Profit and Loss Questions in English

(C) Let the marked price be $₹ x$ and the cost price be $₹ y$.
The discount given is half of the marked price,so the discount $= \frac{x}{2} = 50 \% \text{ of } x$.
The selling price $(SP)$ is the marked price minus the discount,so $SP = x - 0.5x = 0.5x = 50 \% \text{ of } x$.
The shopkeeper incurs a $10 \%$ loss,which means the selling price is $90 \% \text{ of the cost price } (y)$.
Therefore,$50 \% \text{ of } x = 90 \% \text{ of } y$.
$\frac{50}{100} x = \frac{90}{100} y$
$0.5x = 0.9y$
$y = \frac{0.5}{0.9} x = \frac{5}{9} x$.
Thus,the cost price is $\frac{5}{9}$ of the marked price.

(B) The selling price of the saree is $Rs. 7710$ and the discount amount is $Rs. 1285$.
Marked Price = Selling Price + Discount
Marked Price = $7710 + 1285 = Rs. 8995$.
Discount percentage is calculated on the Marked Price:
Discount Percentage = $(\frac{\text{Discount}}{\text{Marked Price}}) \times 100$
Discount Percentage = $(\frac{1285}{8995}) \times 100$
Simplifying the fraction:
$\frac{1285}{8995} = \frac{1}{7}$
Therefore,Discount Percentage = $\frac{1}{7} \times 100 = \frac{100}{7} = 14 \frac{2}{7} \%$.

(A) Let the cost price $(C.P.)$ of the cycle be $₹ x$.
The marked price $(M.P.)$ is $₹ 840$.
The dealer offers a discount of $10 \%$,so the selling price $(S.P.)$ is:
$S.P. = M.P. \times (1 - \text{Discount} \% / 100) = 840 \times (1 - 10/100) = 840 \times 0.9 = ₹ 756$.
The dealer makes a profit of $26 \%$,so the selling price is also given by:
$S.P. = C.P. \times (1 + \text{Profit} \% / 100) = x \times (1 + 26/100) = 1.26x$.
Equating the two expressions for $S.P.$:
$1.26x = 756$
$x = 756 / 1.26$
$x = 600$.
Therefore,the dealer pays $₹ 600$ for the cycle.

(D) Let the marked price of the article be $₹ x$.
According to the question:
Cost Price $(C.P.)$ $= \frac{2x}{5}$.
Since the discount is $10 \%$,the Selling Price $(S.P.)$ is:
$S.P. = x - (10 \% \text{ of } x) = x - 0.1x = 0.9x = ₹ \frac{9x}{10}$.
Gain $= S.P. - C.P. = \frac{9x}{10} - \frac{2x}{5} = \frac{9x - 4x}{10} = \frac{5x}{10} = \frac{x}{2}$.
Gain percent $= \left( \frac{\text{Gain}}{C.P.} \times 100 \right) \%$.
Gain percent $= \left( \frac{x/2}{2x/5} \times 100 \right) \% = \left( \frac{x}{2} \times \frac{5}{2x} \times 100 \right) \% = \left( \frac{5}{4} \times 100 \right) \% = 125 \%$.
Therefore,there will be a $125 \%$ profit.

(B) The cost price of the item is $Rs. 200$.
First,the item is sold at a $10 \%$ loss.
The first selling price $(S.P._1)$ is calculated as:
$S.P._1 = 200 \times (1 - \frac{10}{100}) = 200 \times 0.90 = Rs. 180$.
Next,the price is further reduced by $5 \%$.
The new selling price $(S.P._2)$ is calculated on the previous selling price $(S.P._1)$:
$S.P._2 = 180 \times (1 - \frac{5}{100}) = 180 \times 0.95 = Rs. 171$.
Therefore,the final selling price is $Rs. 171$.

(C) Total number of items bought $= 144$.
Cost price $(C.P.)$ per item $= 90$ paisa $= ₹ 0.90$.
Total $C.P. = 144 \times 0.90 = ₹ 129.60$.
Number of items broken $= 20$.
Remaining items $= 144 - 20 = 124$.
Selling price $(S.P.)$ per item $= ₹ 1.20$.
Total $S.P. = 124 \times 1.20 = ₹ 148.80$.
Gain $= S.P. - C.P. = 148.80 - 129.60 = ₹ 19.20$.
Gain percent $= (\text{Gain} / C.P.) \times 100 = (19.20 / 129.60) \times 100$.
Gain percent $= (1920 / 1296) \% \approx 14.8148 \%$.
Rounding to one decimal place,the gain percent is $14.8 \%$.

(A) Let the cost price $(C.P.)$ of the article be $₹ 100$.
Since the profit is $20 \%$ on the cost price, the profit amount is $₹ 20$.
Therefore, the selling price $(S.P.)$ $= C.P. + \text{Profit} = ₹ 100 + ₹ 20 = ₹ 120$.
Now, the profit percentage calculated on the selling price is given by the formula:
$\text{Profit percentage on } S.P. = \left( \frac{\text{Profit}}{S.P.} \right) \times 100$
$= \left( \frac{20}{120} \right) \times 100$
$= \left( \frac{1}{6} \right) \times 100 = \frac{100}{6} = 16 \frac{2}{3} \%$.

(A) First,calculate the Cost Price $(C.P.)$ of the article.
$C.P. = \frac{100}{100 - \text{loss } \%} \times S.P.$
$C.P. = \frac{100}{100 - 15} \times 102 = \frac{100}{85} \times 102 = 1.17647 \times 102 = Rs. 120$.
Now,the article is sold at $S.P. = Rs. 134.40$.
Since $S.P. > C.P.$,there is a gain.
Gain $= S.P. - C.P. = 134.40 - 120 = Rs. 14.40$.
Gain percent $= \frac{\text{Gain}}{C.P.} \times 100 = \frac{14.40}{120} \times 100 = 12 \%$.

(A) Let the cost price $(C.P.)$ of the first toy be $x$ and the second toy be $y$.
For the first toy: $C.P. \times (1 + 12/100) = 504$
$x \times (112/100) = 504$
$x = (504 \times 100) / 112 = 450$
For the second toy: $C.P. \times (1 - 4/100) = 504$
$y \times (96/100) = 504$
$y = (504 \times 100) / 96 = 525$
Total $C.P. = 450 + 525 = 975$
Total $S.P. = 504 + 504 = 1008$
Since $S.P. > C.P.$,there is a profit.
Profit $= 1008 - 975 = 33$
Profit $\% = (33 / 975) \times 100 = (3300 / 975) = 44 / 13 = 3 \frac{5}{13} \%$

(D) For $A$,the cost price $(C.P.)$ of the horse is calculated as follows:
$C.P. = \frac{4800 \times 100}{100 - 20} = \frac{4800 \times 100}{80} = 6000$.
So,$A$ bought the horse for $Rs. 6000$.
For $B$,the selling price $(S.P.)$ to $C$ is the price that would have given $A$ a profit of $15\%$.
$S.P. = 6000 + (15\% \text{ of } 6000) = 6000 + 900 = 6900$.
$B$ bought the horse from $A$ for $Rs. 4800$ and sold it to $C$ for $Rs. 6900$.
$B$'s gain = $S.P. - C.P. = 6900 - 4800 = 2100$.
Therefore,$B$'s gain is $Rs. 2100$.

(D) Let the original price of the item be $₹ x$ per $Kg$.
The reduction in price is $21\%$,so the new price is $x - 0.21x = 0.79x$ per $Kg$.
According to the problem,the person can buy $3\, Kg$ more for $₹ 100$ at the new price compared to the original price:
$\frac{100}{0.79x} - \frac{100}{x} = 3$
Multiply by $0.79x$ to clear the denominator:
$100 - 0.79(100) = 3(0.79x)$
$100 - 79 = 2.37x$
$21 = 2.37x$
$x = \frac{21}{2.37} \approx 8.8607$
The reduced price is $0.79x = 0.79 \times \frac{21}{2.37} = \frac{0.79 \times 21}{2.37} = \frac{16.59}{2.37} = ₹ 7$ per $Kg$.

(A) Let the total number of cakes be $100$.
Let the cost price $(CP)$ of each cake be $₹ 100$.
Total $CP = 100 \times 100 = ₹ 10000$.
Since the expected profit is $40 \%$, the marked price $(MP)$ of each cake $= 100 + 40 = ₹ 140$.
Cakes damaged completely ($16 \%$ of $100$) $= 16$ cakes. Selling price $(SP)$ $= ₹ 0$.
Cakes slightly damaged ($24 \%$ of $100$) $= 24$ cakes. $SP$ per cake $= 80 \%$ of $CP = 0.80 \times 100 = ₹ 80$.
Total $SP$ for slightly damaged cakes $= 24 \times 80 = ₹ 1920$.
Remaining cakes ($60 \%$ of $100$) $= 60$ cakes. $SP$ per cake $= MP = ₹ 140$.
Total $SP$ for remaining cakes $= 60 \times 140 = ₹ 8400$.
Total $SP = 0 + 1920 + 8400 = ₹ 10320$.
Total Profit $= SP - CP = 10320 - 10000 = ₹ 320$.
Profit percentage $= (\text{Profit} / \text{Total } CP) \times 100 = (320 / 10000) \times 100 = 3.2 \%$.

(A) List price $= 440$
Selling price $(S.P.) = 396$
Discount $= \text{List price} - S.P. = 440 - 396 = 44$
Discount percentage $= (\frac{\text{Discount}}{\text{List price}}) \times 100$
Discount percentage $= (\frac{44}{440}) \times 100 = 0.1 \times 100 = 10 \%$

(C) Let the marked price be $M.P.$
The first discount is $12.5\% = \frac{1}{8}$.
The price after the first discount is $M.P. \times (1 - \frac{1}{8}) = M.P. \times \frac{7}{8}$.
The second discount is $10\% = \frac{1}{10}$.
The price after the second discount is $(M.P. \times \frac{7}{8}) \times (1 - \frac{1}{10}) = M.P. \times \frac{7}{8} \times \frac{9}{10}$.
Given that the final selling price is $Rs. 6,300$,we have:
$M.P. \times \frac{7}{8} \times \frac{9}{10} = 6300$
$M.P. \times \frac{63}{80} = 6300$
$M.P. = 6300 \times \frac{80}{63}$
$M.P. = 100 \times 80 = 8000$
Therefore,the marked price is $Rs. 8000$.

(A) Let the number of oranges bought of each type be $60$ (the $LCM$ of $2$ and $3$ is $6$,so $60$ is a convenient multiple).
Cost Price $(C.P.)$ of the first type of $60$ oranges: Since $2$ oranges cost $1$ rupee,$60$ oranges cost $60 / 2 = 30$ rupees.
Cost Price $(C.P.)$ of the second type of $60$ oranges: Since $3$ oranges cost $1$ rupee,$60$ oranges cost $60 / 3 = 20$ rupees.
Total $C.P.$ for $120$ oranges $(60 + 60)$ $= 30 + 20 = 50$ rupees.
To make a profit of $20 \%$,the total Selling Price $(S.P.)$ should be: $S.P. = C.P. + (20 \% \text{ of } C.P.) = 50 + (0.20 \times 50) = 50 + 10 = 60$ rupees.
This $S.P.$ of $60$ rupees is for $120$ oranges.
Therefore,the $S.P.$ for $1$ dozen ($12$ oranges) $= (60 / 120) \times 12 = 0.5 \times 12 = 6$ rupees.

(A) Let the total number of items be $4$ and the cost price $(C.P.)$ of each item be $100$.
Total $C.P. = 4 \times 100 = 400$.
The marked price $(M.P.)$ is $30 \%$ above $C.P.$, so $M.P. = 100 + 30 = 130$ per item.
$1$. Half the stock ($2$ items) is sold at $M.P.$: $S.P._{1} = 2 \times 130 = 260$.
$2$. One-fourth of the stock ($1$ item) is sold at $15 \%$ discount: $S.P._{2} = 130 - (0.15 \times 130) = 130 - 19.5 = 110.5$.
$3$. The remaining stock ($1$ item) is sold at $30 \%$ discount: $S.P._{3} = 130 - (0.30 \times 130) = 130 - 39 = 91$.
Total $S.P. = 260 + 110.5 + 91 = 461.5$.
Profit $= Total S.P. - Total C.P. = 461.5 - 400 = 61.5$.
Profit $\% = (61.5 / 400) \times 100 = 61.5 / 4 = 15.375 = 15 \frac{3}{8} \%$.

(C) Let the original marked price be $MP_1 = 100x$.
The trader buys the goods at a $20 \%$ discount,so the cost price $CP = 100x - 20x = 80x$.
He wants to make a profit of $25 \%$ on the cost price,so the required selling price $SP = 80x + 25 \% \text{ of } 80x = 80x + 20x = 100x$.
Let the new marked price be $MP_2 = y$. He allows a discount of $20 \%$ on this new marked price,so $SP = y - 20 \% \text{ of } y = 0.8y$.
Equating the two expressions for $SP$: $0.8y = 100x \Rightarrow y = \frac{100x}{0.8} = 125x$.
The percentage increase over the original marked price is $\frac{125x - 100x}{100x} \times 100 = 25 \%$.

(B) Let the marked price $(MP)$ be $100x$.
$A$ discount of $10\%$ is allowed on the $MP$,so the discount amount is $10\% \text{ of } 100x = 10x$.
The selling price $(SP)$ after discount is $100x - 10x = 90x$.
$A$ sales tax of $8\%$ is charged on the discounted price $(90x)$,so the tax amount is $8\% \text{ of } 90x = 0.08 \times 90x = 7.2x$.
The total amount paid by the customer is the sum of the discounted price and the sales tax: $90x + 7.2x = 97.2x$.
Given that the total amount paid is $Rs. 3,402$,we have $97.2x = 3402$.
Solving for $x$: $x = 3402 / 97.2 = 35$.
Therefore,the marked price is $100x = 100 \times 35 = Rs. 3,500$.

(B) Let the total number of oranges be $100$ and the cost price $(C.P.)$ be $₹ 100$.
$S.P.$ of $40$ oranges $= ₹ 100$.
Profit on these $40$ oranges $= 100 - 40 = ₹ 60$.
Profit percentage $= (60 / 40) \times 100 = 150 \%$.
Remaining oranges $= 100 - 40 = 60$.
He sells $80 \%$ of the remaining oranges,which is $0.80 \times 60 = 48$ oranges.
He sells these at half the previous rate of profit,so the new profit rate is $150 \% / 2 = 75 \%$.
$S.P.$ of $48$ oranges $= 48 + 75 \% \text{ of } 48 = 48 + 36 = ₹ 84$.
Total $S.P.$ $= 100 + 84 = ₹ 184$.
Total Profit $= 184 - 100 = ₹ 84$.
Overall profit percentage $= (84 / 100) \times 100 = 84 \%$.

(B) The discount percentage offered by the merchant is $10 \%$.
The total cost of the items purchased is the sum of the prices of the cooker,the heater,and the bag.
Total cost $= Rs. 650 + Rs. 500 + Rs. 65 = Rs. 1215$.
The total discount is calculated as $10 \%$ of the total cost.
Total discount $= 10 \% \text{ of } 1215 = \frac{10}{100} \times 1215 = Rs. 121.50$.

(B) Given that the selling price $(S.P._1)$ is $Rs. 450$ and the loss is $10\%$.
Let the cost price be $C.P.$
Since $Loss\% = \frac{C.P. - S.P.}{C.P.} \times 100$,we have $10 = \frac{C.P. - 450}{C.P.} \times 100$.
$0.1 \times C.P. = C.P. - 450$.
$0.9 \times C.P. = 450$.
$C.P. = \frac{450}{0.9} = Rs. 500$.
Now,if the new selling price $(S.P._2)$ is $Rs. 540$,the gain is $S.P._2 - C.P. = 540 - 500 = Rs. 40$.
$Gain\% = \frac{Gain}{C.P.} \times 100 = \frac{40}{500} \times 100 = 8\%$.

(A) Let the initial amount of money be $x$.
He loses $20 \frac{1}{2} \% = 20.5 \%$ of his money.
The amount remaining after the loss is $x - 0.205x = 0.795x$.
He spends $80 \%$ of the remainder,so he is left with $20 \%$ of the remainder.
Therefore,$0.20 \times 0.795x = 159$.
$0.159x = 159$.
$x = \frac{159}{0.159} = 1000$.
Thus,he had $Rs. 1000$ at first.

(C) Given: Cost Price $(C.P.)$ = $Rs. 3,200$.
Desired profit percentage = $25\%$.
Selling Price $(S.P.)$ = $C.P. + (Profit\% \text{ of } C.P.)$
$S.P. = 3200 + (0.25 \times 3200) = 3200 + 800 = Rs. 4,000$.
Let the Marked Price $(M.P.)$ be $x$.
Discount = $20\%$ of $M.P. = 0.20x$.
$S.P. = M.P. - \text{Discount} = x - 0.20x = 0.80x$.
Since $S.P. = 4000$,we have $0.80x = 4000$.
$x = \frac{4000}{0.80} = 5000$.
Therefore,the marked price is $Rs. 5,000$.

(B) Given,Cost Price $(CP) = 210 \text{ Rs.}$
Profit percentage $= 20 \%$.
Selling Price $(SP) = CP + \text{Profit} = 210 + (20 \% \text{ of } 210) = 210 + 42 = 252 \text{ Rs.}$
Let the Marked Price $(MP)$ be $x$.
Discount is given as $12.5 \%$ on the Marked Price.
So,$SP = MP - \text{Discount} = x - (12.5 \% \text{ of } x) = x(1 - 0.125) = 0.875x$.
Equating the Selling Price: $0.875x = 252$.
$x = 252 / 0.875 = 252 / (7/8) = (252 \times 8) / 7 = 36 \times 8 = 288$.
Therefore,the marked price is $288 \text{ Rs.}$

(A) Let the marked price $(M.P.)$ be $100$.
Since a discount of $10 \%$ is allowed,the selling price $(S.P.)$ is $100 - 10 = 90$.
We want a profit of $17 \%$,so the selling price must be $117 \%$ of the cost price $(C.P.)$.
$S.P. = 1.17 \times C.P.$
$90 = 1.17 \times C.P.$
$C.P. = \frac{90}{1.17} = \frac{9000}{117} \approx 76.92$.
Alternatively,using the formula: $M.P. = C.P. \times \frac{100 + \text{Profit } \%}{100 - \text{Discount } \%}$.
$M.P. = C.P. \times \frac{100 + 17}{100 - 10} = C.P. \times \frac{117}{90} = C.P. \times 1.3$.
This means the marked price is $130 \%$ of the cost price.
Therefore,the goods must be marked $30 \%$ above the cost price.

(B) Let the cost price $(C.P.)$ be $x$.
Given that the marked price $(M.P.)$ is $30 \%$ above the cost price,we have: $M.P. = x + 0.30x = 1.30x$.
Given $M.P. = Rs. 286$,so $1.30x = 286$.
$x = 286 / 1.30 = 220$.
Thus,the cost price $(C.P.)$ is $Rs. 220$.
The seller allows a $10 \%$ discount on the marked price.
Discount $= 10 \% \text{ of } 286 = 0.10 \times 286 = Rs. 28.60$.
Selling price $(S.P.)$ $= M.P. - \text{Discount} = 286 - 28.60 = Rs. 257.40$.
Gain $= S.P. - C.P. = 257.40 - 220 = Rs. 37.40$.

(C) Let the cost price $(C.P.)$ of the article be $₹ 100$.
Since the tradesman marks the goods $30 \%$ above the cost price,the marked price $(M.P.)$ is $₹ 130$.
The discount allowed is $6 \frac{1}{4} \% = \frac{25}{4} \% = 6.25 \%$.
The selling price $(S.P.)$ is calculated as: $S.P. = M.P. \times \left(1 - \frac{\text{Discount} \%}{100}\right)$.
$S.P. = 130 \times \left(1 - \frac{25/4}{100}\right) = 130 \times \left(1 - \frac{25}{400}\right) = 130 \times \left(1 - \frac{1}{16}\right) = 130 \times \frac{15}{16}$.
$S.P. = \frac{130 \times 15}{16} = \frac{1950}{16} = ₹ 121.875 = ₹ 121 \frac{7}{8}$.
Gain percent is calculated as: $\text{Gain} \% = \left(\frac{S.P. - C.P.}{C.P.}\right) \times 100$.
$\text{Gain} \% = \left(\frac{121.875 - 100}{100}\right) \times 100 = 21.875 \% = 21 \frac{7}{8} \%$.

(A) Step $1$: Calculate the cost price after successive discounts.
First discount of $15 \%$ on $₹ 600 = 600 - (600 \times 0.15) = 600 - 90 = ₹ 510$.
Second discount of $20 \%$ on $₹ 510 = 510 - (510 \times 0.20) = 510 - 102 = ₹ 408$.
Step $2$: Add transportation cost to the purchase price.
Total Cost Price $(C.P.) = 408 + 28 = ₹ 436$.
Step $3$: Calculate gain percent.
Selling Price $(S.P.) = ₹ 545$.
Gain $= S.P. - C.P. = 545 - 436 = ₹ 109$.
Gain Percent $= (\text{Gain} / C.P.) \times 100 = (109 / 436) \times 100 = 0.25 \times 100 = 25 \%$.

(A) The marked price of the piano is $Rs. 15,000$.
Successive discounts are $20\%$,$10\%$,and $10\%$.
The sale price can be calculated as:
$\text{Sale Price} = 15000 \times \left(1 - \frac{20}{100}\right) \times \left(1 - \frac{10}{100}\right) \times \left(1 - \frac{10}{100}\right)$
$\text{Sale Price} = 15000 \times \frac{80}{100} \times \frac{90}{100} \times \frac{90}{100}$
$\text{Sale Price} = 15000 \times 0.8 \times 0.9 \times 0.9$
$\text{Sale Price} = 15000 \times 0.648 = 9720$
Thus,the sale price is $Rs. 9720$.

(A) Let the selling price $(S.P.)$ of $1$ $m$ of cloth be $ 1$.
Then,the $S.P.$ of $25$ $m$ of cloth $= 25$.
And the $S.P.$ of $5$ $m$ of cloth $= 5$.
Gain $= S.P. - C.P. = S.P.$ of $5$ $m$ of cloth $= 5$.
Therefore,$C.P.$ of $25$ $m$ of cloth $= S.P.$ of $25$ $m$ of cloth $-$ Gain $= 25 - 5 = 20$.
Gain percentage $= (\text{Gain} / C.P.) \times 100$.
Gain percentage $= (5 / 20) \times 100 = 25 \%$.

(C) Let the cost price $(C.P.)$ of the suitcase for $A$ be $₹ x$.
According to the problem,$A$ sells it to $B$ at a $10\%$ profit,so the selling price for $A$ (which is the cost price for $B$) is $x \times (1 + \frac{10}{100}) = x \times \frac{110}{100}$.
$B$ then sells it to $C$ at a $30\%$ profit. Therefore,the selling price for $B$ (which is the cost price for $C$) is $(x \times \frac{110}{100}) \times (1 + \frac{30}{100}) = x \times \frac{110}{100} \times \frac{130}{100}$.
Given that $C$ pays $₹ 2,860$,we have:
$x \times \frac{110}{100} \times \frac{130}{100} = 2860$
$x \times \frac{11}{10} \times \frac{13}{10} = 2860$
$x \times \frac{143}{100} = 2860$
$x = \frac{2860 \times 100}{143}$
$x = 20 \times 100 = ₹ 2000$.
Thus,the price at which $A$ bought the suitcase is $₹ 2000$.

(B) Total Cost Price $(C.P.)$ $= ₹ 96,000$.
Total expected Selling Price $(S.P.)$ for $10\%$ profit $= 96,000 \times \frac{110}{100} = ₹ 1,05,600$.
$C.P.$ of the first part $= \frac{2}{5} \times 96,000 = ₹ 38,400$.
$S.P.$ of the first part at $6\%$ loss $= 38,400 \times \frac{94}{100} = ₹ 36,096$.
$C.P.$ of the remaining part $= 96,000 - 38,400 = ₹ 57,600$.
Required $S.P.$ of the remaining part $= 1,05,600 - 36,096 = ₹ 69,504$.
Gain on the remaining part $= 69,504 - 57,600 = ₹ 11,904$.
Gain percentage $= \left( \frac{11,904}{57,600} \right) \times 100 = \frac{11,904}{576} = 20 \frac{2}{3}\%$.

(C) Let the cost price $(C.P.)$ of the article be $₹ x$.
According to the problem,the initial selling price at a $15 \%$ gain is $x + 0.15x = 1.15x$.
If the article is sold for $₹ 27$ more,the new selling price becomes $1.15x + 27$.
At this new price,the profit is $20 \%$,so the new selling price is $x + 0.20x = 1.20x$.
Equating the two expressions for the new selling price:
$1.20x = 1.15x + 27$
Subtracting $1.15x$ from both sides:
$0.05x = 27$
Solving for $x$:
$x = \frac{27}{0.05} = \frac{2700}{5} = 540$.
Therefore,the cost price of the article is $₹ 540$.

(D) Let the cost price $(C.P.)$ of one ball be $₹x$.
The $C.P.$ of $17$ balls $= 17x$.
The selling price $(S.P.)$ of $17$ balls is given as $₹720$.
According to the problem, the loss is equal to the $C.P.$ of $5$ balls, which is $5x$.
We know that $\text{Loss} = C.P. - S.P.$
Substituting the values: $5x = 17x - 720$.
Rearranging the terms: $17x - 5x = 720$.
$12x = 720$.
$x = 720 / 12 = 60$.
Therefore, the cost price of one ball is $₹60$.

(C) Let the cost price $(C.P.)$ of items $A$ and $B$ be $₹x$ and $₹y,$ respectively.
According to the problem,the profit amount is the same for both items.
Profit on $A = 10 \% \text{ of } x = 0.10x$
Profit on $B = 15 \% \text{ of } y = 0.15y$
Since the profits are equal,we have $0.10x = 0.15y$.
Dividing both sides by $0.05$,we get $2x = 3y$,which implies $\frac{x}{y} = \frac{3}{2}$.
Thus,the ratio of the cost prices $x:y$ is $3:2$.
Checking the options,for option $C$,the ratio is $3000:2000 = 3:2$.
Therefore,the cost prices of $A$ and $B$ are $₹3000$ and $₹2000$ respectively.

(B) Let the cost price of the computer be $CP = 100$.
Arun marks up the price by $20 \%$,so the marked price $MP = 100 + (20 \% \text{ of } 100) = 120$.
He sells the computer at a discount of $15 \%$ on the marked price.
Selling price $SP = MP - (15 \% \text{ of } MP) = 120 - (0.15 \times 120) = 120 - 18 = 102$.
Net gain $= SP - CP = 102 - 100 = 2$.
Net gain percent $= (\text{Gain} / CP) \times 100 = (2 / 100) \times 100 = 2 \%$.
Alternatively,using the formula for successive percentage change: $\text{Net gain } \% = (x + y + \frac{xy}{100}) \%$,where $x = 20$ and $y = -15$.
Net gain $\% = 20 - 15 + \frac{20 \times (-15)}{100} = 5 - 3 = 2 \%$.

(A) The single equivalent discount for successive discounts of $20 \%$ and $10 \%$ is calculated as:
$D_{eq} = \left(20 + 10 - \frac{20 \times 10}{100}\right) \% = (30 - 2) \% = 28 \%$.
The cost price $(C.P.)$ of the table after discounts is:
$C.P. = 1500 \times \left(1 - \frac{28}{100}\right) = 1500 \times 0.72 = Rs. 1080$.
Including the transportation cost,the total cost price $(C.P._{total})$ is:
$C.P._{total} = 1080 + 20 = Rs. 1100$.
To earn a profit of $20 \%$,the selling price $(S.P.)$ is:
$S.P. = C.P._{total} \times \left(1 + \frac{20}{100}\right) = 1100 \times 1.2 = Rs. 1320$.

(C) Let the cost price $(C.P.)$ for $A$ be $₹ x$.
According to the problem,the successive gains are $20 \%$,$10 \%$,and $12 \frac{1}{2} \%$ (which is $12.5 \%$ or $\frac{25}{2} \%$).
The selling price for $A$ is $x \times (1 + \frac{20}{100}) = x \times \frac{120}{100}$.
The selling price for $B$ is $(x \times \frac{120}{100}) \times (1 + \frac{10}{100}) = x \times \frac{120}{100} \times \frac{110}{100}$.
The selling price for $C$ (which is the price $D$ pays) is $(x \times \frac{120}{100} \times \frac{110}{100}) \times (1 + \frac{12.5}{100}) = x \times \frac{120}{100} \times \frac{110}{100} \times \frac{112.5}{100}$.
Given that $D$ pays $₹ 29.70$,we have:
$x \times \frac{120}{100} \times \frac{110}{100} \times \frac{112.5}{100} = 29.70$
$x \times \frac{6}{5} \times \frac{11}{10} \times \frac{9}{8} = 29.70$
$x \times \frac{594}{400} = 29.70$
$x = \frac{29.70 \times 400}{594} = \frac{11880}{594} = 20$.
Thus,$A$ purchased the article for $₹ 20$.

(A) First,calculate the Cost Price $(C.P.)$ of $80$ ball pens.
Given that selling $80$ pens for $₹ 140$ results in a $30 \%$ loss,the selling price is $70 \%$ of the $C.P.$
$C.P. = 140 \times \frac{100}{70} = ₹ 200$.
Now,to make a profit of $30 \%$,the new selling price $(S.P.)$ for $80$ pens must be:
$S.P. = 200 \times \frac{130}{100} = ₹ 260$.
This means $80$ ball pens are sold for $₹ 260$ to gain $30 \%$.
To find how many pens can be sold for $₹ 104$:
Number of pens $= \frac{80}{260} \times 104 = \frac{8}{26} \times 104 = 8 \times 4 = 32$.
Therefore,the retailer should sell $32$ ball pens.

(A) When two items are sold at the same selling price,one at a profit of $x \%$ and the other at a loss of $x \%$,there is always a net loss in the transaction.
The formula for the net loss percentage is given by: $\text{Loss } \% = \left(\frac{x}{10}\right)^2$.
Here,$x = 12$.
Substituting the value of $x$ into the formula:
$\text{Loss } \% = \left(\frac{12}{10}\right)^2 = (1.2)^2 = 1.44 \%$.
Converting $1.44$ into a fraction:
$1.44 = \frac{144}{100} = \frac{36}{25} = 1 \frac{11}{25} \%$.
Therefore,the shopkeeper incurs a loss of $1 \frac{11}{25} \%$.

(A) Let the marked price of the trouser be $₹ x$.
According to the problem,the discount amount is equal to the cost of the shirt,which is $₹ 320$.
Therefore,$40 \%$ of $x = 320$.
$\frac{40}{100} \times x = 320$.
$x = \frac{320 \times 100}{40} = 800$.
The marked price of the trouser is $₹ 800$.
The selling price $(S.P.)$ of the trouser is the marked price minus the discount.
$S.P. = 800 - 320 = ₹ 480$.
Thus,Ajit paid $₹ 480$ for the trouser.

(B) Let the marked price be $M$.
Given that the discount is $15 \%$,the cost price $(C.P.)$ for Rahim is $85 \%$ of $M$.
$0.85 \times M = 510$
$M = \frac{510}{0.85} = 600$
So,the marked price is $Rs. 600$.
Rahim sells the item at $5 \%$ above the marked price.
Selling price $(S.P.)$ $= M + (5 \% \text{ of } M) = 600 + (0.05 \times 600) = 600 + 30 = 630$.
The profit earned is $S.P. - C.P. = 630 - 510 = 120$.
Thus,the profit earned is $Rs. 120$.

(C) Let the total cost price $(C.P.)$ of $100$ articles be $₹ 100$.
Each article costs $₹ 1$.
The marked price $(M.P.)$ of each article is $100 \% + 40 \% = 140 \%$ of $C.P. = ₹ 1.40$.
He sells $\frac{3}{4}$ of the goods (i.e.,$75$ articles) at the marked price:
$S.P._1 = 75 \times 1.40 = ₹ 105$.
He sells the remaining $\frac{1}{4}$ of the goods (i.e.,$25$ articles) at a $40 \%$ discount on the marked price:
Discounted price per article $= 1.40 \times (100 \% - 40 \%) = 1.40 \times 0.60 = ₹ 0.84$.
$S.P._2 = 25 \times 0.84 = ₹ 21$.
Total selling price $(S.P.)$ $= 105 + 21 = ₹ 126$.
Since $S.P. > C.P.$,there is a gain.
Gain $= S.P. - C.P. = 126 - 100 = ₹ 26$.
Gain percentage $= (\text{Gain} / C.P.) \times 100 = (26 / 100) \times 100 = 26 \%.$

(B) Total cost price $(C.P.)$ of $240$ bananas $= \frac{240}{12} \times 48 = 20 \times 48 = ₹ 960$.
To get a profit of $25\%$,the total selling price $(S.P.)$ must be $= 960 \times (1 + 0.25) = 960 \times 1.25 = ₹ 1200$.
He sells half of the bananas ($120$ bananas) at $₹ 5$ each. Amount received $= 120 \times 5 = ₹ 600$.
Remaining bananas $= 240 - 120 = 120$. Out of these,$\frac{1}{6}$ are rotten,so rotten bananas $= 120 \times \frac{1}{6} = 20$.
Good remaining bananas $= 120 - 20 = 100$.
Required $S.P.$ for the remaining $100$ bananas $= 1200 - 600 = ₹ 600$.
Price per banana for the remaining stock $= \frac{600}{100} = ₹ 6$ per banana.

(D) The ratio of capital invested by $A$ and $B$ is $3,50,000 : 1,40,000 = 5 : 2$.
Let the total yearly profit be $x$.
$A$ receives $20\%$ of the profit for management,which is $0.2x = \frac{x}{5}$.
The remaining profit is $x - \frac{x}{5} = \frac{4x}{5}$.
This remaining profit is divided between $A$ and $B$ in the ratio $5 : 2$.
$A$'s share from the remaining profit $= \frac{5}{7} \times \frac{4x}{5} = \frac{4x}{7}$.
$B$'s share from the remaining profit $= \frac{2}{7} \times \frac{4x}{5} = \frac{8x}{35}$.
Total share of $A = \frac{x}{5} + \frac{4x}{7} = \frac{7x + 20x}{35} = \frac{27x}{35}$.
Total share of $B = \frac{8x}{35}$.
The difference between $A$'s and $B$'s share is $\frac{27x}{35} - \frac{8x}{35} = \frac{19x}{35}$.
Given that the difference is $Rs. 38,000$,we have $\frac{19x}{35} = 38,000$.
$x = \frac{38,000 \times 35}{19} = 2,000 \times 35 = 70,000$.
Thus,the total profit is $Rs. 70,000$.

(A) Let the original marked price be $₹ x$.
First discount is $10 \%$,so the price after the first discount is $x \times (1 - 0.10) = 0.9x$.
Second discount is $6 \%$ on the discounted price,so the final selling price is $0.9x \times (1 - 0.06) = 0.9x \times 0.94$.
$0.9 \times 0.94 = 0.846$.
Given that the final selling price is $Rs. 846$,we have $0.846x = 846$.
$x = \frac{846}{0.846} = 1000$.
Therefore,the original marked price is $Rs. 1000$.

(C) Let the $C.P.$ (Cost Price) of the article be $₹ 100$ and the marked price be $₹ x$.
The trade discount is $20 \%$ and the cash discount is $6 \frac{1}{4} \% = 6.25 \%$.
The single equivalent discount is calculated as:
$D = \left( d_1 + d_2 - \frac{d_1 \times d_2}{100} \right) \% = \left( 20 + 6.25 - \frac{20 \times 6.25}{100} \right) \% = (26.25 - 1.25) \% = 25 \%$.
Selling Price $(S.P.)$ after discount $= x \times (100 - 25) \% = x \times 0.75$.
Given that the trader gets a net gain of $20 \%$,so $S.P. = C.P. + 20 \% \text{ of } C.P. = 100 + 20 = ₹ 120$.
Equating the two expressions for $S.P.$:
$x \times 0.75 = 120$
$x = \frac{120}{0.75} = \frac{12000}{75} = 160$.
The marked price is $₹ 160$,which is $60 \%$ above the cost price of $₹ 100$.

(A) Let the marked price be $100$.
After a $10 \%$ discount,the price becomes $100 - 10 = 90$.
After a $20 \%$ discount on $90$,the price becomes $90 - (0.20 \times 90) = 90 - 18 = 72$.
After a $40 \%$ discount on $72$,the price becomes $72 - (0.40 \times 72) = 72 - 28.8 = 43.2$.
The total discount is $100 - 43.2 = 56.8 \%$.
Alternatively,using the formula for two successive discounts $x$ and $y$: $D = x + y - \frac{xy}{100}$.
For $10 \%$ and $20 \%$: $10 + 20 - \frac{10 \times 20}{100} = 30 - 2 = 28 \%$.
For $28 \%$ and $40 \%$: $28 + 40 - \frac{28 \times 40}{100} = 68 - 11.2 = 56.8 \%$.

(C) Let the marked price of the $TV$ be $₹ x$.
According to the problem,the difference between the two discount scenarios is $₹ 500$.
Discount at $20 \%$ means the selling price is $80 \%$ of $x$,which is $0.8x$.
Discount at $25 \%$ means the selling price is $75 \%$ of $x$,which is $0.75x$.
The difference in price is given as:
$0.8x - 0.75x = 500$
$0.05x = 500$
$x = \frac{500}{0.05} = 10000$
The marked price is $₹ 10000$.
Tarun bought the $TV$ at a $20 \%$ discount,so the cost price is:
$Cost Price = 10000 - (20 \% \text{ of } 10000) = 10000 - 2000 = ₹ 8000$.

(C) Let the cost price for the manufacturer be $₹ x$.
$1$. The manufacturer sells it at $10 \%$ profit,so the selling price is $x \times (1 + \frac{10}{100}) = x \times \frac{110}{100}$.
$2$. The wholesale dealer sells it at $20 \%$ profit,so the selling price is $(x \times \frac{110}{100}) \times (1 + \frac{20}{100}) = x \times \frac{110}{100} \times \frac{120}{100}$.
$3$. The shopkeeper sells it at $15 \%$ loss for $₹ 56,100$,so the final equation is:
$x \times \frac{110}{100} \times \frac{120}{100} \times \frac{85}{100} = 56,100$
$4$. Simplifying the equation:
$x \times \frac{11}{10} \times \frac{6}{5} \times \frac{17}{20} = 56,100$
$x \times \frac{11 \times 6 \times 17}{1000} = 56,100$
$x \times \frac{1122}{1000} = 56,100$
$x = \frac{56,100 \times 1000}{1122} = 50,000$
Thus,the cost price for the manufacturer is $₹ 50,000$.