SP × 100 / (100 − Loss%)

These four formulas handle 90% of all profit and loss questions. The trick is recognising WHICH two pieces of information the question gives you, then picking the right formula. Always label what you KNOW and what you NEED before you start calculating.

=== WORKED PROBLEMS — ALL 4 FORMULA TYPES ===

─────────────────────────────────────────────
PROBLEM 1: Find SP
  "A shopkeeper buys a bag for ₹800 and wants
   to make a 25% profit. What should he sell it for?"

  Known: CP = ₹800, Profit% = 25
  Need:  SP = ?

  Formula: SP = CP × (1 + Profit%/100)
         SP = 800 × (1 + 25/100)
         SP = 800 × (1 + 0.25)
         SP = 800 × 1.25
         SP = ₹1,000

  ✓ Check: Profit = 1000 − 800 = 200
            Profit% = (200/800) × 100 = 25% ✓
─────────────────────────────────────────────
PROBLEM 2: Find SP (Loss)
  "A TV bought for ₹15,000 is sold at a 10% loss.
   Find the selling price."

  Known: CP = ₹15,000, Loss% = 10
  Need:  SP = ?

  Formula: SP = CP × (1 − Loss%/100)
         SP = 15,000 × (1 − 10/100)
         SP = 15,000 × 0.90
         SP = ₹13,500
─────────────────────────────────────────────
PROBLEM 3: Find CP (given SP and Profit%)
  "A watch is sold for ₹1,320 at a profit of 10%.
   What was the cost price?"

  Known: SP = ₹1,320, Profit% = 10
  Need:  CP = ?

  Formula: CP = SP × 100 / (100 + Profit%)
         CP = 1,320 × 100 / (100 + 10)
         CP = 132,000 / 110
         CP = ₹1,200

  ✓ Check: Profit = 1320 − 1200 = 120
            Profit% = (120/1200) × 100 = 10% ✓
─────────────────────────────────────────────
PROBLEM 4: Find CP (given SP and Loss%)
  "A laptop is sold for ₹34,000 at a loss of 15%.
   Find the original cost price."

  Known: SP = ₹34,000, Loss% = 15
  Need:  CP = ?

  Formula: CP = SP × 100 / (100 − Loss%)
         CP = 34,000 × 100 / (100 − 15)
         CP = 3,400,000 / 85
         CP = ₹40,000

  ✓ Check: Loss = 40,000 − 34,000 = 6,000
            Loss% = (6,000/40,000) × 100 = 15% ✓

Marked Price, Discounts and the Real-World Shop Problem

Real interviews love adding one more layer: the concept of Marked Price (MP) and Discount. Here's where it gets interesting.

Marked Price (MP): The price tag on the shelf — what the seller ADVERTISES. Also called List Price.

Discount: A reduction given on the Marked Price. Discount is always calculated on the MP — not on CP.

SP = MP × (1 − Discount%/100)

The juicy interview question combines all three: a shopkeeper marks up the price above CP, then gives a discount — and you need to figure out the final profit or loss.

Here's the mental model: Imagine a shirt costs ₹500 (CP). The shopkeeper writes ₹800 on the tag (MP — a 60% markup). Then there's a sale: 25% discount on ₹800. The customer pays ₹600 (SP). The shopkeeper still made ₹100 profit on a ₹500 shirt. He gave a discount AND still profited. That's the trick these questions test.

=== MARKED PRICE + DISCOUNT — COMBINED PROBLEM ===

PROBLEM:
  "A shopkeeper marks a bicycle 40% above its cost
   price of ₹2,500 and then gives a discount of 20%.
   Find: (a) Marked Price  (b) Selling Price
         (c) Profit or Loss  (d) Profit/Loss %"

─────────────────────────────────────────────
Given:
  Cost Price (CP)  = ₹2,500
  Markup           = 40% above CP
  Discount         = 20% on Marked Price

Step 1 — Find Marked Price (MP):
  MP = CP × (1 + Markup%/100)
     = 2,500 × (1 + 40/100)
     = 2,500 × 1.40
     = ₹3,500

Step 2 — Find Selling Price (SP):
  SP = MP × (1 − Discount%/100)
     = 3,500 × (1 − 20/100)
     = 3,500 × 0.80
     = ₹2,800

Step 3 — Profit or Loss?
  SP (2,800) > CP (2,500)  →  PROFIT
  Profit = SP − CP = 2,800 − 2,500 = ₹300

Step 4 — Profit %:
  Profit % = (Profit / CP) × 100
           = (300 / 2,500) × 100
           = 12%

─────────────────────────────────────────────
SHORTCUT FORMULA (for combined markup + discount):

  Net % change = M − D − (M × D)/100
  where M = markup%, D = discount%

  = 40 − 20 − (40 × 20)/100
  = 40 − 20 − 800/100
  = 40 − 20 − 8
  = +12%    ← positive means PROFIT

  Same answer! Use the shortcut in exams
  when time is tight.
─────────────────────────────────────────────

Tricky Problem Types That Appear in Every Aptitude Test

Knowing the formulas is only half the battle. Aptitude tests throw a few classic curveballs that trip up even prepared candidates. Here are the three most common trap-style questions, fully worked through.

Type 1 — Two items, same SP, one profit one loss: Whenever a problem says 'two articles each sold at the same price, one at X% profit and one at X% loss' — there is ALWAYS a net loss. Always. This is a mathematical certainty.

Net Loss % = (Common %/10)² = X²/100

Type 2 — Dishonest shopkeeper using false weights: If a shopkeeper uses a 900g weight and claims it's 1000g, he's effectively gaining 100g per kg. Profit% = (True weight − False weight) / False weight × 100.

Type 3 — Successive profit/loss (buying then reselling): If you buy at CP, sell at a profit to Person B, and Person B resells at another profit, the final price chains multiplicatively — not additively.

=== CLASSIC TRAP PROBLEMS — FULLY WORKED ===

─────────────────────────────────────────────
TRAP PROBLEM 1: Two items, same SP, X% profit & X% loss

  "Two mobiles are sold at ₹9,900 each.
   One at 10% profit, another at 10% loss.
   Is there an overall profit or loss? By how much?"

  Many people say ZERO. It's NOT zero. There is always a LOSS.

  For the item sold at 10% PROFIT:
    CP₁ = SP × 100/(100 + Profit%)
        = 9,900 × 100/110
        = ₹9,000

  For the item sold at 10% LOSS:
    CP₂ = SP × 100/(100 − Loss%)
        = 9,900 × 100/90
        = ₹11,000

  Total CP  = 9,000 + 11,000 = ₹20,000
  Total SP  = 9,900 + 9,900  = ₹19,800

  Overall   → Loss = 20,000 − 19,800 = ₹200

  Shortcut: Loss % = X²/100 = 10²/100 = 1%
  Loss amount = 1% of Total CP = 1% of 20,000 = ₹200 ✓

─────────────────────────────────────────────
TRAP PROBLEM 2: False weight shopkeeper

  "A shopkeeper uses a weight of 800g instead of
   1kg (1000g) but sells goods at the listed CP.
   What is his profit %?"

  He gives 800g but charges for 1000g.
  He gains 200g on every 800g he actually gives out.

  Profit % = (Gain / False weight) × 100
           = (200 / 800) × 100
           = 25%

  Intuition check: For every 800g he spends,
  he gets paid for 1000g — that's a 200g gain
  on an 800g investment = 25%.

─────────────────────────────────────────────
TRAP PROBLEM 3: Successive transactions

  "A buys a guitar for ₹5,000 and sells to B
   at 20% profit. B sells to C at 10% profit.
   What does C pay?"

  A's SP = B's CP = 5,000 × 1.20 = ₹6,000
  B's SP = C's CP = 6,000 × 1.10 = ₹6,600

  C pays ₹6,600.

  WRONG approach (don't do this!):
  Adding 20% + 10% = 30%  →  5,000 × 1.30 = 6,500 ✗
  Percentages in successive steps multiply, not add.

Tricks and Shortcuts for Timed Exams

When the clock is ticking, longhand arithmetic is your enemy. Here are the mental math shortcuts that top scorers use to solve profit and loss problems in under 30 seconds.

Shortcut 1 — Net % for Markup + Discount: Net % = M − D − (M×D)/100. If positive, profit; if negative, loss.

Shortcut 2 — Loss when same SP and equal %: Net loss % = X²/100. Plug in X, get the loss percent instantly.

Shortcut 3 — False weight profit: Profit% = (Error/False weight) × 100, where Error = True weight − False weight.

Shortcut 4 — Fraction-to-percent conversion table: Memorise common fractions: 1/2 = 50%, 1/3 ≈ 33.33%, 1/4 = 25%, 1/5 = 20%, 1/10 = 10%. Many profit% values are simple fractions of CP.

Shortcut 5 — Single discount equivalent to multiple discounts: If two successive discounts d₁% and d₂%, single discount = d₁ + d₂ − (d₁×d₂)/100.

Practice these with real numbers until they become reflex. In an exam, the difference between a good score and a great score is often just how fast you apply these.

=== SHORTCUTS REFERENCE ===

1. Net % (markup M + discount D):
   Net% = M − D − (M×D)/100

2. Net loss (same SP, equal % X profit & loss):
   Loss% = X²/100

3. False weight profit:
   Profit% = (TrueWt − FalseWt) / FalseWt × 100

4. Successive discounts d1, d2:
   Single discount% = d1 + d2 − (d1×d2)/100

5. Common fraction → profit% mapping:
   1/5 = 20%   1/4 = 25%   1/3 ≈ 33.33%
   1/2 = 50%   2/3 ≈ 66.67%   3/4 = 75%

Use these in place of full calculation when
speed is critical and numbers are round.

The Hidden Tax in Successive Discounts — Why Two 10% Offs Aren't 20% Off

Every junior dev thinks successive discounts add up. They don't. A 10% discount followed by another 10% discount is not 20% off. It's 19% off. Why? Because the second discount applies to the already-reduced price, not the original. This is the classic 'successive percentage' trap. It wrecks profit calculations in e-commerce pricing engines and inventory systems. The formula is simple: net discount = A + B - (A * B / 100). For 10% and 10%: 10 + 10 - (100/100) = 19%. Always compute the effective discount chain before setting final price. Your margin depends on it.

# io.thecodeforge.com — Successive Discount Calculator
def net_discount(d1: float, d2: float) -> float:
    """Return net discount percentage from two successive discounts."""
    return d1 + d2 - (d1 * d2 / 100)

# Real-world: '10% off' + '10% off' coupon stack
print(f"Net discount: {net_discount(10, 10):.2f}%")
print(f"Effective price: {100 - net_discount(10, 10):.2f}% of original")

The False Profit of Dishonest Weights — How 900g as 1kg Bakes Your Margins

Selling 900 grams as a kilogram is the oldest trick in the book. It's also the most common coding mistake in point-of-sale systems that convert weight. The gain percent isn't the difference between 1000 and 900. It's (Error / Actual Given) 100. So for 900g: (1000 - 900) / 900 100 = 11.11%. Not 10%. This is the 'false weight' formula. It's asymmetric. Cheating by 100g on a 1kg sack yields a higher profit percentage than a 10% markup on the cost. Why? Because the loss is borne by the customer's quantity, not your cost. If you're building a billing system for a grocery chain, hardcode this check. It's a compliance audit flag.

# io.thecodeforge.com — Dishonest Weight Profit Calculator
def false_weight_profit(claimed: float, actual: float) -> float:
    """Return profit percentage when seller gives less than claimed weight."""
    return ((claimed - actual) / actual) * 100

# 1kg claimed, but only 900g delivered
print(f"Profit %: {false_weight_profit(1000, 900):.2f}%")

# What about 750g as 1kg?
print(f"Profit %: {false_weight_profit(1000, 750):.2f}%")

Marked Price, Discount, and Profit Percentage Chain

In retail, the marked price (MP) is the price printed on the tag, while the selling price (SP) is what the customer actually pays after discounts. The profit percentage is usually calculated on the cost price (CP), but sometimes on SP. This section builds a chain: MP → Discount → SP → Profit% (on CP or SP).

Example: A shopkeeper marks an item at ₹2,00,000 and offers a 10% discount. The SP = MP × (1 - Discount%) = 2,00,000 × 0.9 = ₹1,80,000. If the CP is ₹1,50,000, profit = SP - CP = ₹30,000. Profit% on CP = (30,000/1,50,000) × 100 = 20%. Profit% on SP = (30,000/1,80,000) × 100 ≈ 16.67%.

Chain formula: SP = MP × (1 - d) where d is discount rate. Then profit% on CP = (SP - CP)/CP × 100. If profit% on SP is given, profit = (profit% on SP) × SP.

Tricky case: If a shopkeeper wants a profit of 20% on SP after giving a 10% discount, find MP. Let CP = ₹100. Desired profit on SP = 20%, so SP = CP/(1 - 0.2) = 100/0.8 = ₹125. Then MP = SP/(1 - 0.1) = 125/0.9 ≈ ₹138.89. So MP should be about 38.89% above CP.

def find_mp(cp, discount_rate, profit_rate_on_sp):
    sp = cp / (1 - profit_rate_on_sp)
    mp = sp / (1 - discount_rate)
    return mp

cp = 150000
discount = 0.10
profit_on_sp = 0.20
mp = find_mp(cp, discount, profit_on_sp)
print(f"Marked Price: ₹{mp:.2f}")

Partnership and Profit Sharing Problems

Partnership problems involve two or more persons investing money in a business and sharing profits according to their investment and time. The profit is divided in the ratio of (capital × time).

Example: A invests ₹2,00,000 for 12 months, B invests ₹3,00,000 for 8 months. Profit at year-end is ₹1,20,000. A's share = (2,00,000 × 12) : (3,00,000 × 8) = 24,00,000 : 24,00,000 = 1:1. So each gets ₹60,000.

Variations: If partners join later or withdraw, adjust time accordingly. For example, A invests ₹2,00,000 for 12 months, B joins after 4 months with ₹3,00,000. A's capital-time = 2,00,000 × 12 = 24,00,000; B's = 3,00,000 × 8 = 24,00,000; ratio 1:1.

Tricky problem: A and B invest in ratio 3:2. A's capital is for 10 months, B's for 8 months. Profit is ₹50,000. Find each share. Let A's capital = 3k, B's = 2k. A's capital-time = 3k × 10 = 30k; B's = 2k × 8 = 16k; ratio = 30:16 = 15:8. A's share = (15/23)×50,000 ≈ ₹32,608.70; B's = ₹17,391.30.

Shortcut: If time is same, profit ratio = investment ratio. If investment same, profit ratio = time ratio.

def profit_share(capitals, times, total_profit):
    weighted = [c * t for c, t in zip(capitals, times)]
    total_weight = sum(weighted)
    shares = [p * w / total_weight for w in weighted]
    return shares

capitals = [200000, 300000]
times = [12, 8]
profit = 120000
shares = profit_share(capitals, times, profit)
for i, s in enumerate(shares):
    print(f"Partner {i+1}: ₹{s:.2f}")

Faulty Weights and Dishonest Shop Problems

Dishonest shopkeepers use faulty weights to cheat customers, selling less quantity than claimed. This creates hidden profit. The key is to compare the actual cost and selling price per unit weight.

Example: A shopkeeper uses a 900g weight as 1kg. He buys goods at ₹100/kg and sells at ₹120/kg (using faulty weight). What is his profit percentage?

Method: For every 1kg he claims to sell, he actually gives only 900g. His cost for 900g = 0.9 × 100 = ₹90. He sells that 900g at the price of 1kg = ₹120. So profit = 120 - 90 = ₹30. Profit% on cost = (30/90)×100 = 33.33%.

Formula: If the shopkeeper uses a weight of w grams instead of 1000g, and buys at CP per kg, sells at SP per kg, then profit% = [(SP - CP × (w/1000)) / (CP × (w/1000))] × 100. Alternatively, if he sells at CP but uses faulty weight, profit% = [(1000/w) - 1] × 100.

Example: He sells at cost price but uses 800g as 1kg. Profit% = (1000/800 - 1)×100 = (1.25 - 1)×100 = 25%.

Tricky case: He claims to sell at a discount but uses faulty weight. For instance, he marks MP at ₹150/kg, gives 10% discount, but uses 900g weight. CP = ₹100/kg. Find profit%. Effective SP per 900g = 150 × 0.9 = ₹135. Cost for 900g = ₹90. Profit = 45, profit% = 50%.

def faulty_weight_profit(cp_per_kg, sp_per_kg, actual_weight_g):
    cost = cp_per_kg * (actual_weight_g / 1000)
    selling = sp_per_kg  # he sells at price of 1kg
    profit = selling - cost
    profit_pct = (profit / cost) * 100
    return profit_pct

cp = 100
sp = 120
weight = 900
print(f"Profit%: {faulty_weight_profit(cp, sp, weight):.2f}%")

# Selling at cost with faulty weight
print(f"Profit% at CP: {(1000/weight - 1)*100:.2f}%")