Profit and Loss Concepts

1. Cost Price, Selling Price & Profit/Loss Basics

• Formula:

$$\text{Profit}=SP-CP$$ $$\text{Loss}=CP-SP$$ \text{Profit%}=\frac{\text{Profit}}{CP}\times100 \text{Loss%}=\frac{\text{Loss}}{CP}\times100

• Example: An article is bought for ₹800 and sold for ₹920. Find the profit percentage. Solution:

$$\text{Profit}=920-800=₹120$$ \text{Profit%}=\frac{120}{800}\times100 =\boxed{15%}

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2. Finding SP from CP and Profit %

• Formula:

$$SP=CP\left(1+\frac{r}{100}\right)$$

For a loss of (r%):

$$SP=CP\left(1-\frac{r}{100}\right)$$

• Example: An article costs ₹1,200. It is sold at a profit of 15%. Find the selling price. Solution:

$$SP=1200\left(1+\frac{15}{100}\right)$$ $$=1200(1.15)=\boxed{₹1380}$$

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3. Finding CP from SP and Profit/Loss %

• Formula:

CP=\frac{SP\times100}{100+\text{Profit%}} CP=\frac{SP\times100}{100-\text{Loss%}}

• Example: An article is sold for ₹1,380 at a 15% profit. Find its cost price. Solution:

$$CP=\frac{1380\times100}{115}$$ $$=\boxed{₹1200}$$

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4. Dishonest Dealer / False Weights

• Formula: If a dealer gives (w) units while charging for 1 unit at cost price:

\text{Profit%}=\frac{1-w}{w}\times100

If the dealer gives (x) grams instead of 1000 grams:

\text{Profit%}=\frac{1000-x}{x}\times100

• Example: A dealer uses 900 g instead of 1 kg while charging the cost price of 1 kg. Find the profit percentage. Solution: Actual quantity given:

$$900\text{ g}$$

Error:

$$1000-900=100\text{ g}$$

Therefore:

\text{Profit%}=\frac{100}{900}\times100 =\boxed{11.11%}

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5. Marked Price and Discount

• Formula:

$$\text{Discount}=MP-SP$$ \text{Discount%}=\frac{MP-SP}{MP}\times100 $$SP=MP\left(1-\frac{d}{100}\right)$$

• Example: An article marked at ₹2,500 is sold at a 20% discount. Find the selling price. Solution:

$$SP=2500(1-0.20)$$ $$=2500(0.8)=\boxed{₹2000}$$

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6. Marked Price from SP and Discount

• Formula:

$$MP=\frac{SP\times100}{100-d}$$

• Example: An article is sold for ₹1,600 after a 20% discount. Find its marked price. Solution:

$$MP=\frac{1600\times100}{80}$$ $$=\boxed{₹2000}$$

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7. Profit After Discount

• Formula:

$$SP=MP\left(1-\frac d{100}\right)$$

Then:

\text{Profit%}=\frac{SP-CP}{CP}\times100

• Example: An article costs ₹1,000 and is marked 40% above CP. A discount of 10% is offered. Find the profit percentage. Solution:

$$MP=1000(1.4)=₹1400$$

After 10% discount:

$$SP=1400(0.9)=₹1260$$

Profit:

$$1260-1000=₹260$$

Therefore:

\text{Profit%}=\frac{260}{1000}\times100 =\boxed{26%}

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8. Successive Discounts

• Formula: For discounts (a%) and (b%):

$$\text{Net discount}=a+b-\frac{ab}{100}$$

• Example: An article is offered successive discounts of 20% and 10%. Find the equivalent discount. Solution:

$$20+10-\frac{20\times10}{100}$$ =30-2=\boxed{28%}

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9. Multiple Transaction Chains

• Formula: For successive profits:

$$SP=CP\left(1+\frac{r_1}{100}\right) \left(1+\frac{r_2}{100}\right)\cdots$$

For losses, use (1-\frac r{100}).

• Example: A manufacturer sells an article costing ₹100 to a wholesaler at 20% profit. The wholesaler sells it to a retailer at 10% profit. Find the retailer’s cost price. Solution: Manufacturer’s SP:

$$100(1.20)=₹120$$

Wholesaler’s SP:

$$120(1.10)=₹132$$

Retailer’s CP:

$$\boxed{₹132}$$

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10. Overall Profit/Loss in Multiple Transactions

• Formula:

$$\frac{SP_{\text{final}}-CP_{\text{initial}}}{CP_{\text{initial}}}\times100$$

For successive percentage changes, multiply the factors.

• Example: A product passes through two sellers who make profits of 20% and 25%. Find the overall profit percentage over the original cost price. Solution:

$$SP=CP(1.20)(1.25)$$ $$=1.5CP$$

Thus final SP is 150% of original CP.

Overall profit:

\boxed{50%}

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11. GST with Discount

• Formula:

$$SP=MP\left(1-\frac d{100}\right)$$ $$\text{Final price}=SP\left(1+\frac{\text{GST}}{100}\right)$$

• Example: An item has MP ₹2,000, a 20% discount and 12% GST. Find the final price. Solution: Discounted price:

$$2000(0.8)=₹1600$$

GST:

$$1600(0.12)=₹192$$

Final price:

$$1600+192=\boxed{₹1792}$$

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12. GST Included in Selling Price

• Formula: If final price includes (r%) GST:

$$\text{Price before GST}=\frac{\text{Final price}\times100}{100+r}$$

GST amount:

$$\text{Final price}-\text{Price before GST}$$

• Example: An item costs ₹2,360 including 18% GST. Find its price before GST. Solution:

$$\text{Price before GST} =\frac{2360\times100}{118}$$ $$=\boxed{₹2000}$$

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13. Profit and Loss with Equal Selling Prices

• Formula: If two articles are sold at the same SP, one at (x%) profit and the other at (x%) loss:

\text{Overall loss%}=\frac{x^2}{100}

• Example: Two articles are sold for ₹1,200 each. One gives a 20% profit and the other a 20% loss. Find the overall loss percentage. Solution: Using the shortcut:

\text{Loss%}=\frac{20^2}{100} =\boxed{4%}

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14. Equal Cost Prices with Different Profit/Loss

• Formula: If two articles have the same CP:

$$\text{Overall profit/loss} =\frac{\text{Total SP}-\text{Total CP}}{\text{Total CP}}\times100$$

• Example: Two articles each cost ₹1,000. One is sold at 20% profit and the other at 10% loss. Find the overall profit percentage. Solution: First SP:

$$1000(1.2)=₹1200$$

Second SP:

$$1000(0.9)=₹900$$

Total CP:

$$₹2000$$

Total SP:

$$₹2100$$

Profit:

$$₹100$$

Therefore:

\text{Profit%}=\frac{100}{2000}\times100 =\boxed{5%}

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15. Required Marked Price for a Target Profit

• Formula: If discount is (d%) and desired profit is (p%):

$$MP=\frac{CP(100+p)}{100-d}$$

• Example: An article costs ₹800. It must be sold at 20% profit after giving a 20% discount. Find the required marked price. Solution:

$$MP=\frac{800(120)}{80}$$ $$=\boxed{₹1200}$$

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16. Finding Discount from MP, CP and Desired Profit

• Formula: Desired SP:

$$SP=CP\left(1+\frac p{100}\right)$$

Then:

$$d=\frac{MP-SP}{MP}\times100$$

• Example: An article costs ₹1,000 and is marked at ₹1,500. What discount can be offered while still making a 20% profit? Solution: Required SP:

$$1000(1.2)=₹1200$$

Discount:

$$1500-1200=₹300$$

Discount percentage:

\frac{300}{1500}\times100 =\boxed{20%}

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17. Profit Percentage Based on Selling Price

• Formula: If profit is (p%) of SP:

$$CP=SP\left(1-\frac p{100}\right)$$

More directly:

\text{Profit% on CP} =\frac{p}{100-p}\times100

• Example: A trader’s profit is 20% of the selling price. Find the profit percentage on cost price. Solution: Assume:

$$SP=100$$

Profit:

$$20$$

Therefore:

$$CP=100-20=80$$

Profit percentage on CP:

\frac{20}{80}\times100 =\boxed{25%}

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18. Loss Percentage Based on Selling Price

• Formula: If loss is (l%) of SP:

$$CP=SP\left(1+\frac l{100-l}\right)$$

Equivalently:

\text{Loss% on CP} =\frac{l}{100-l}\times100

• Example: A loss is 20% of the selling price. Find the loss percentage on cost price. Solution: Assume:

$$SP=100$$

Loss:

$$20$$

Therefore:

$$CP=120$$

Loss percentage:

\frac{20}{120}\times100 =\boxed{16.67%}

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19. Dishonest Dealer with False Weight and Price Change

• Formula: If a dealer gives (w) units instead of 1 unit while charging for 1 unit, the effective profit factor is:

$$\frac1w$$

If the dealer also charges (r%) above/below CP, combine the factors multiplicatively.

• Example: A dealer gives 800 g instead of 1 kg and charges 10% below the cost price of 1 kg. Find the effective profit percentage. Solution: Assume CP of 1 kg = ₹100.

He charges:

$$100(0.9)=₹90$$

for 800 g.

CP of 800 g:

$$₹80$$

Profit:

$$90-80=₹10$$

Profit percentage:

\frac{10}{80}\times100 =\boxed{12.5%}

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Advanced Variants

20. Two Successive Profits/Losses

• Formula: For successive changes (a%) and (b%):

$$\text{Net change}=a+b+\frac{ab}{100}$$

Treat losses as negative.

• Example: A trader makes a 20% profit and then suffers a 10% loss on the new value. Find the net change. Solution:

$$20-10+\frac{20(-10)}{100}$$ =20-10-2 =\boxed{8%}

Overall result = 8% profit.

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21. Markup Followed by Successive Discounts

• Formula:

$$SP=CP\left(1+\frac m{100}\right) \left(1-\frac{d_1}{100}\right) \left(1-\frac{d_2}{100}\right)$$

• Example: An article is marked 50% above CP and given discounts of 20% and 10%. Find the overall profit percentage. Solution:

$$SP=CP(1.5)(0.8)(0.9)$$ $$=1.08CP$$

Therefore:

\boxed{8%\text{ profit}}

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22. Profit Sharing in Partnerships

• Formula:

$$\text{Profit share}\propto\text{Capital}\times\text{Time}$$

Hence:

$$A:B=I_AT_A:I_BT_B$$

• Example: A invests ₹5,000 for 12 months and B invests ₹6,000 for 8 months. If the total profit is ₹2,700, find their shares. Solution:

$$A:B=5000(12):6000(8)$$ $$=60000:48000=5:4$$

Total parts:

$$9$$

A’s share:

$$2700\times\frac59=\boxed{₹1500}$$

B’s share:

$$2700\times\frac49=\boxed{₹1200}$$

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23. Partnership with Change in Capital

• Formula:

$$\text{Profit ratio}=I_1T_1:I_2T_2$$

If investment changes during the year, divide the period into intervals and calculate:

$$\text{Effective investment}=\sum(\text{Investment}\times\text{Time})$$

• Example: A invests ₹10,000 for 12 months. B invests ₹8,000 for the first 6 months and ₹12,000 for the next 6 months. Find the profit-sharing ratio. Solution: A:

$$10000\times12=120000$$

B:

$$8000\times6+12000\times6 =48000+72000=120000$$

Therefore:

$$\boxed{A:B=1:1}$$

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24. Required False Weight for a Given Profit

• Formula: If a dealer wants (p%) profit while charging CP:

$$\text{Weight given} =\frac{100}{100+p}\times\text{True weight}$$

• Example: A dealer wants to make a 25% profit while claiming to sell at cost price. How much weight should he give instead of 1 kg? Solution:

$$\text{Weight}=\frac{100}{125}\times1000$$ $$=\boxed{800\text{ g}}$$

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25. Buy (x), Get (y) Free

• Formula: If a customer pays for (x) items and receives (y) additional items free, effective discount:

\text{Discount%}=\frac{y}{x+y}\times100

• Example: A shop offers “Buy 4, Get 1 Free.” Find the effective discount percentage. Solution: Customer pays for 4 but receives 5:

\text{Discount%}=\frac{1}{5}\times100 =\boxed{20%}

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26. Buy (x), Get (y) Free with Profit

• Formula: If each item has CP (C), customer pays for (x) items at price (SP) and receives (x+y) items:

$$\text{Total CP}=(x+y)C$$ \text{Profit%}= \frac{xSP-(x+y)C}{(x+y)C}\times100

• Example: An article costs ₹80. A shopkeeper sells 5 articles for ₹500 and gives 1 article free. Find his profit percentage. Solution: Total articles:

$$5+1=6$$

Total CP:

$$6\times80=₹480$$

Revenue:

$$₹500$$

Profit:

$$500-480=₹20$$

Profit percentage:

\frac{20}{480}\times100 =\boxed{4.17%}

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27. Profit/Loss When Selling Price Changes

• Formula: If the selling price is increased by (x%), the new profit can be calculated from:

$$SP_{\text{new}}=SP\left(1+\frac{x}{100}\right)$$

Then compare with CP.

• Example: An article is sold at a 20% profit. If its selling price is increased by 10%, what is the new profit percentage? Solution: Assume:

$$CP=100$$

Original SP:

$$120$$

New SP:

$$120(1.1)=132$$

New profit:

$$132-100=32$$

Therefore:

\boxed{32%}

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28. Same SP with Different Profit and Loss Percentages

• Formula: If the same SP gives (x%) profit on one article and (y%) loss on another:

$$CP_1=\frac{SP\times100}{100+x}$$ $$CP_2=\frac{SP\times100}{100-y}$$

Compare (CP_1+CP_2) with (2SP).

• Example: Two articles are sold for ₹1,200 each. One gives 20% profit and the other 10% loss. Find the overall profit/loss. Solution: First CP:

$$\frac{1200\times100}{120}=₹1000$$

Second CP:

$$\frac{1200\times100}{90}=₹1333.33$$

Total CP:

$$₹2333.33$$

Total SP:

$$₹2400$$

Profit:

$$2400-2333.33=₹66.67$$

Profit percentage:

\frac{66.67}{2333.33}\times100 =\boxed{2.86%\text{ profit}}