Profit, Loss and Discount

Q1: Multiple Choice Type:

i. A box is sold for ₹18 at a loss of 10%. When sold at a profit of 15%; its S.P. will be:

Step 1: Let the Cost Price (C.P.) = ₹\(x\). Loss = 10%, so
Selling Price (S.P.) = C.P. − 10% of C.P. = \(x – \frac{10}{100}x = \frac{90}{100}x\)
Given S.P. = ₹18, so
\(\frac{90}{100}x = 18 \Rightarrow x = \frac{18 \times 100}{90} = 20\)
Step 2: Now, profit = 15%, so new S.P. = C.P. + 15% of C.P.
= \(20 + \frac{15}{100} \times 20 = 20 + 3 = ₹23\)
Answer: d. ₹23

ii. C.P. of 20 articles is equal to S.P. of 16 articles. The profit or loss as percent is:

Step 1: Let C.P. per article = ₹\(x\), S.P. per article = ₹\(y\).
Given, total C.P. for 20 articles = total S.P. for 16 articles
\(\Rightarrow 20x = 16y \Rightarrow y = \frac{20}{16}x = 1.25x\)
Step 2: Since S.P. per article \(>\) C.P. per article, there is profit.
Profit per article = \(y – x = 1.25x – x = 0.25x\)
Profit % = \(\frac{0.25x}{x} \times 100 = 25\%\)
Answer: a. 25% profit

iii. Some goods are sold at a discount of 20%. If the same goods are sold without discount, their price will change by:

Step 1: Let the marked price = ₹100.
Selling price with 20% discount = \(100 – 20 = ₹80\)
Step 2: Selling without discount means price changes from ₹80 to ₹100.
Percentage change = \(\frac{100 – 80}{80} \times 100 = 25\%\) increase.
Answer: a. 25% increase

iv. The marked price of an article is ₹400. If tax on it increases from 10% to 15%, the amount of it will increase by:

Step 1: Tax at 10% = \(\frac{10}{100} \times 400 = ₹40\)
Tax at 15% = \(\frac{15}{100} \times 400 = ₹60\)
Step 2: Increase in tax amount = ₹60 − ₹40 = ₹20
Answer: d. None of these

v. If the rate of GST on an inter-state sale is 18%, the total amount for a service of ₹200 is:

Step 1: GST = \(\frac{18}{100} \times 200 = ₹36\)
Step 2: Total amount = ₹200 + ₹36 = ₹236
Answer: c. ₹236

vi. Statement 1: In case of profit (i.e. if S.P. > C.P.), \(C.P. = S.P. \times \left(\frac{100 \times Profit\%}{100}\right)\).
Statement 2: In case of loss (i.e. if C.P. > S.P.), \(S.P. = C.P. \times \left(\frac{100 \times C.P}{100 – Loss\%}\right)\).
Which of the following options is correct?

Step 1: Both statements are incorrect as their formulas are not right.
Correct formulas:
– For profit: \(S.P. = C.P. \times \left(1 + \frac{Profit\%}{100}\right)\)
– For loss: \(S.P. = C.P. \times \left(1 – \frac{Loss\%}{100}\right)\)
Answer: b. Both the statements are false.

vii. Assertion (A): Two successive discounts of 10% and 5% are equal to a single discount of \(14\frac{1}{2}\)%.
Reason (R): Rate of discount = \(\frac{Discount}{S.P.} \times 100\%\).

Step 1: Successive discount = \(1 – (1 – \frac{10}{100})(1 – \frac{5}{100})\)
\(= 1 – (0.90 \times 0.95) = 1 – 0.855 = 0.145 = 14.5\%\)
Step 2: Reason: Generally discount % is based on marked price, not selling price.
Answer: c. A is true, but R is false.

viii. Assertion (A): If selling price of an article is ₹400 gaining \(\frac{1}{4}\) of its C.P., then gain % = 25%.
Reason (R): Loss = \(\frac{C.P. \times Loss\%}{100}\).

Step 1: Understanding Assertion (A)
Selling Price (S.P.) = ₹400
Gain = \(\frac{1}{4}\) of C.P.
Let Cost Price = ₹x
Then, Gain = \(\frac{1}{4}x\)
So, Selling Price = C.P. + Gain \[ 400 = x + \frac{1}{4}x = \frac{5x}{4} \\ \Rightarrow x = \frac{400 \times 4}{5} = ₹320 \]Gain = ₹400 – ₹320 = ₹80
Gain % = \(\frac{Gain}{C.P.} \times 100 = \frac{80}{320} \times 100 = 25\%\)
Assertion (A) is True.
Step 2: Understanding Reason (R)
Reason states: \[ \text{Loss} = \frac{C.P. \times \text{Loss\%}}{100} \] This is a standard formula and is always true.
Reason (R) is True.
Step 3: Is Reason the Correct Explanation of Assertion?
The assertion talks about gain, not loss. Although Reason is true, it is not the correct explanation of the Assertion.
Answer: b. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).

ix. Assertion (A): If S.P. is ₹1200 and sales tax is 20% then amount of the bill = ₹1440.
Reason (R): S.P. is the taxable amount, hence amount of the bill = \(S.P. \times \left(1 + \frac{Rate\ of\ sales\ tax}{100}\right)\).

Step 1: Amount of bill = \(1200 \times \left(1 + \frac{20}{100}\right) = 1200 \times 1.2 = ₹1440\), which is correct.
Step 2: Reason correctly states that S.P. is taxable amount and formula is accurate.
Answer: a. Both A and R are correct, and R is the correct explanation for A.

x. Assertion (A): 12 articles are bought for one rupee and 8 of them are sold for one rupee. Then gain% = 50%.
Reason (R): Profit% = \(\frac{Profit}{C.P.} \times 100\%\) and Loss% = \(\frac{Loss}{C.P.} \times 100\%\).

Step 1: Analyzing the Assertion:
Cost Price (C.P.) of 12 articles = ₹1
⇒ C.P. of 1 article = ₹1 ÷ 12 = ₹1/12
Selling Price (S.P.) of 8 articles = ₹1
⇒ S.P. of 1 article = ₹1 ÷ 8 = ₹1/8
Profit on 1 article = S.P. – C.P.
= ₹1/8 − ₹1/12
Find LCM of 8 and 12 = 24
So, ₹1/8 = ₹3/24 and ₹1/12 = ₹2/24
⇒ Profit = ₹(3/24 − 2/24) = ₹1/24
Profit% = (Profit ÷ C.P.) × 100%
= (1/24 ÷ 1/12) × 100%
= (1/24 × 12/1) × 100%
= (12/24) × 100% = 0.5 × 100% = 50%
Assertion is TRUE.
Step 2:Analyzing the Reason:
The formula for Profit% is indeed:
Profit% = (Profit ÷ C.P.) × 100%
and similarly, Loss% = (Loss ÷ C.P.) × 100%
Reason is also TRUE.
Step 3: Reason: Both the formulas of Profit percentage and Loss percentatge are correct but both doesnot explain the Asserion.Answer: b. Both A and R are correct, and R is not the correct explanation for A.

Q2: A man sold his bicycle for ₹405 losing one-tenth of its cost price. Find:

i. its cost price;

Step 1: Let the Cost Price (C.P.) be ₹\(x\).
Step 2: Loss = one-tenth of C.P. = \(\frac{1}{10} \times x = \frac{x}{10}\)
Step 3: Selling Price (S.P.) = C.P. − Loss = \(x – \frac{x}{10} = \frac{9x}{10}\)
Step 4: Given S.P. = ₹405, so \[ \frac{9x}{10} = 405 \\ x = \frac{405 \times 10}{9} = 450 \]Answer: Cost Price = ₹450

ii. the loss percent.

Step 5: Loss = C.P. − S.P. = ₹450 − ₹405 = ₹45
Step 6: Loss % = \(\frac{Loss}{C.P.} \times 100 = \frac{45}{450} \times 100 = 10\%\)
Answer: Loss percent = 10%

Q3: A man sold a radio set for ₹250 and gained one-ninth of its cost price. Find:

i. its cost price;

Step 1: Let the Cost Price (C.P.) be ₹\(x\).
Step 2: Profit = \(\frac{1}{9} \times x = \frac{x}{9}\)
Step 3: Selling Price (S.P.) = C.P. + Profit = \(x + \frac{x}{9} = \frac{10x}{9}\)
Step 4: Given S.P. = ₹250, so \[ \frac{10x}{9} = 250 \\ x = \frac{250 \times 9}{10} = 225 \]Answer: Cost Price = ₹225

ii. the profit percent.

Step 5: Profit = S.P. − C.P. = ₹250 − ₹225 = ₹25
Step 6: Profit % = \(\frac{Profit}{C.P.} \times 100 = \frac{25}{225} \times 100 = 11.\overline{11}\%\)
Answer: Profit percent = \(11\frac{1}{9}\%\)

Q4: Mr. Sinha sold two tape recorders for ₹990 each, gaining 10% on one and losing 10% on the other. Find his total loss or gain, as percent, on the whole transaction.

Step 1: Let the cost price of the first tape recorder be ₹\(x\).
Step 2: Since Mr. Sinha gains 10% on the first tape recorder,
Selling Price (S.P.) = ₹990 = C.P. + 10% of C.P. \[ 990 = x + \frac{10}{100}x = \frac{110}{100} x = \frac{11}{10} x \\ \Rightarrow x = \frac{990 \times 10}{11} = 900 \]Step 3: Cost price of first tape recorder = ₹900
Step 4: Let the cost price of the second tape recorder be ₹\(y\).
Step 5: Mr. Sinha loses 10% on the second tape recorder,
Selling Price = ₹990 = C.P. − 10% of C.P. \[ 990 = y – \frac{10}{100} y = \frac{90}{100} y = \frac{9}{10} y \\ \Rightarrow y = \frac{990 \times 10}{9} = 1100 \]Step 6: Cost price of second tape recorder = ₹1100
Step 7: Total Cost Price = \(900 + 1100 = ₹2000\)
Total Selling Price = \(990 + 990 = ₹1980\)
Step 8: Net loss = Total C.P. − Total S.P. = ₹2000 − ₹1980 = ₹20
Step 9: Loss % on whole transaction = \(\frac{20}{2000} \times 100 = 1\%\)
Answer: Total loss on the whole transaction = 1%

Q5: A tape recorder is sold for ₹2,760 at a gain of 15% and a C.D. player is sold for ₹3,240 at a loss of 10%. Find:

i. the C.P.of the tape recorder.

Step 1: Let the C.P. of the tape recorder be ₹\(x\).
Step 2: Given gain = 15%, so Selling Price (S.P.) = C.P. + 15% of C.P. \[ S.P. = x + \frac{15}{100}x = \frac{115}{100}x = \frac{23}{20} x \]Step 3: Given S.P. = ₹2760, so \[ \frac{23}{20} x = 2760 \\ x = \frac{2760 \times 20}{23} = 2400 \]Answer: C.P. of tape recorder = ₹2400

ii. the C.P. of the C.D. player.

Step 4: Let the C.P. of the C.D. player be ₹\(y\).
Step 5: Given loss = 10%, so S.P. = C.P. − 10% of C.P. \[ S.P. = y – \frac{10}{100} y = \frac{90}{100} y = \frac{9}{10} y \]Step 6: Given S.P. = ₹3240, so \[ \frac{9}{10} y = 3240 \\ y = \frac{3240 \times 10}{9} = 3600 \]Answer: C.P. of C.D. player = ₹3600

iii. the total C.P. of both.

Step 7: Total C.P. = ₹2400 + ₹3600 = ₹6000
Answer: Total Cost Price = ₹6000

iv. the total S.P. of both.

Step 8: Total S.P. = ₹2760 + ₹3240 = ₹6000
Answer: Total Selling Price = ₹6000

v. the gain % or the loss % on the whole.

Step 9: Total C.P. = ₹6000 and Total S.P. = ₹6000
Step 10: Since Total S.P. = Total C.P., there is no profit or loss.
Answer: No gain, no loss on the whole transaction

Q6: John sold an article to Peter at 20% profit and Peter sold it to Mohan at 5% loss. If Mohan paid ₹912 for the article, find how much did John pay for it?

Step 1: Let the cost price of John be ₹\(x\).
Step 2: John sold the article to Peter at 20% profit,
So, Selling Price for John = Cost Price for Peter = \(x + 20\%\ of\ x = \frac{120}{100} x = \frac{6}{5} x\)
Step 3: Peter sold the article to Mohan at 5% loss,
Selling Price for Peter = Cost Price for Mohan = ₹912 = Peter’s C.P. − 5% of Peter’s C.P. \[ 912 = \text{Peter’s C.P.} \times \left(1 – \frac{5}{100}\right) = \text{Peter’s C.P.} \times \frac{95}{100} \\ \Rightarrow \text{Peter’s C.P.} = \frac{912 \times 100}{95} = 960 \]Step 4: But Peter’s C.P. = John’s S.P. = \(\frac{6}{5} x\) \[ \frac{6}{5} x = 960 \\ x = \frac{960 \times 5}{6} = 800 \]Answer: John paid ₹800 for the article

Q7: By selling an article for ₹1,200, Rohit loses one-fifth of its cost price. For how much should he sell it in order to gain 30%?

Step 1: Let the cost price (C.P.) of the article be ₹\(x\).
Step 2: Rohit loses one-fifth of the cost price by selling at ₹1,200.
Loss = \(\frac{1}{5} \times x = \frac{x}{5}\)
Therefore, Selling Price (S.P.) = C.P. − Loss = \(x – \frac{x}{5} = \frac{4}{5} x\)
Step 3: Given S.P. = ₹1,200, \[ \frac{4}{5} x = 1200 \\ x = \frac{1200 \times 5}{4} = 1500 \]Step 4: To gain 30%, Selling Price should be: \[ S.P. = C.P. + 30\% \text{ of } C.P. = x + \frac{30}{100} x = \frac{130}{100} x = \frac{13}{10} \times 1500 = 1950 \]Answer: Rohit should sell the article for ₹1,950 to gain 30%

Q8: By selling an article for ₹1,200; Rohit gains one-fifth of its cost price. What should be the selling price of the article when he sells it at 30% gain?

Step 1: Let the cost price (C.P.) of the article be ₹\(x\).
Step 2: Gain = one-fifth of cost price = \(\frac{x}{5}\)
So, Selling Price = C.P. + Gain = \(x + \frac{x}{5} = \frac{6x}{5}\)
Step 3: Given that selling price = ₹1,200, \[ \frac{6x}{5} = 1200 \\ x = \frac{1200 \times 5}{6} = 1000 \]Step 4: New selling price at 30% gain = \(x + \frac{30}{100}x = \frac{130}{100}x\) \[ S.P. = \frac{130}{100} \times 1000 = ₹1300 \]Answer: Rohit should sell the article for ₹1,300 to gain 30%

Q9: 25% of the cost price of an article is ₹600. Find its selling price when it is sold at a profit of 25%.

Step 1: Let the cost price (C.P.) of the article be ₹\(x\).
Step 2: Given: 25% of the cost price is ₹600. \[ \frac{25}{100} \times x = 600 \\ x = \frac{600 \times 100}{25} = 2400 \]Step 3: Profit = 25% of C.P. \[ \text{Selling Price (S.P.)} = C.P. + 25\% \text{ of C.P.} = \frac{125}{100} \times 2400 = ₹3000 \]Answer: The selling price of the article is ₹3,000

Q10: A man sold a bicycle at 5% profit. If the cost had been 30% less and the selling price ₹63 less, he would have made a profit of 30%. What was the cost price of the bicycle?

Step 1: Let the original cost price (C.P.) be ₹\(x\)
Step 2: Profit = 5%, so original selling price (S.P.) = ₹\(\frac{105}{100}x = \frac{21}{20}x\)
Step 3: New cost price = 30% less of original cost price \[ = x – \frac{30}{100}x = \frac{70}{100}x = \frac{7}{10}x \]Step 4: New selling price = original S.P. − ₹63 = \(\frac{21}{20}x – 63\)
According to the question, new profit = 30%, so: \[ \text{New S.P.} = \text{New C.P.} + 30\% \text{ of New C.P.} = \frac{130}{100} \times \frac{7}{10}x = \frac{91}{100}x \]So, \[ \frac{21}{20}x – 63 = \frac{91}{100}x \]Step 5: Solve the equation:
Take LHS and RHS: \[ \frac{21}{20}x – \frac{91}{100}x = 63 \] Take LCM 100: \[ \left( \frac{105x – 91x}{100} \right) = 63 \Rightarrow \frac{14x}{100} = 63 \\ x = \frac{63 \times 100}{14} = 450 \]Answer: The cost price of the bicycle is ₹450

Q11: Renu sold an article at a loss of 8 percent. Had she bought it at 10% less and sold for ₹36 more, she would have gained 20%. Find the cost price of the article.

Step 1: Let the original cost price (C.P.) of the article be ₹\(x\).
Step 2: She sold at 8% loss, so selling price (S.P.) = \[ S.P. = x – \frac{8}{100}x = \frac{92}{100}x = \frac{23}{25}x \]Step 3: New cost price if bought at 10% less = \[ x – \frac{10}{100}x = \frac{90}{100}x = \frac{9}{10}x \]Step 4: New selling price = Original S.P. + ₹36 = \[ \frac{23}{25}x + 36 \]Given: At this new price, she makes a gain of 20% on the new cost price, so: \[ \text{New S.P.} = \text{New C.P.} + 20\% \text{ of New C.P.} = \frac{120}{100} \times \frac{9}{10}x = \frac{108}{100}x = \frac{27}{25}x \]Step 5: Equating the two expressions for new S.P.: \[ \frac{23}{25}x + 36 = \frac{27}{25}x \]Subtract \(\frac{23}{25}x\) from both sides: \[ 36 = \frac{27x – 23x}{25} = \frac{4x}{25} \]Now solve for \(x\): \[ x = \frac{36 \times 25}{4} = \frac{900}{4} = 225 \]Answer: The cost price of the article is ₹225

Q12: The cost price of an article is 25% below the marked price. If the article is available at 15% discount and its cost price is ₹2,400, find:

i. its marked price

Step 1: Let the marked price be ₹\(x\).
Given: Cost price is 25% below marked price. \[ \text{C.P.} = x – \frac{25}{100}x = \frac{75}{100}x = \frac{3}{4}x \]Step 2: It is given that the cost price is ₹2,400 \[ \frac{3}{4}x = 2400 \Rightarrow x = \frac{2400 \times 4}{3} = 3200 \]Answer: ₹3,200 

ii. its selling price

Step 3: Discount = 15%, so selling price = \[ S.P. = x – \frac{15}{100}x = \frac{85}{100}x = \frac{85}{100} \times 3200 = ₹2720 \]Answer: ₹2,720

iii. the profit percent.

Step 4: Cost Price = ₹2,400 and Selling Price = ₹2,720
Profit = \(2720 – 2400 = ₹320\) \[ \text{Profit \%} = \frac{320}{2400} \times 100 = \frac{4}{3} \times 10 = \frac{40}{3} = 13 \frac{1}{3}\% \] Answer: \(13\frac{1}{3}\)% profit

Q13: Find a single discount (as percent) equivalent to following successive discounts:

i. 20% and 12%

Step 1: Use the successive discount formula: \[ D = A + B – \frac{AB}{100} \] Where A = 20%, B = 12% \[ D = 20 + 12 – \frac{20 \times 12}{100} = 32 – 2.4 = 29.6\% \]Answer: 29.6%

ii. 10%, 20%, and 20%

Step 1: Apply first two: A = 10%, B = 20% \[ D_1 = 10 + 20 – \frac{10 \times 20}{100} = 30 – 2 = 28\% \]Step 2: Now apply third discount (20%) on D₁ = 28% \[ D = 28 + 20 – \frac{28 \times 20}{100} = 48 – 5.6 = 42.4\% \]Answer: 42.4%

iii. 20%, 10%, and 5%

Step 1: Apply first two: A = 20%, B = 10% \[ D_1 = 20 + 10 – \frac{20 \times 10}{100} = 30 – 2 = 28\% \]Step 2: Apply third discount (5%) on D₁ = 28% \[ D = 28 + 5 – \frac{28 \times 5}{100} = 33 – 1.4 = 31.6\% \]Answer: 31.6%

Q14: When the rate of Tax is decreased from 9% to 6% for a coloured T.V.; Mrs Geeta will save ₹780 in buying this T.V. Find the list price of the T.V.

Step 1: Let the list price of the T.V. be ₹x.
Step 2: When the tax rate is 9%, tax = \(\frac{9}{100} \times x = \frac{9x}{100}\)
When the tax rate is 6%, tax = \(\frac{6}{100} \times x = \frac{6x}{100}\)
Step 3: The difference in tax is the saving: \[ \frac{9x}{100} – \frac{6x}{100} = \frac{3x}{100} \]Step 4: It is given that this difference equals ₹780: \[ \frac{3x}{100} = 780 \\ \Rightarrow x = \frac{780 \times 100}{3} = \frac{78000}{3} = 26,000 \]Answer: The list price of the T.V. is ₹26,000.

Q15: A shopkeeper sells an article for ₹21,384 including 10% tax. However, the actual rate of Tax is 8%. Find the extra profit made by the dealer.

Step 1: Let the selling price **excluding tax** be ₹x.
Tax at 10% means: \[ \text{Total bill} = x + 10\%\text{ of }x = x + \frac{10x}{100} = \frac{110x}{100} \]Step 2: Total bill is given as ₹21,384: \[ \frac{110x}{100} = 21384 \Rightarrow x = \frac{21384 \times 100}{110} = 19440 \] So, the **price charged before tax** = ₹19,440
Step 3: Actual tax rate is 8%, so actual tax payable by shopkeeper = \[ \frac{8}{100} \times 19440 = 1555.20 \]Step 4: The dealer collected tax at 10%: \[ \frac{10}{100} \times 19440 = 1944 \]Step 5: Extra tax (extra profit) made by the dealer = \[ 1944 – 1555.20 = ₹388.80 \]Answer: The extra profit made by the dealer is ₹388.80.

Q16: An article is purchased for ₹1,792 which includes a discount of 30% and 28% GST. Find the marked price of the article.

Step 1: Let the marked price be ₹x.
Discount = 30%, so selling price before tax = \[ x – 30\% \text{ of } x = x – \frac{30x}{100} = \frac{70x}{100} \]Step 2: GST is 28%, so final price = \[ \text{Selling price} + 28\% \text{ of selling price} = \frac{70x}{100} + \frac{28}{100} \times \frac{70x}{100} = \frac{70x}{100} \times \left(1 + \frac{28}{100}\right) = \frac{70x}{100} \times \frac{128}{100} \]Step 3: Final price is given as ₹1792: \[ \frac{70x}{100} \times \frac{128}{100} = 1792 \\ \Rightarrow \frac{8960x}{10000} = 1792 \\ \Rightarrow x = \frac{1792 \times 10000}{8960} = \frac{17920000}{8960} = 2000 \]Answer: The marked price of the article is ₹2,000.