Profit and Loss Questions in English

(A) Let the cost price $(C.P.)$ of the article be $₹ x$.
Initial loss is $19 \%$,so the initial selling price $(S.P._1)$ is $x - 0.19x = 0.81x$.
New profit is $17 \%$,so the new selling price $(S.P._2)$ is $x + 0.17x = 1.17x$.
According to the problem,the difference between the two selling prices is $₹ 162$:
$1.17x - 0.81x = 162$
$0.36x = 162$
$x = \frac{162}{0.36}$
$x = \frac{16200}{36} = 450$
Therefore,the cost price of the article is $₹ 450$.

(D) Total cost price $(C.P.)$ of $150$ pens $= 150 \times 12 = ₹ 1800$.
Required total gain $= 15 \%$.
Required total selling price $(S.P.)$ $= 1800 \times \frac{115}{100} = ₹ 2070$.
$S.P.$ of the first $50$ pens at $10 \%$ gain $= 50 \times 12 \times \frac{110}{100} = ₹ 660$.
Remaining pens $= 150 - 50 = 100$.
$C.P.$ of remaining $100$ pens $= 100 \times 12 = ₹ 1200$.
Let the required gain percentage on the remaining pens be $x \%$.
$S.P.$ of remaining $100$ pens $= 1200 \times \frac{(100 + x)}{100} = 12(100 + x) = 1200 + 12x$.
Total $S.P. = 660 + 1200 + 12x = 2070$.
$1860 + 12x = 2070$.
$12x = 2070 - 1860 = 210$.
$x = \frac{210}{12} = 17.5 = 17 \frac{1}{2} \%$.

(D) When two items are sold at the same selling price $(S.P.)$,and there is a profit of $x \%$ on one and a loss of $x \%$ on the other,the overall result is always a loss.
The formula for the net loss percentage is given by: $\text{Loss } \% = \left( \frac{x}{10} \right)^2 = \frac{x^2}{100}$.
Given $x = 20$,the loss percentage is:
$\text{Loss } \% = \frac{20^2}{100} = \frac{400}{100} = 4 \%$.
Therefore,the dealer incurred a $4 \%$ loss in the entire transaction.

(B) Let the cost price of $1$ article be $x$.
Then,the cost price of $40$ articles is $40x$.
According to the problem,the selling price of $25$ articles is equal to the cost price of $40$ articles,so the selling price of $25$ articles is $40x$.
The cost price of $25$ articles is $25x$.
Gain = Selling Price - Cost Price = $40x - 25x = 15x$.
Gain percent = $\left( \frac{\text{Gain}}{\text{Cost Price of } 25 \text{ articles}} \right) \times 100 = \left( \frac{15x}{25x} \right) \times 100 = \frac{15}{25} \times 100 = 60\%$.

(B) Let the cost price $(C.P.)$ of the article for $A$ be $₹ x$.
$A$ sells it to $B$ at a profit of $\frac{1}{5}$ of his outlay,so the selling price for $A$ (which is the $C.P.$ for $B$) is $x \times (1 + \frac{1}{5}) = x \times \frac{6}{5}$.
$B$ sells it to $C$ at a gain of $20\%$,so the selling price for $B$ (which is the $C.P.$ for $C$) is $(x \times \frac{6}{5}) \times (1 + \frac{20}{100}) = x \times \frac{6}{5} \times \frac{6}{5} = x \times \frac{36}{25}$.
$C$ sells it for $₹ 600$ and incurs a loss of $\frac{1}{6}$ of his outlay. Thus,the selling price for $C$ is $(C.P. \text{ of } C) \times (1 - \frac{1}{6}) = 600$.
Substituting the $C.P.$ of $C$:
$(x \times \frac{36}{25}) \times \frac{5}{6} = 600$
$x \times \frac{6}{5} = 600$
$x = 600 \times \frac{5}{6} = 500$.
Therefore,the cost price of $A$ is $₹ 500$.

(D) Let the total amount be $₹ x$.
According to the problem:
$1$. Amount spent on article = $20\% \text{ of } x = 0.2x$.
$2$. Remaining amount = $x - 0.2x = 0.8x$.
$3$. Amount spent on transport = $5\% \text{ of } 0.8x = 0.05 \times 0.8x = 0.04x$.
$4$. After gifting $Rs. 120$,the remaining amount is $Rs. 1,400$.
Equation: $x - 0.2x - 0.04x - 120 = 1400$
$0.76x = 1520$
$x = \frac{1520}{0.76} = 2000$.
Expenditure on transport = $0.04 \times 2000 = Rs. 80$.

(A) Cost Price $(CP)$ $= ₹ 78,350$.
Marked Price $(MP)$ $= CP \times (1 + \frac{30}{100}) = 78,350 \times 1.3 = ₹ 1,01,855$.
Selling Price $(SP)$ $= MP \times (1 - \frac{20}{100}) = 1,01,855 \times 0.8 = ₹ 81,484$.
Profit $= SP - CP = 81,484 - 78,350 = ₹ 3,134$.
Profit Percentage $= (\frac{\text{Profit}}{CP}) \times 100 = (\frac{3,134}{78,350}) \times 100 = 4 \%$.

(B) Let the selling price $(S.P.)$ be $₹100$.
Given that the loss is $20 \%$ of the $S.P.$
Loss $= 20 \% \text{ of } ₹100 = ₹20$.
Since $\text{Loss} = C.P. - S.P.$,we have $20 = C.P. - 100$.
Therefore,the cost price $(C.P.)$ $= 100 + 20 = ₹120$.
The loss percentage on the cost price is given by the formula: $\frac{\text{Loss}}{C.P.} \times 100$.
$\text{Loss } \% = \frac{20}{120} \times 100 = \frac{1}{6} \times 100 = \frac{50}{3} = 16 \frac{2}{3} \%$.

$(C)$ Let the Cost Price $(CP)$ of the first article be $CP_1$ and the second be $CP_2$.
Given that the Selling Price $(SP)$ of each article is $Rs. 4,000$.
Total $SP = 4,000 + 4,000 = Rs. 8,000$.
Since there is no loss and no gain, Total $CP = Total SP = Rs. 8,000$.
For the first article, $SP_1 = 4,000$ and Gain $= 25\%$.
$CP_1 = \frac{SP_1 \times 100}{100 + \text{Gain}\%} = \frac{4,000 \times 100}{125} = 3,200$.
Since $CP_1 + CP_2 = 8,000$, then $CP_2 = 8,000 - 3,200 = 4,800$.
For the second article, $CP_2 = 4,800$ and $SP_2 = 4,000$.
Loss $= CP_2 - SP_2 = 4,800 - 4,000 = 800$.
Loss $\% = \frac{\text{Loss}}{CP_2} \times 100 = \frac{800}{4,800} \times 100 = \frac{1}{6} \times 100 = 16 \frac{2}{3}\%$.

(C) Let the number of eggs be a multiple of the Least Common Multiple of $3$ and $5$,which is $15$.
Cost Price $(C.P.)$ of $3$ eggs $= ₹ 5$.
Therefore,$C.P.$ of $15$ eggs $= (5/3) \times 15 = ₹ 25$.
Selling Price $(S.P.)$ of $5$ eggs $= ₹ 12$.
Therefore,$S.P.$ of $15$ eggs $= (12/5) \times 15 = ₹ 36$.
Gain on $15$ eggs $= S.P. - C.P. = 36 - 25 = ₹ 11$.
If the gain is $₹ 11$,the number of eggs is $15$.
If the gain is $₹ 143$,the number of eggs $= (15 / 11) \times 143$.
$= 15 \times 13 = 195$ eggs.

(A) Let the marked price of the article be $₹100$.
Given that the cost price $(C.P.)$ is $64 \%$ of the marked price:
$C.P. = 0.64 \times 100 = ₹64$.
$A$ discount of $12 \%$ is allowed on the marked price:
Discount $= 12 \% \text{ of } ₹100 = ₹12$.
Therefore,the selling price $(S.P.)$ is:
$S.P. = \text{Marked Price} - \text{Discount} = 100 - 12 = ₹88$.
The profit is calculated as:
$\text{Profit} = S.P. - C.P. = 88 - 64 = ₹24$.
The gain percentage is calculated as:
$\text{Gain } \% = \left( \frac{\text{Profit}}{C.P.} \right) \times 100 = \left( \frac{24}{64} \right) \times 100 = \frac{3}{8} \times 100 = 37.5 \%$.

(B) Let the cost price $(C.P.)$ of the article be $₹ x$.
Given that the percentage gain is equal to the cost price,the gain percentage is $x\%$.
We know that $Gain = S.P. - C.P. = 144 - x$.
The formula for gain percentage is $\frac{Gain}{C.P.} \times 100 = Gain\%$.
Substituting the values: $\frac{144 - x}{x} \times 100 = x$.
$(144 - x) \times 100 = x^2$.
$14400 - 100x = x^2$.
$x^2 + 100x - 14400 = 0$.
Factoring the quadratic equation: $x^2 + 180x - 80x - 14400 = 0$.
$x(x + 180) - 80(x + 180) = 0$.
$(x - 80)(x + 180) = 0$.
Since the cost price cannot be negative,$x = 80$.
Therefore,the cost price of the article is $₹ 80$.

(B) Total Selling Price $(S.P.)$ of two articles $= 5000 + 5000 = ₹ 10000$.
Since there is no gain or loss in the deal,Total Cost Price $(C.P.)$ $= ₹ 10000$.
$C.P.$ of the first article $= 5000 \times \frac{100}{125} = ₹ 4000$.
$C.P.$ of the second article $= 10000 - 4000 = ₹ 6000$.
Since the $S.P.$ of the second article is $₹ 5000$,the loss on the second article $= 6000 - 5000 = ₹ 1000$.
Loss percentage $= \frac{\text{Loss}}{C.P.} \times 100 = \frac{1000}{6000} \times 100 = \frac{100}{6} = 16 \frac{2}{3} \%$.

(A) Step $1$: Find the $LCM$ of the quantities $8$ and $12$ to standardize the number of oranges.
$LCM(8, 12) = 24$.
Step $2$: Calculate the Cost Price $(C.P.)$ for $24$ oranges.
$C.P. = (34 / 8) \times 24 = 34 \times 3 = Rs. 102$.
Step $3$: Calculate the Selling Price $(S.P.)$ for $24$ oranges.
$S.P. = (57 / 12) \times 24 = 57 \times 2 = Rs. 114$.
Step $4$: Calculate the profit on $24$ oranges.
$Profit = S.P. - C.P. = 114 - 102 = Rs. 12$.
Step $5$: Determine the number of oranges required for a profit of $Rs. 45$.
Since a profit of $Rs. 12$ is earned by selling $24$ oranges,a profit of $Rs. 1$ is earned by selling $24 / 12 = 2$ oranges.
Therefore,for a profit of $Rs. 45$,the number of oranges to be sold is $2 \times 45 = 90$ oranges.

(D) Let the advertised price be $₹ x$.
Since the commission is $23 \%$,the Selling Price $(S.P.)$ is $x - 0.23x = 0.77x$.
Given that the profit is $10 \%$ on the Cost Price $(C.P.)$ and the profit amount is $₹ 56$,we have:
$Profit = S.P. - C.P. = 56$
$C.P. = S.P. - 56 = 0.77x - 56$.
Since profit is $10 \%$ of $C.P.$,we have $Profit = 0.10 \times C.P.$
$56 = 0.10 \times (0.77x - 56)$
$56 = 0.077x - 5.6$
$56 + 5.6 = 0.077x$
$61.6 = 0.077x$
$x = \frac{61.6}{0.077} = \frac{61600}{77} = 800$.
Therefore,the advertised price is $₹ 800$.

(D) Initial cost price $(CP_1) = Rs.\, 9600$.
Selling price after $5\%$ loss $(SP_1) = 9600 \times (1 - 0.05) = 9600 \times 0.95 = Rs.\, 9120$.
Now, she purchases another item with this amount, so $CP_2 = Rs.\, 9120$.
Selling price after $5\%$ gain $(SP_2) = 9120 \times (1 + 0.05) = 9120 \times 1.05 = Rs.\, 9576$.
Overall loss = Initial $CP - \text{Final } SP = 9600 - 9576 = Rs.\, 24$.
Therefore, there is an overall loss of $Rs.\, 24$.

(A) Let the cost price $(CP)$ be $₹ 100$ and the marked price $(MP)$ be $x$.
According to the problem,the selling price $(SP)$ is $\frac{3}{4}$ of the marked price,so $SP = \frac{3}{4}x$.
Since there is a gain of $25 \%$,the selling price is also $125 \%$ of the cost price:
$SP = 1.25 \times 100 = ₹ 125$.
Equating the two expressions for $SP$:
$\frac{3}{4}x = 125$
Solving for $x$:
$x = \frac{125 \times 4}{3} = \frac{500}{3}$.
Now,we find the ratio of the marked price $(MP)$ to the cost price $(CP)$:
$Ratio = \frac{MP}{CP} = \frac{500/3}{100} = \frac{500}{300} = \frac{5}{3}$.
Therefore,the ratio is $5:3$.

(B) Let the initial marked price be $100$.
After a $10 \%$ discount,the price becomes $100 - 10 = 90$.
After a $20 \%$ discount on $90$,the price becomes $90 - (20 \% \text{ of } 90) = 90 - 18 = 72$.
After a $50 \%$ discount on $72$,the price becomes $72 - (50 \% \text{ of } 72) = 72 - 36 = 36$.
The final price is $36$.
The total discount is $100 - 36 = 64 \%$.

(D) To find the minimum selling price,we must identify the scheme that offers the maximum total discount.
$I.$ Equivalent discount $= 10 + 10 - \frac{10 \times 10}{100} = 20 - 1 = 19 \%$
$II.$ Equivalent discount $= 12 + 8 - \frac{12 \times 8}{100} = 20 - 0.96 = 19.04 \%$
$III.$ Equivalent discount $= 15 + 5 - \frac{15 \times 5}{100} = 20 - 0.75 = 19.25 \%$
$IV.$ Equivalent discount $= 20 \%$
Comparing the discounts: $19 \% < 19.04 \% < 19.25 \% < 20 \%$.
Since scheme $IV$ offers the highest discount of $20 \%$,the selling price will be the minimum under scheme $IV$.

(B) The formula for depreciation is $A = P(1 - \frac{R}{100})^T$,where $A$ is the present value,$P$ is the initial value,$R$ is the rate of depreciation,and $T$ is the time in years.
Given $A = 729$,$R = 10$,and $T = 3$.
Substituting the values: $729 = P(1 - \frac{10}{100})^3$.
$729 = P(1 - 0.1)^3$.
$729 = P(0.9)^3$.
$729 = P \times 0.729$.
$P = \frac{729}{0.729} = 1000$.
Therefore,the worth of the article $3$ years ago was $Rs. 1000$.

(A) Let the number of dozens of oranges bought at each price be $x$.
The cost price $(C.P.)$ for the first batch is $40x$ and for the second batch is $30x$.
Total $C.P. = 40x + 30x = 70x$.
The total number of dozens bought is $2x$. He sold these at $Rs. 45$ per dozen.
Total selling price $(S.P.)$ $= 2x \times 45 = 90x$.
Profit is given as $Rs. 480$.
Profit $= S.P. - C.P. = 90x - 70x = 20x$.
Given $20x = 480$,so $x = 24$.
The total number of dozens bought $= 2x = 2 \times 24 = 48$ dozens.

(D) Let the cost price $(C.P.)$ of the first chair be $₹ x$ and the $C.P.$ of the second chair be $₹(900 - x)$.
According to the problem,the selling price $(S.P.)$ of the first chair is $\frac{4}{5}x$ (a loss of $\frac{1}{5}x$) and the $S.P.$ of the second chair is $\frac{5}{4}(900 - x)$ (a profit of $\frac{1}{4}(900 - x)$).
The total profit is given as $₹90$.
Total $S.P. - Total C.P. = Profit$
$(\frac{4}{5}x + \frac{5}{4}(900 - x)) - 900 = 90$
$\frac{4}{5}x + 1125 - \frac{5}{4}x = 990$
$1125 - 990 = \frac{5}{4}x - \frac{4}{5}x$
$135 = \frac{25x - 16x}{20}$
$135 = \frac{9x}{20}$
$x = \frac{135 \times 20}{9} = 15 \times 20 = 300$.
The cost of the first chair is $₹300$ and the cost of the second chair is $900 - 300 = ₹600$.
The lower priced chair costs $₹300$.

(B) Let the Selling Price $(S.P.)$ of $100$ oranges be $₹ x$.
Given that the gain is equal to the $S.P.$ of $20$ oranges.
$\therefore$ Gain $= S.P.$ of $20$ oranges $= ₹ \frac{x \times 20}{100} = ₹ \frac{x}{5}$.
We know that $Cost Price (C.P.) = S.P. - Gain$.
$\therefore C.P. = x - \frac{x}{5} = ₹ \frac{4x}{5}$.
Gain percent is calculated as: $\text{Gain percent} = \left( \frac{\text{Gain}}{C.P.} \right) \times 100$.
$\therefore \text{Gain percent} = \left( \frac{x/5}{4x/5} \right) \times 100 = \frac{1}{4} \times 100 = 25 \%$.

(C) Let the cost price $(C.P.)$ of the article be $₹ 100$ and its selling price $(S.P.)$ be $₹ x$.
According to the given condition:
$60 \% \text{ of } C.P. = 50 \% \text{ of } S.P.$
$0.60 \times 100 = 0.50 \times x$
$60 = 0.5x$
$x = \frac{60}{0.5} = 120$
Since the selling price $(₹ 120)$ is greater than the cost price $(₹ 100)$,there is a profit.
$\text{Profit} = S.P. - C.P. = 120 - 100 = ₹ 20$.
$\text{Profit } \% = \left( \frac{\text{Profit}}{C.P.} \times 100 \right) \% = \left( \frac{20}{100} \times 100 \right) \% = 20 \%$.

(D) Total Cost Price $(C.P.)$ of two horses $= 2 \times 40000 = 80000$.
Total loss on the whole transaction $= 3600$.
Total Selling Price $(S.P.)$ of two horses $= C.P. - \text{Loss} = 80000 - 3600 = 76400$.
$S.P.$ of the first horse sold at $15\%$ gain $= 40000 \times (1 + 15/100) = 40000 \times 1.15 = 46000$.
$S.P.$ of the second horse $= \text{Total } S.P. - S.P. \text{ of the first horse} = 76400 - 46000 = 30400$.

(D) Let the total number of guavas bought be $xy$.
Cost Price $(C.P.)$ of $x$ guavas $= Rs. y$.
Therefore,$C.P.$ of $xy$ guavas $= xy \times \frac{y}{x} = y^{2}$.
Selling Price $(S.P.)$ of $y$ guavas $= Rs. x$.
Therefore,$S.P.$ of $xy$ guavas $= xy \times \frac{x}{y} = x^{2}$.
Since $x > y$,then $x^{2} > y^{2}$,which implies a gain.
Gain $= S.P. - C.P. = x^{2} - y^{2}$.
Gain $\% = \left( \frac{\text{Gain}}{C.P.} \right) \times 100 = \left( \frac{x^{2} - y^{2}}{y^{2}} \right) \times 100 \%$.

(D) Total amount received by all officers $= 45 \times 25,000 = 11,25,000$.
The ratio of the amount received by each officer to each clerk is $5:3$. Therefore,the amount received by each clerk $= \frac{3}{5} \times 25,000 = 15,000$.
Total amount received by all clerks $= 80 \times 15,000 = 12,00,000$.
Total profit earned $= 11,25,000 + 12,00,000 = 23,25,000$.
Converting to lakhs,the total profit is $23.25$ lakhs.

(D) Let the cost price of the articles be $₹ 100$.
To earn a profit of $30 \%$,he labeled them at $₹ 130$.
After giving a discount of $10 \%$ on the labeled price,the selling price of the articles $= 130 - (10 \% \text{ of } 130) = 130 - 13 = ₹ 117$.
So,the actual profit percentage $= \frac{(\text{Selling Price} - \text{Cost Price})}{\text{Cost Price}} \times 100 = \frac{(117 - 100)}{100} \times 100 = 17 \%$.

(A) Initial Cost Price $(CP_1) = Rs.\, 46,000$.
Loss percentage $= 12\%$.
Selling Price $(SP_1) = CP_1 \times (1 - \frac{12}{100}) = 46,000 \times 0.88 = Rs.\, 40,480$.
Now,she purchases another item with this amount,so $CP_2 = Rs.\, 40,480$.
She sells this item at a gain of $12\%$.
Selling Price $(SP_2) = CP_2 \times (1 + \frac{12}{100}) = 40,480 \times 1.12 = Rs.\, 45,337.60$.
Overall Loss $= CP_1 - SP_2 = 46,000 - 45,337.60 = Rs.\, 662.40$.
Alternatively,using the net percentage change formula for successive loss and gain of $x\%$: Net change $= -\frac{x^2}{100} = -\frac{12^2}{100} = -1.44\%$.
Loss $= 1.44\% \text{ of } 46,000 = \frac{1.44}{100} \times 46,000 = 14.4 \times 46 = Rs.\, 662.40$.

(D) Initial cost price of the bike $= Rs. 54,000$.
Selling price after $8\%$ loss $= 54,000 \times (1 - 0.08) = 54,000 \times 0.92 = Rs. 49,680$.
Rehaan uses this amount to purchase another bike, so the new cost price $= Rs. 49,680$.
Selling price after $10\%$ profit $= 49,680 \times (1 + 0.10) = 49,680 \times 1.1 = Rs. 54,648$.
Overall profit or loss $= \text{Final Selling Price} - \text{Initial Cost Price} = 54,648 - 54,000 = Rs. 648$.
Since the result is positive, it is a profit of $Rs. 648$.

(A) Let the printed price (marked price) of the book be $₹ 100$.
Since the discount is $10 \%$,the selling price is $100 - 10 = ₹ 90$.
Let the cost price be $CP$.
The shopkeeper earns a profit of $12 \%$,so the selling price is $CP \times (1 + \frac{12}{100}) = CP \times 1.12$.
Equating the selling prices: $1.12 \times CP = 90$.
$CP = \frac{90}{1.12} = \frac{9000}{112}$.
Simplifying the fraction by dividing by $4$: $CP = \frac{2250}{28} = \frac{1125}{14}$.
The ratio of cost price to printed price is $\frac{1125}{14} : 100 = \frac{1125}{1400}$.
Dividing both by $25$: $\frac{45}{56}$.
Thus,the ratio is $45:56$.

(D) Cost price $(CP) = 5600$
Selling price $(SP) = CP \times \frac{3}{4} = 5600 \times \frac{3}{4} = 4200$
Since $SP < CP$,there is a loss.
Loss $= CP - SP = 5600 - 4200 = 1400$
Loss percentage $= \left( \frac{\text{Loss}}{CP} \right) \times 100 = \left( \frac{1400}{5600} \right) \times 100 = \frac{1}{4} \times 100 = 25 \%$
Therefore,Ravi incurred a loss of $25 \%$.

(D) The shopkeeper sells $10$ notebooks in a day,so in two weeks ($14$ days),he sells $14 \times 10 = 140$ notebooks.
Commission per notebook $= 45 \times \frac{4}{100} = Rs.\, 1.80$.
Total commission from notebooks $= 140 \times 1.80 = Rs.\, 252$.
He sells $6$ pencil boxes in a day,so in two weeks ($14$ days),he sells $14 \times 6 = 84$ pencil boxes.
Commission per pencil box $= 80 \times \frac{20}{100} = Rs.\, 16$.
Total commission from pencil boxes $= 84 \times 16 = Rs.\, 1344$.
Total commission earned $= 252 + 1344 = Rs.\, 1596$.

(D) Total cost price of $30\, Kg$ wheat $= 30 \times 45 = ₹ 1350$.
To make an overall profit of $25\%$,the total selling price must be $= 1350 \times 1.25 = ₹ 1687.50$.
Quantity sold initially $= 40\% \text{ of } 30\, Kg = 12\, Kg$.
Selling price of $12\, Kg$ wheat $= 12 \times 50 = ₹ 600$.
Remaining quantity $= 30 - 12 = 18\, Kg$.
Required selling price for the remaining $18\, Kg = 1687.50 - 600 = ₹ 1087.50$.
Selling price per $Kg$ for the remaining quantity $= \frac{1087.50}{18} \approx ₹ 60.41$.
Rounding to the nearest option,the price is $₹ 60$ per $Kg$.

(C) Let the original cost price $(C.P.)$ of the horse be $₹ x$.
The initial selling price $(S.P.)$ at a gain of $15 \%$ is $S.P. = x + 0.15x = 1.15x$.
If the horse were bought for $25 \%$ less,the new cost price $(C.P.')$ would be $x - 0.25x = 0.75x$.
If the horse were sold for $₹ 60$ less,the new selling price $(S.P.')$ would be $1.15x - 60$.
According to the problem,the new profit is $32 \%$,so $S.P.' = C.P.' \times (1 + 32/100) = 0.75x \times 1.32$.
Calculating the new $S.P.'$: $0.75x \times 1.32 = 0.99x$.
Equating the two expressions for $S.P.'$: $1.15x - 60 = 0.99x$.
Rearranging the terms: $1.15x - 0.99x = 60$.
$0.16x = 60$.
$x = 60 / 0.16 = 375$.
Therefore,the cost price of the horse was $₹ 375$.

(A) Let the cost price for $A$ be $₹ x$.
$A$ sells the article to $B$ at a gain of $25 \%$,so $B$'s cost price is $x \times (1 + 0.25) = 1.25x$.
$B$ sells the article to $C$ at a gain of $20 \%$,so $C$'s cost price is $1.25x \times (1 + 0.20) = 1.25x \times 1.2 = 1.5x$.
$C$ sells the article to $D$ at a gain of $10 \%$,so $D$'s cost price is $1.5x \times (1 + 0.10) = 1.5x \times 1.1 = 1.65x$.
Given that $D$ pays $₹ 330$,we have the equation:
$1.65x = 330$
Solving for $x$:
$x = \frac{330}{1.65} = \frac{33000}{165} = 200$.
Therefore,the cost price for $A$ was $₹ 200$.

(A) Let the cost price of the article be $₹ x$.
Given that the percentage loss is equal to the cost price,the loss percentage is $x\%$.
The selling price is given by: $\text{Selling Price} = \text{Cost Price} \times \left(1 - \frac{\text{Loss}\%}{100}\right)$.
Substituting the given values: $21 = x \times \left(1 - \frac{x}{100}\right)$.
$21 = x - \frac{x^2}{100}$.
Multiplying the entire equation by $100$: $2100 = 100x - x^2$.
Rearranging into a quadratic equation: $x^2 - 100x + 2100 = 0$.
Factoring the quadratic equation: $(x - 30)(x - 70) = 0$.
Thus,$x = 30$ or $x = 70$.
Therefore,the cost price of the article can be either $₹ 30$ or $₹ 70$.

(C) Let the cost price of each item be $x$.
There are $100$ items in total,so $50$ items were sold at $20 \%$ profit and $50$ items were sold at $40 \%$ profit.
In the first case,the total profit $= (50 \times 0.20x) + (50 \times 0.40x) = 10x + 20x = 30x$.
In the second case,if all $100$ items were sold at $25 \%$ profit,the total profit $= 100 \times 0.25x = 25x$.
According to the problem,$30x - 25x = 100$.
$5x = 100$.
$x = 20$.
Therefore,the cost price of each item is $Rs. 20$.

(C) Let the second successive discount be $x \%$.
According to the problem,the selling price after two successive discounts is given by:
$3200 \times (1 - \frac{10}{100}) \times (1 - \frac{x}{100}) = 2448$
Simplifying the equation:
$3200 \times \frac{90}{100} \times (1 - \frac{x}{100}) = 2448$
$2880 \times (1 - \frac{x}{100}) = 2448$
Now,solve for $x$:
$1 - \frac{x}{100} = \frac{2448}{2880}$
$1 - \frac{x}{100} = 0.85$
$\frac{x}{100} = 1 - 0.85 = 0.15$
$x = 15$
Therefore,the second discount is $15 \%$.

(A) Let the cost price $(C.P.)$ be $₹100$.
The marked price $(M.P.)$ is $30\%$ above the cost price,so $M.P. = 100 + (30\% \text{ of } 100) = ₹130$.
$A$ discount of $15\%$ is allowed on the marked price,so the selling price $(S.P.)$ is $130 - (15\% \text{ of } 130) = 130 - 19.5 = ₹110.5$.
The profit is calculated as $S.P. - C.P. = 110.5 - 100 = ₹10.5$.
If the profit is $₹10.5$,the $C.P.$ is $₹100$.
If the profit is $₹84$,the $C.P.$ is $\frac{84}{10.5} \times 100 = 8 \times 100 = ₹800$.

(B) Let the marked price be $₹ x$.
The selling price after giving a $5 \%$ commission on the marked price is:
$SP = x \times \left( \frac{100 - 5}{100} \right) = \frac{95x}{100} = 0.95x$.
The cost price is given as $₹ 95$,and the shopkeeper wants a profit of $10 \%$. The selling price can also be calculated as:
$SP = CP \times \left( \frac{100 + \text{Profit } \%}{100} \right) = 95 \times \left( \frac{100 + 10}{100} \right) = 95 \times 1.1 = ₹ 104.5$.
Equating the two expressions for the selling price:
$0.95x = 104.5$
$x = \frac{104.5}{0.95} = 110$.
Therefore,the marked price is $₹ 110$.

(A) Let the cost price of $1 \ kg$ $(1000 \ g)$ of sugar be $₹ x$.
Therefore,the cost price of $950 \ g$ of sugar $= ₹ \frac{950x}{1000} = ₹ 0.95x$.
According to the question,the selling price of $950 \ g$ of sugar $= ₹ x$.
Profit $= \text{Selling Price} - \text{Cost Price} = x - 0.95x = 0.05x$.
Profit $\% = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100$.
Profit $\% = \left( \frac{0.05x}{0.95x} \right) \times 100 = \frac{5}{95} \times 100 = \frac{1}{19} \times 100 = \frac{100}{19} = 5 \frac{5}{19} \%$.

(C) Let the cost price of the horse be $₹ x$.
Then,the cost price of the carriage is $₹ (20000 - x)$.
According to the problem,the horse is sold at $20 \%$ profit and the carriage at $10 \%$ loss,resulting in an overall profit of $2 \%$ on the total cost price of $₹ 20000$.
The total selling price is $20000 \times (1 + 0.02) = 20400$.
The equation is:
$1.20x + 0.90(20000 - x) = 20400$
$1.20x + 18000 - 0.90x = 20400$
$0.30x = 2400$
$x = \frac{2400}{0.30} = 8000$.
Therefore,the cost price of the horse is $₹ 8000$.

(A) Let the cost price of the article for $A$ be $₹ x$.
$A$ sells it to $B$ at $15 \%$ profit,so the selling price for $A$ (which is the cost price for $B$) is $x \times (1 + \frac{15}{100}) = 1.15x$.
$B$ sells it to $C$ at $10 \%$ loss,so the selling price for $B$ (which is the cost price for $C$) is $1.15x \times (1 - \frac{10}{100}) = 1.15x \times 0.9 = 1.035x$.
Given that $C$ pays $₹ 517.50$,we have the equation: $1.035x = 517.50$.
Solving for $x$: $x = \frac{517.50}{1.035} = 500$.
Therefore,$A$ purchased the article at $₹ 500$.

(C) Let the original selling price be $x$.
Let the cost price be $C.P$.
When the article is sold at $\frac{2}{3}x$,there is a loss of $10 \%$.
Therefore,the selling price is $90 \%$ of $C.P$.
$\frac{2}{3}x = 0.9 \times C.P.$
$C.P. = \frac{2}{3 \times 0.9}x = \frac{2}{2.7}x = \frac{20}{27}x$.
Now,the gain when sold at the original price $x$ is:
$\text{Gain} = x - \frac{20}{27}x = \frac{7}{27}x$.
$\text{Gain } \% = \left( \frac{\text{Gain}}{C.P.} \right) \times 100 = \left( \frac{\frac{7}{27}x}{\frac{20}{27}x} \right) \times 100 = \frac{7}{20} \times 100 = 35 \%$.

(C) Let the cost price of the article for $A$ be $₹ x$.
Since $A$ sells it to $B$ at a loss of $10 \%$,the selling price is $x - 0.10x = 0.90x$.
Given that $0.90x = 45000$,we find $x = \frac{45000}{0.90} = ₹ 50000$.
$B$ sells the article to $C$ at a price that would have given $A$ a profit of $10 \%$. This selling price is $1.10 \times 50000 = ₹ 55000$.
$B$ bought the article for $₹ 45000$ and sold it for $₹ 55000$.
Profit for $B = 55000 - 45000 = ₹ 10000$.
Profit percentage for $B = \left( \frac{10000}{45000} \right) \times 100 \% = \left( \frac{10}{45} \right) \times 100 \% = \left( \frac{2}{9} \right) \times 100 \% = \frac{200}{9} \%$.

(C) Let the marked price be $ 100$.
Since the cost price is $80 \%$ of the marked price,the cost price $= 0.80 \times 100 = 80$.
The discount allowed is $12 \%$ on the marked price.
Therefore,the selling price $= 100 - 12 = 88$.
Profit $= \text{Selling Price} - \text{Cost Price} = 88 - 80 = 8$.
Profit percentage $= (\frac{\text{Profit}}{\text{Cost Price}}) \times 100 = (\frac{8}{80}) \times 100 = 10 \%$.

(B) Let the list price (marked price) of the wrist watch be $₹ x$.
The selling price after allowing a discount of $10 \%$ is given by:
$SP = x \times \left(1 - \frac{10}{100}\right) = x \times 0.9 = \frac{9x}{10}$.
The cost price $(CP)$ is $₹ 450$. The merchant earns a profit of $20 \%$.
Therefore,the selling price is also given by:
$SP = CP \times \left(1 + \frac{\text{Profit} \%}{100}\right) = 450 \times \left(1 + \frac{20}{100}\right) = 450 \times 1.2 = 540$.
Equating the two expressions for the selling price:
$\frac{9x}{10} = 540$.
Solving for $x$:
$x = \frac{540 \times 10}{9} = 60 \times 10 = 600$.
Thus,the list price of the wrist watch is $₹ 600$.

(C) Let the total profit be $P$.
$A$ received $\frac{1}{3}P$ and $B$ received $\frac{1}{4}P$.
The share of $C$ is given by $P - (\frac{1}{3}P + \frac{1}{4}P) = P - (\frac{4+3}{12})P = P - \frac{7}{12}P = \frac{5}{12}P$.
Given that the share of $C$ is $Rs. 5000$,we have:
$\frac{5}{12}P = 5000$
$P = 5000 \times \frac{12}{5} = 1000 \times 12 = 12000$.
Therefore,the total profit is $Rs. 12000$.
The amount received by $A$ is $\frac{1}{3} \times 12000 = Rs. 4000$.

(A) Total number of days in two weeks $= 2 \times 7 = 14 \text{ days}$.
Total notebooks sold $= 14 \times 10 = 140$.
Commission per notebook $= 4\% \text{ of } 457 = 0.04 \times 457 = Rs. 18.28$.
Total commission from notebooks $= 140 \times 18.28 = Rs. 2559.2$.
Total pencil boxes sold $= 14 \times 6 = 84$.
Commission per pencil box $= 20\% \text{ of } 80 = 0.20 \times 80 = Rs. 16$.
Total commission from pencil boxes $= 84 \times 16 = Rs. 1344$.
Total commission earned $= 2559.2 + 1344 = Rs. 3903.2$.