Profit & Loss, Discount & Tax

Q1: Find the gain or loss per cent, when:

i. C.P. = ₹750, S.P. = ₹875

Step 1: S.P. > C.P. ⇒ There is a gain.
Step 2: Gain = S.P. − C.P. = ₹875 − ₹750 = ₹125
Step 3: Gain % = (Gain ÷ C.P.) × 100
⇒ Gain % = (125 ÷ 750) × 100
⇒ Gain % = \(\frac{1}{6} \times 100 = 16 \frac{2}{3}\%\)
Answer: Gain = \(16 \frac{2}{3}\)%

ii. C.P. = ₹126, S.P. = ₹94.50

Step 1: S.P. < C.P. ⇒ There is a loss.
Step 2: Loss = C.P. − S.P. = ₹126 − ₹94.50 = ₹31.50
Step 3: Loss % = (Loss ÷ C.P.) × 100
⇒ Loss % = (31.50 ÷ 126) × 100
⇒ Loss % = 0.25 × 100 = 25%
Answer: Loss = 25%

iii. C.P. = ₹80.40, S.P. = ₹68.34

Step 1: S.P. < C.P. ⇒ There is a loss.
Step 2: Loss = C.P. − S.P. = ₹80.40 − ₹68.34 = ₹12.06
Step 3: Loss % = (Loss ÷ C.P.) × 100
⇒ Loss % = (12.06 ÷ 80.40) × 100
⇒ Loss % = 15%
Answer: Loss = 15%

iv. C.P. = ₹58.75, S.P. = ₹51.70

Step 1: S.P. < C.P. ⇒ There is a loss.
Step 2: Loss = C.P. − S.P. = ₹58.75 − ₹51.70 = ₹7.05
Step 3: Loss % = (Loss ÷ C.P.) × 100
⇒ Loss % = (7.05 ÷ 58.75) × 100
⇒ Loss % = 12%
Answer: Loss = 12%

Q2: Ranjit purchased an almirah fort ₹5248 and paid ₹127 for its transportation. He sold it for ₹6020. Find his gain or loss per cent.

Step 1: Total Cost Price = Cost of almirah + Transportation charges
= ₹5248 + ₹127 = ₹5375
Step 2: Selling Price (S.P.) = ₹6020
Step 3: Since S.P. > C.P., there is a gain.
Step 4: Gain = S.P. − C.P. = ₹6020 − ₹5375 = ₹645
Step 5: Gain % = (Gain ÷ C.P.) × 100
= (645 ÷ 5375) × 100
Step 6: Gain % = \(\frac{645 \times 100}{5375} = 12%\)
Answer: Gain = 12%

Q3: Ahmed purchased an old scooter for ₹14625 and spent ₹3225 on its repairs. Then, he sold it for ₹16422. Find his gain or loss per cent.

Step 1: Total Cost Price = Purchase Price + Repair Cost
= ₹14625 + ₹3225 = ₹17850
Step 2: Selling Price (S.P.) = ₹16422
Step 3: Since S.P. < C.P., there is a loss.
Step 4: Loss = C.P. − S.P. = ₹17850 − ₹16422 = ₹1428
Step 5: Loss % = (Loss ÷ C.P.) × 100
= (1428 ÷ 17850) × 100
Step 6: Loss % = \(\frac{1428 \times 100}{17850} = 8\%\)
Answer: Loss = 8%

Q4: A man buys two cricket bats, one for ₹1360 and the other for ₹1040. He sells the first bat at a gain of 15% and the second one at a loss of 15%. Find his gain or loss per cent in the whole transaction.

Step 1: Cost Price (C.P.) of first bat = ₹1360
Step 2: Gain on first bat = 15% of ₹1360 = \(\frac{15}{100} \times 1360 = ₹204\)
Step 3: Selling Price of first bat = ₹1360 + ₹204 = ₹1564
Step 4: Cost Price (C.P.) of second bat = ₹1040
Step 5: Loss on second bat = 15% of ₹1040 = \(\frac{15}{100} \times 1040 = ₹156\)
Step 6: Selling Price of second bat = ₹1040 − ₹156 = ₹884
Step 7: Total Cost Price = ₹1360 + ₹1040 = ₹2400
Step 8: Total Selling Price = ₹1564 + ₹884 = ₹2448
Step 9: Since Total S.P. > Total C.P., there is a gain.
Step 10: Total Gain = ₹2448 − ₹2400 = ₹48
Step 11: Gain % = (Gain ÷ Total C.P.) × 100
= (48 ÷ 2400) × 100 = 2%
Answer: Gain = 2%

Q5: Nandlal bought 20 dozen notebooks at ₹156 per dozen. He sold 8 dozens of them at 10% gain and the remaining 12 dozens at 20% gain. What is his gain per cent in the whole transaction?

Step 1: Cost Price per dozen = ₹156
Step 2: Total C.P. = 20 dozens × ₹156 = ₹3120
Step 3: First part: 8 dozens sold at 10% gain
Selling Price = C.P. + 10% of C.P. = ₹156 + 10% of ₹156 = ₹156 + ₹15.60 = ₹171.60 per dozen
Total S.P. for 8 dozens = ₹171.60 × 8 = ₹1372.80
Step 4: Remaining 12 dozens sold at 20% gain
Selling Price = ₹156 + 20% of ₹156 = ₹156 + ₹31.20 = ₹187.20 per dozen
Total S.P. for 12 dozens = ₹187.20 × 12 = ₹2246.40
Step 5: Total Selling Price = ₹1372.80 + ₹2246.40 = ₹3619.20
Step 6: Total Cost Price = ₹3120
Step 7: Gain = Total S.P. − Total C.P. = ₹3619.20 − ₹3120 = ₹499.20
Step 8: Gain % = (Gain ÷ Total C.P.) × 100
= (499.20 ÷ 3120) × 100
= \(\frac{499.20 \times 100}{3120} = 16\%\)
Answer: Gain = 16%

Q6: Heera bought 25 kg of rice at ₹48 per kg and 35 kg of rice at ₹60 per kg. He sold the mixture at ₹66 per kg. Find his gain per cent.

Step 1: Cost Price of 25 kg rice = 25 × ₹48 = ₹1200
Step 2: Cost Price of 35 kg rice = 35 × ₹60 = ₹2100
Step 3: Total Cost Price = ₹1200 + ₹2100 = ₹3300
Step 4: Total Quantity of Rice = 25 kg + 35 kg = 60 kg
Step 5: Selling Price per kg = ₹66
Total Selling Price = 60 × ₹66 = ₹3960
Step 6: Gain = S.P. − C.P. = ₹3960 − ₹3300 = ₹660
Step 7: Gain % = (Gain ÷ C.P.) × 100
= (660 ÷ 3300) × 100 = 20%
Answer: Gain = 20%

Q7: If the selling price of an article is \(\frac{4}{5}\)th of its cost price, find the loss percent.

Step 1: Let the Cost Price (C.P.) = ₹100 (Assume for simplicity)
Step 2: Given, Selling Price (S.P.) = \(\frac{4}{5}\) × C.P. = \(\frac{4}{5} \times 100 = ₹80\)
Step 3: Since S.P. < C.P., there is a loss.
Step 4: Loss = C.P. − S.P. = ₹100 − ₹80 = ₹20
Step 5: Loss % = (Loss ÷ C.P.) × 100
= (20 ÷ 100) × 100 = 20%
Answer: Loss = 20%

Q8: If the selling price of an article is \(1\frac{1}{3}\) of its cost price, find the gain per cent.

Step 1: Convert the mixed fraction to improper fraction:
\(1\frac{1}{3} = \frac{4}{3}\)
Step 2: Let the Cost Price (C.P.) = ₹100 (Assume for ease of calculation)
Step 3: Selling Price (S.P.) = \(\frac{4}{3} \times 100 = ₹133.33\)
Step 4: Since S.P. > C.P., there is a gain.
Step 5: Gain = S.P. − C.P. = ₹133.33 − ₹100 = ₹33.33
Step 6: Gain % = (Gain ÷ C.P.) × 100
= (33.33 ÷ 100) × 100 = 33.33%
Answer: Gain = 33.33%

Q9: A man sold a table for ₹2250 and gained one-ninth of its cost price. Find:

i. the cost price of the table

Step 1: Let the Cost Price (C.P.) be ₹x
Step 2: Given gain = \(\frac{1}{9}\) of C.P. = \(\frac{1}{9}x\)
Step 3: Selling Price = C.P. + Gain = \(x + \frac{1}{9}x = \frac{10x}{9}\)
Step 4: Given S.P. = ₹2250
So, \(\frac{10x}{9} = 2250\)
Step 5: Multiply both sides by 9:
10x = 2250 × 9 = 20250
Step 6: Divide both sides by 10:
x = \(\frac{20250}{10} = ₹2025\)
Answer: Cost Price = ₹2025

ii. the gain per cent earned by the man

Step 1: Gain = S.P. − C.P. = ₹2250 − ₹2025 = ₹225
Step 2: Gain % = (Gain ÷ C.P.) × 100
= (225 ÷ 2025) × 100
= \(\frac{1}{9} \times 100 = 11 \frac{1}{9}\%\)
Answer: Gain % = \(11 \frac{1}{9}\)%

Q10: By selling a pen for ₹195, a man loses one-sixteenth of what it costs him. Find:

i. the cost price of the pen

Step 1: Let the Cost Price (C.P.) be ₹x
Step 2: Given: Loss = \(\frac{1}{16}\) of C.P. = \(\frac{1}{16}x\)
Step 3: Selling Price = C.P. − Loss = \(x – \frac{1}{16}x = \frac{15x}{16}\)
Step 4: Given: S.P. = ₹195
So, \(\frac{15x}{16} = 195\)
Step 5: Multiply both sides by 16:
15x = 195 × 16 = 3120
Step 6: Divide both sides by 15:
x = \(\frac{3120}{15} = ₹208\)
Answer: Cost Price = ₹208

ii. loss per cent

Step 1: Loss = C.P. − S.P. = ₹208 − ₹195 = ₹13
Step 2: Loss % = (Loss ÷ C.P.) × 100
= (13 ÷ 208) × 100
Step 3: Loss % = \(\frac{13 \times 100}{208} = \frac{25}{4} = 6.25\%\)
Answer: Loss % = 6.25%

Q11: A cycle was sold at a gain of 10%. Had it been sold for ₹99 more, the gain would have been 12%. Find the cost price of the cycle.

Step 1: Let the Cost Price (C.P.) be ₹x
Step 2: Selling Price at 10% gain = \(x + \frac{10}{100}x = \frac{110x}{100} = \frac{11x}{10}\)
Step 3: Selling Price at 12% gain = \(x + \frac{12}{100}x = \frac{112x}{100} = \frac{28x}{25}\)
Step 4: Given: Difference between the two selling prices = ₹99
So, \(\frac{28x}{25} – \frac{11x}{10} = 99\)
Step 5: Take LCM of 25 and 10 = 50
\(\Rightarrow \frac{56x – 55x}{50} = \frac{x}{50} = 99\)
Step 6: Multiply both sides by 50:
x = 99 × 50 = ₹4950
Answer: Cost Price = ₹4950

Q12: A bucket was sold at a loss of 8%. Had it been sold for ₹56 more, there would have been a gain of 8%. What is the cost price of the bucket?

Step 1: Let the Cost Price (C.P.) be ₹x
Step 2: Selling Price at 8% loss = \(\frac{92}{100}x = \frac{23}{25}x\)
Step 3: Selling Price at 8% gain = \(\frac{108}{100}x = \frac{27}{25}x\)
Step 4: Difference between the two S.P.s = ₹56
So, \(\frac{27x}{25} – \frac{23x}{25} = 56\)
Step 5: \(\frac{4x}{25} = 56\)
Step 6: Multiply both sides by 25:
4x = 56 × 25 = 1400
Step 7: Divide both sides by 4:
x = \(\frac{1400}{4} = ₹350\)
Step 8: Mixed Calculation Check (Optional):
Loss S.P. = \(\frac{92}{100} \times 350 = ₹322\)
Gain S.P. = \(\frac{108}{100} \times 350 = ₹378\)
Difference = ₹378 − ₹322 = ₹56
Answer: Cost Price = ₹350

Q13: The selling price of 18 books is equal to the cost price of 21 books. Find the gain or loss per cent.

Step 1: Let the Cost Price of 1 book = ₹1 (for easy calculation)
Then, Cost Price of 21 books = ₹21
Step 2: Given: S.P. of 18 books = C.P. of 21 books = ₹21
So, Selling Price of 18 books = ₹21
⇒ Selling Price of 1 book = ₹21 ÷ 18 = ₹\(\frac{7}{6}\)
Step 3: Cost Price of 1 book = ₹1
Selling Price of 1 book = ₹\(\frac{7}{6}\) = ₹1.166…
Since S.P. > C.P., there is a gain.
Step 4: Gain = S.P. − C.P. = \(\frac{7}{6} – 1 = \frac{1}{6}\)
Step 5: Gain % = \(\frac{\text{Gain}}{\text{C.P.}} \times 100 = \frac{1}{6} \times 100 = 16\frac{2}{3}\%\)
Answer: Gain = \(16 \frac{2}{3}\)%

Q14: The cost price of 12 fans is equal to the selling price of 16 fans. Find the gain or loss per cent.

Step 1: Let the Selling Price of 1 fan = ₹1 (for easy calculation)
Then, Selling Price of 16 fans = ₹16
Step 2: Given: Cost Price of 12 fans = ₹16
So, Cost Price of 1 fan = ₹16 ÷ 12 = ₹\(\frac{4}{3} = ₹1.33\)
Step 3: Selling Price of 1 fan = ₹1
Cost Price of 1 fan = ₹\(\frac{4}{3}\)
Since S.P. < C.P., there is a loss.
Step 4: Loss = C.P. − S.P. = \(\frac{4}{3} – 1 = \frac{1}{3}\)
Step 5: Loss % = \(\frac{\text{Loss}}{\text{C.P.}} \times 100 = \frac{1}{3} \div \frac{4}{3} \times 100 = \frac{1}{4} \times 100 = 25\%\)
Answer: Loss = 25%

Q15: On selling 250 cassettes, a man had a gain equal to the selling price of 25 cassettes. Find his gain per cent.

Step 1: Let the Selling Price of 1 cassette = ₹1 (assumption for simplicity)
Step 2: Selling Price of 250 cassettes = 250 × ₹1 = ₹250
Step 3: Gain = S.P. of 25 cassettes = 25 × ₹1 = ₹25
Step 4: Therefore, Cost Price = S.P. − Gain = ₹250 − ₹25 = ₹225
Step 5: Gain % = \(\frac{\text{Gain}}{\text{C.P.}} \times 100 = \frac{25}{225} \times 100\)
= \(\frac{1}{9} \times 100 = 11\frac{1}{9}\%\)
Answer: Gain = \(11 \frac{1}{9}\)%

Q16: On selling 36 oranges, a vendor loses the selling price of 4 oranges. Find his loss per cent.

Step 1: Let the Selling Price of 1 orange = ₹1 (assumption)
Step 2: Selling Price of 36 oranges = ₹36
Step 3: Loss = S.P. of 4 oranges = 4 × ₹1 = ₹4
Step 4: Therefore, Cost Price = S.P. + Loss = ₹36 + ₹4 = ₹40
Step 5: Loss % = \(\frac{\text{Loss}}{\text{C.P.}} \times 100 = \frac{4}{40} \times 100 = 10\%\)
Answer: Loss = 10%

Q17: Toffees are bought at 2 for a rupee and sold at 5 for ₹3. Find the gain or loss per cent.

Step 1: Cost Price of 2 toffees = ₹1 ⇒ Cost Price of 1 toffee = ₹\(\frac{1}{2}\)
Step 2: Selling Price of 5 toffees = ₹3 ⇒ Selling Price of 1 toffee = ₹\(\frac{3}{5}\)
Step 3: Compare C.P. and S.P. of 1 toffee:
C.P. = ₹\(\frac{1}{2} = ₹0.50\), S.P. = ₹\(\frac{3}{5} = ₹0.60\)
Step 4: Since S.P. > C.P., there is a gain.
Gain = ₹0.60 − ₹0.50 = ₹0.10
Step 5: Gain % = \(\frac{\text{Gain}}{\text{C.P.}} \times 100 = \frac{0.10}{0.50} \times 100 = 20\%\)
Answer: Gain = 20%

Q18: Coffee costing ₹450 per kg was mixed with Chicory costing ₹225 per kg in the ratio 5 : 2 for a certain blend. If the mixture was sold at 405 per kg; find the gain or loss per cent.

Step 1: Total Cost Price of mixture = \((₹450 \times 5) + (₹225 \times 2)\) = ₹ 2700
Step 2: Total Selling Price of mixture = \(₹405 \times 7\) = ₹ 2835
Step 3: As S.P. is greater than C.P., so there is a gain.
Step 4: Gain = S.P. – C.P. = 2835 – 2700 = ₹135
Step 5: Gain % = \(\frac{135}{2700} \times 100 = 5\%\)
Answer: Gain = 5%