📘 Important Formulas
Profit = Selling Price − Cost Price
Loss = Cost Price − Selling Price
Profit % = \(\frac{\text{Profit}}{\text{C.P.}}\times100\)
Loss % = \(\frac{\text{Loss}}{\text{C.P.}}\times100\)
When gain is \(r%\):
\[
S.P.=C.P.\times\frac{100+r}{100}
\]
When loss is \(r%\):
\[
S.P.=C.P.\times\frac{100-r}{100}
\]
To find C.P. when gain is given:
\[
C.P.=\frac{S.P.\times100}{100+\text{Gain%}}
\]
To find C.P. when loss is given:
\[
C.P.=\frac{S.P.\times100}{100-\text{Loss%}}
\]
1. Find the gain or loss per cent.
2. Find the selling price.
3. Find the cost price.
4. A boy buys 9 oranges for ₹9.60 and sells them at 11 for ₹12. Find his gain or loss per cent.
Step 1: Find the cost price of 1 orange.
\[ C.P.=\frac{₹9.60}{9} =₹\frac{32}{30} =₹\frac{16}{15} \]Step 2: Find the selling price of 1 orange.
\[ S.P.=\frac{₹12}{11} \]Step 3: Find the gain.
\[ \text{Gain} = \frac{12}{11}-\frac{16}{15} = \frac{4}{165} \]Step 4: Find gain percentage.
\[ \text{Gain%} = \frac{\frac{4}{165}} {\frac{16}{15}} \times100 \] \[ =\frac{1}{44}\times100 = 2\frac{3}{22}% \]5. A retailer buys a cooler for ₹1200 and overhead expenses on it are ₹40. If he sells the cooler for ₹1550, determine his profit per cent.
Step 1: Find the total cost price.
\[ \text{Total C.P.} = ₹1200+₹40 =₹1240 \]Step 2: Find the profit.
\[ \text{Profit} = ₹1550-₹1240 =₹310 \]Step 3: Find profit percentage.
\[ \text{Profit%} = \frac{310}{1240}\times100 =25% \]6. A dealer buys a wrist watch for ₹225 and spends ₹15 on its repairs. If he sells the watch for ₹300, find his profit per cent.
Step 1: Find the total cost.
\[ \text{Total C.P.} = ₹225+₹15 =₹240 \]Step 2: Find the profit.
\[ \text{Profit} = ₹300-₹240 =₹60 \]Step 3: Find profit percentage.
\[ \text{Profit%} = \frac{60}{240}\times100 =25% \]7. Coffee costing ₹250 per kg was mixed with chicory costing ₹75 per kg in the ratio of 5:2. If the mixture was sold at ₹230 per kg, find the gain or loss per cent.
Step 1: Assume 5 kg coffee and 2 kg chicory.
| Item | Quantity | Rate | Cost |
|---|---|---|---|
| Coffee | 5 kg | ₹250/kg | ₹1250 |
| Chicory | 2 kg | ₹75/kg | ₹150 |
Total cost = ₹1250 + ₹150 = ₹1400
Total quantity = 5 + 2 = 7 kg
Cost price of 1 kg mixture:
\[ C.P.=\frac{1400}{7}=₹200 \]Step 2: Find the profit.
\[ \text{Profit}=₹230-₹200=₹30 \]Step 3: Find profit percentage.
\[ \text{Profit%} = \frac{30}{200}\times100 =15% \]8. Rishabh bought two boxes for ₹1300. He sold one box at a profit of 20% and the other box at a loss of 12%. If the selling price of both boxes is the same, find the cost price of each box.
Let the cost price of the first box be ₹x.
Therefore, cost price of the second box = ₹(1300 − x).
First box is sold at 20% profit:
\[ S.P._1=120%\text{ of }x =1.20x \]Second box is sold at 12% loss:
\[ S.P._2=88%\text{ of }(1300-x) \]Since both selling prices are equal:
\[ 1.20x=0.88(1300-x) \] \[ 1.20x=1144-0.88x \] \[ 2.08x=1144 \] \[ x=550 \]Therefore, cost price of the second box:
\[ 1300-550=₹750 \]Verification:
\[ 20%\text{ profit on ₹550}=₹110 \] \[ S.P.=₹550+₹110=₹660 \] \[ 12%\text{ loss on ₹750}=₹90 \] \[ S.P.=₹750-₹90=₹660 \] Both selling prices are indeed ₹660.9. The cost price of 12 balls is equal to the selling price of 15 balls. Find the loss per cent.
Let the cost price of 1 ball be ₹1.
Then C.P. of 12 balls = ₹12.
Therefore, S.P. of 15 balls = ₹12.
\[ S.P.\text{ of 1 ball} = \frac{12}{15} = ₹0.80 \]Loss on 1 ball:
\[ 1-0.80=₹0.20 \] \[ \text{Loss%} = \frac{0.20}{1}\times100 =20% \]10. If the cost price of 18 pens is equal to the selling price of 16 pens, find the gain or loss per cent.
Let the cost price of 1 pen be ₹1.
C.P. of 18 pens = ₹18.
Therefore, S.P. of 16 pens = ₹18.
\[ S.P.\text{ of 1 pen} = \frac{18}{16} =₹1.125 \]Gain on 1 pen:
\[ 1.125-1=₹0.125 \] \[ \text{Gain%} = \frac{0.125}{1}\times100 =12.5% \] \[ =12\frac{1}{2}% \]11. A man purchased some eggs at 3 for ₹5 and sold them at 5 for ₹12. Thus, he gained ₹143 in all. How many eggs did he purchase?
Step 1: Find C.P. of one egg.
\[ C.P.=\frac{₹5}{3} \]Step 2: Find S.P. of one egg.
\[ S.P.=\frac{₹12}{5} \]Step 3: Find gain on one egg.
\[ \text{Gain} = \frac{12}{5}-\frac{5}{3} \] \[ = \frac{36-25}{15} = \frac{11}{15} \]Therefore, gain on one egg = \(\frac{11}{15}\) rupee.
Total gain = ₹143.
Number of eggs:
\[ n\times\frac{11}{15}=143 \] \[ n=143\times\frac{15}{11} \] \[ n=13\times15 =195 \]12. By selling a book for ₹258, a bookseller gains 20%. For how much should he sell it to gain 30%?
At 20% gain, ₹258 represents 120% of the cost price.
\[ C.P. = \frac{258\times100}{120} =₹215 \]For a gain of 30%, selling price will be 130% of C.P.
\[ S.P. = \frac{130}{100}\times215 \] \[ =₹279.50 \]13. A defective briefcase costing ₹800 is being sold at a loss of 8%. If the price is further reduced by 5%, find its selling price.
Step 1: Find the selling price after 8% loss.
\[ S.P. = 92%\text{ of ₹800} \] \[ =\frac{92}{100}\times800 =₹736 \]Step 2: Reduce this price by another 5%.
\[ 5%\text{ of ₹736} = \frac{5}{100}\times736 =₹36.80 \]New selling price:
\[ ₹736-₹36.80 =₹699.20 \]14. A dealer gets ₹56 more instead of selling a table at a loss of 10%, it is sold at a gain of 10%. Find the cost price of the table.
Let the cost price of the table be ₹x.
When sold at a loss of 10%:
\[ S.P._1=90%\text{ of }x =0.90x \]When sold at a gain of 10%:
\[ S.P._2=110%\text{ of }x =1.10x \]Difference between the two selling prices = ₹56.
\[ 1.10x-0.90x=56 \] \[ 0.20x=56 \] \[ x=\frac{56}{0.20} =₹280 \]📋 Quick Answer Check
| Question | Answer |
|---|---|
| 1(a) | Gain = 15% |
| 1(b) | Loss = \(13\frac{1}{23}%\) |
| 1(c) | Loss = \(13\frac{8}{9}%\) |
| 1(d) | Gain = 4% |
| 2(a) | ₹1638 |
| 2(c) | ₹559 |
| 3(a) | ₹1425 |
| 3(b) | ₹255 |
| 3(c) | ₹685 |
| 4 | Gain = \(2\frac{3}{22}%\) ≈ 2.27% |
| 5 | Profit = 25% |
| 6 | Profit = 25% |
| 7 | Gain = 15% |
| 8 | ₹550 and ₹750 |
| 9 | Loss = 20% |
| 10 | Gain = \(12\frac{1}{2}%\) |
| 11 | 195 eggs |
| 12 | ₹279.50 |
| 13 | ₹699.20 |
| 14 | ₹280 |
Always remember that profit or loss percentage is calculated on the Cost Price (C.P.), unless the question specifically tells you otherwise.
C.P. → compare with S.P. → find Profit/Loss → divide by C.P. → multiply by 100.
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