Profit and Loss

1. Core Concepts & Theoretical Blueprint

Profit and Loss questions test the relationship between three core values: Cost Price (CP), Selling Price (SP), and the resulting Profit or Loss, always measured as a percentage of CP unless explicitly stated otherwise.

Absolute Core Formulas:

Selling Price Formulas:

Marked Price (MP) and Discount:

False Weight Trick:

The Universal Trap: 1) Profit%/Loss% always on CP, never SP unless stated. 2) MP (discount base) vs CP (profit base) confusion. 3) Successive discounts wrongly averaged.

2. Exhaustive Question Typology

                            PROFIT AND LOSS
                                   |
      ---------------------------------------------------------------
      |            |              |               |                |
  Type 1:       Type 2:        Type 3:        Type 4:          Type 5:
  Basic CP-SP   Marked Price   Successive     Faulty/False     Equal SP,
  Profit/Loss   & Discount     Discounts       Weight/          Different
  % computation  problems      (2 or more)     Measure          Profit-Loss
                                                Problems         (find CP ratio)
      |            |              |
  Type 6:       Type 7:        Type 8:
  Cost Price    Article sold   Mixed
  from multiple at Profit%     articles /
  articles sold at  and Loss%   averaging
  same SP,      to two         profit%
  overall gain/  different
  loss          buyers

Type 1 — Basic CP–SP profit/loss percentage:

• Governing Equation: $\text{Profit\%/Loss\%} = \frac{|SP-CP|}{CP}\times 100$

Type 2 — Marked Price and Discount problems:

• Governing Equation: $SP = MP\left(1-\frac{d}{100}\right) = CP\left(1+\frac{p}{100}\right)$

Type 3 — Successive discounts:

• Governing Equation:
  $$\text{Net Discount \%} = d_1+d_2-\frac{d_1d_2}{100}$$

Type 4 — Faulty weight/measure problems:

• Governing Equation: $\text{Profit\%} = \frac{1000-x}{x}\times 100$

Type 5 — Same SP, different profit/loss on two articles:

• Governing Equation: Overall result is always a loss:
  $$\text{Loss\%} = \left(\frac{p}{10}\right)^2$$

Type 6 — Cost Price of multiple articles, overall gain/loss:

• Governing Equation: Compute effective CP/unit and SP/unit, then apply Type 1.

Type 7 — Article sold to two buyers at different % profit/loss:

• Governing Equation: Weighted average using each portion's CP fraction.

Type 8 — "CP equals SP of n articles" ratio-based problems:

• Governing Equation:
  $$\text{Profit\%} = \frac{x-y}{y}\times 100$$

3. Type-wise Practice MCQs with Full Solutions

Type 1 — Basic CP-SP Profit/Loss %

MCQ 1. An article is bought for ₹450 and sold for ₹540. Find the profit percentage. (A) 20% (B) 18% (C) 22% (D) 25%

Correct Answer: (A) Solution: Profit$=90$. $\%=\dfrac{90}{450}\times100=20\%$.

MCQ 2. A book bought for ₹320 is sold at a loss of ₹40. Find the loss percentage. (A) 12.5% (B) 10% (C) 15% (D) 13%

Correct Answer: (A) Solution: $\%=\dfrac{40}{320}\times100=12.5\%$.

MCQ 3. A trader buys an article for ₹800 and wants a profit of 15%. Find the required SP. (A) ₹920 (B) ₹900 (C) ₹930 (D) ₹910

Correct Answer: (A) Solution: $SP=800\times1.15=920$.

Type 2 — Marked Price and Discount

MCQ 1. A shopkeeper marks an item 30% above CP and gives a 10% discount. Find the profit %. (A) 17% (B) 20% (C) 15% (D) 18%

Correct Answer: (A) Solution: Net%$=30-10-\dfrac{300}{100}=20-3=17\%$.

MCQ 2. An article marked at ₹500 is sold for ₹425. Find the discount percentage. (A) 15% (B) 12% (C) 18% (D) 14%

Correct Answer: (A) Solution: Discount$=75$. $\%=\dfrac{75}{500}\times100=15\%$.

MCQ 3. A shopkeeper allows a discount of 20% on the marked price and still makes a profit of 12%. If CP is ₹500, find the marked price. (A) ₹700 (B) ₹650 (C) ₹680 (D) ₹720

Correct Answer: (A) Solution: $SP=500\times1.12=560$. $MP=SP/0.8=560/0.8=700$.

Type 3 — Successive Discounts

MCQ 1. Find the net discount on successive discounts of 20% and 10%. (A) 28% (B) 30% (C) 25% (D) 27%

Correct Answer: (A) Solution: $20+10-\dfrac{200}{100}=30-2=28\%$.

MCQ 2. Two successive discounts of 15% and 5% are equivalent to a single discount of: (A) 19.25% (B) 20% (C) 18% (D) 19%

Correct Answer: (A) Solution: $15+5-\dfrac{75}{100}=20-0.75=19.25\%$.

MCQ 3. A trader offers successive discounts of 25% and 20% on an item marked at ₹800. Find the final SP. (A) ₹480 (B) ₹500 (C) ₹460 (D) ₹520

Correct Answer: (A) Solution: $800\times0.75\times0.8=480$.

Type 4 — Faulty Weight/Measure Problems

MCQ 1. A dishonest dealer professes to sell at CP but uses a weight of 800g instead of 1000g. Find his profit %. (A) 25% (B) 20% (C) 22% (D) 28%

Correct Answer: (A) Solution: $\%=\dfrac{1000-800}{800}\times100=25\%$.

MCQ 2. A shopkeeper uses a 900g weight for a kg and sells at cost price. Find his profit %. (A) 11.11% (B) 10% (C) 12% (D) 9%

Correct Answer: (A) Solution: $\%=\dfrac{100}{900}\times100=11.11\%$.

MCQ 3. A dealer uses a false weight and gains 20% while selling at cost price. Find the weight he actually uses (per 1000g). (A) 833.33 g (B) 800 g (C) 850 g (D) 820 g

Correct Answer: (A) Solution: $\dfrac{1000-x}{x}=0.20\Rightarrow1000-x=0.2x\Rightarrow1000=1.2x\Rightarrow x=833.33$ g.

Type 5 — Same SP, Different Profit/Loss

MCQ 1. A man sells two articles at the same SP, one at 20% profit and the other at 20% loss. Find the overall loss %. (A) 4% (B) 0% (C) 2% (D) 5%

Correct Answer: (A) Solution: Loss%$=(20/10)^2=4\%$.

MCQ 2. Two horses are sold at the same price, one at 10% gain, the other at 10% loss. Find the overall result. (A) 1% loss (B) No loss or gain (C) 1% gain (D) 2% loss

Correct Answer: (A) Solution: Loss%$=(10/10)^2=1\%$.

MCQ 3. A trader sells two items at the same price, one at 25% profit, the other at 25% loss. Find the overall loss %. (A) 6.25% (B) 5% (C) 7% (D) 6%

Correct Answer: (A) Solution: Loss%$=(25/10)^2=6.25\%$.

Type 6 — Cost Price of Multiple Articles

MCQ 1. A man buys 12 articles for ₹8 and sells 8 articles for ₹12. Find his profit %. (A) 125% (B) 100% (C) 110% (D) 120%

Correct Answer: (A) Solution: CP/unit$=8/12=2/3$. SP/unit$=12/8=3/2$. Profit%$=\dfrac{(3/2)-(2/3)}{2/3}\times100=\dfrac{5/6}{2/3}\times100=\dfrac54\times100=125\%$.

MCQ 2. By selling 45 lemons for ₹40, a man loses 20%. How many should he sell for ₹24 to gain 20% (find lemons per rupee first)? (A) 18 lemons (B) 20 lemons (C) 15 lemons (D) 16 lemons

Correct Answer: (A) Solution: SP for 45 lemons=₹40 represents 80% of CP (20% loss), so CP for 45 lemons$=40/0.8=50$. For 20% profit, SP should be $50\times1.2=60$ for 45 lemons, i.e., rate$=45/60=0.75$ lemons per rupee... reframe: for ₹24 at gain 20%, find lemons: New SP rate = 60 rupees per 45 lemons → for ₹24: $45\times(24/60)=18$ lemons.

MCQ 3. A man buys oranges at 6 for ₹5 and sells at 5 for ₹6. Find his profit %. (A) 44% (B) 40% (C) 42% (D) 38%

Correct Answer: (A) Solution: CP/unit$=5/6$. SP/unit$=6/5$. Profit%$=\dfrac{(6/5)-(5/6)}{5/6}\times100=\dfrac{36/30-25/30}{25/30}\times100=\dfrac{11/30}{25/30}\times100=\dfrac{11}{25}\times100=44\%$.

Type 7 — Article Sold at Different % to Two Buyers

MCQ 1. A man sells half his goods at 20% profit and the rest at 10% loss. If his overall profit is ₹500, find the total CP. (A) ₹10,000 (B) ₹9,500 (C) ₹10,500 (D) ₹9,800

Correct Answer: (A) Solution: Overall%$=\dfrac{20-10}{2}=5\%$ (simple average since equal split). $5\%$ of CP$=500\Rightarrow$CP$=10000$.

MCQ 2. A dealer sells 1/3 of his stock at 10% profit, 1/3 at 15% profit, and the rest at 20% profit. Find his overall profit %. (A) 15% (B) 14% (C) 16% (D) 15.5%

Correct Answer: (A) Solution: Equal thirds: average$=\dfrac{10+15+20}{3}=15\%$.

MCQ 3. A shopkeeper sells 60% of his goods at 25% profit and the remaining at 10% loss. Find his overall profit %. (A) 11% (B) 10% (C) 12% (D) 13%

Correct Answer: (A) Solution: Weighted: $0.6\times25+0.4\times(-10)=15-4=11\%$.

Type 8 — "CP Equals SP of n Articles" Problems

MCQ 1. The cost price of 20 articles equals the selling price of 16 articles. Find the profit %. (A) 25% (B) 20% (C) 22% (D) 28%

Correct Answer: (A) Solution: $\%=\dfrac{20-16}{16}\times100=25\%$.

MCQ 2. The cost price of 25 articles equals the selling price of 20 articles. Find the profit %. (A) 25% (B) 20% (C) 22% (D) 24%

Correct Answer: (A) Solution: $\%=\dfrac{25-20}{20}\times100=25\%$.

MCQ 3. The cost price of 18 articles equals the selling price of 24 articles. Find the loss %. (A) 25% (B) 20% (C) 22% (D) 24%

Correct Answer: (A) Solution: Since CP(18) = SP(24), and 18<24, this indicates a loss: $\%=\dfrac{24-18}{24}\times100=25\%$ loss.

4. High-Yield Speed Tricks & Shortcut Mental Models

Shortcut 1 — The "Same SP, Equal % Profit and Loss" Instant Loss Formula

• Application: Any Type 5 question.
• Mental Model: Overall result is always a loss, equal to $(p/10)^2\%$.

Shortcut 2 — CP = 100 Assumption Method

• Application: Any multi-step profit/loss/discount chain.
• Mental Model: Assume CP (or MP) = ₹100 to convert all subsequent multiplications into simple percentage arithmetic.

5. Deep-Dive: Most Frequently Asked Questions

Problem 1 (SSC/RRB Standard): A shopkeeper marks his goods 40% above cost price and allows a discount of 25%. Find his profit percentage.

Traditional Method (Slow): CP=100. MP=140. Discount=35. SP=105. Profit=5%. (~35-40 seconds.)

Exam Shortcut (Fast): Net% = $40-25-10=5\%$. (Under 10 seconds.)

Problem 2 (UPSC/Banking Advanced): A trader sells two horses for ₹9,975 each. On one he gains 15% and on the other he loses 15%. Find his overall gain or loss percent.

Step-by-Step Breakdown:

1. Loss% = $(15/10)^2=2.25\%$.
2. $CP_1=9975/1.15=8673.91$; $CP_2=9975/0.85=11735.29$.
3. Total CP=20409.20; Total SP=19950.
4. Loss=459.20 ≈2.25% of total CP.
5. Answer: Overall loss = 2.25%, i.e., ~₹459.20.

6. Chapter Checklist for Students

• I always compute Profit%/Loss% on CP, never SP.
• I clearly separate MP (discount base) from CP (profit base).
• I apply the successive-change formula for multi-discount and markup-discount problems.
• I instantly recognize "same SP, equal %" pattern and apply $(p/10)^2\%$ loss.
• I use the CP=100 assumption method for chained problems.