Profit and Loss Questions in English

(C) Let the costs of the two articles be $x$ and $y$ respectively.
According to the problem,the discount on the first article is equal to the discount on the second article.
Therefore,$15 \% \text{ of } x = 20 \% \text{ of } y$.
$\frac{15}{100} \times x = \frac{20}{100} \times y$
$\frac{x}{y} = \frac{20}{15} = \frac{4}{3}$.
This means the ratio of the costs of the two articles must be $4:3$.
Checking the options:
For option $C$,$80:60 = 4:3$.
Thus,the costs can be $Rs. 80$ and $Rs. 60$.

(B) The list price of the article is $Rs. 2000$.
The first discount is $20 \%$. After the first discount,the price becomes $2000 - (20 \% \text{ of } 2000) = 2000 - 400 = Rs. 1600$.
The second discount is $10 \%$ on the remaining price. So,the second discount is $10 \% \text{ of } 1600 = Rs. 160$.
The net selling price is $1600 - 160 = Rs. 1440$.
Alternatively,using the successive discount formula: $\text{Net Selling Price} = 2000 \times (1 - 0.20) \times (1 - 0.10) = 2000 \times 0.80 \times 0.90 = 2000 \times 0.72 = Rs. 1440$.

(C) Marked Price $(MP) = Rs. 720$
Selling Price $(SP) = Rs. 550.80$
First discount $= 10\%$
Price after first discount $= 720 \times (1 - 0.10) = 720 \times 0.90 = Rs. 648$
Let the second discount be $x\%$.
Price after second discount $= 648 \times (1 - \frac{x}{100}) = 550.80$
$1 - \frac{x}{100} = \frac{550.80}{648}$
$1 - \frac{x}{100} = 0.85$
$\frac{x}{100} = 1 - 0.85 = 0.15$
$x = 15\%$
Therefore,the second discount rate is $15\%$.

(D) Let the marked price be $M$ and the cost price be $C$.
Given that the selling price $S = \frac{2}{5} M$.
Since there is a loss of $25 \%$,the selling price is $75 \%$ of the cost price,i.e.,$S = 0.75 C = \frac{3}{4} C$.
Equating the two expressions for $S$: $\frac{2}{5} M = \frac{3}{4} C$.
Therefore,the ratio of the marked price to the cost price is $\frac{M}{C} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$.
Thus,the ratio is $15:8$.

$(A)$ Let the marked price $(M.P.)$ of the article be $100$.
Given that the cost price $(C.P.)$ is $64 \%$ of the marked price, so $C.P. = 64$.
$A$ discount of $12 \%$ is allowed on the marked price.
Therefore, the selling price $(S.P.)$ $= M.P. - (12 \% \text{ of } M.P.) = 100 - 12 = 88$.
Gain $= S.P. - C.P. = 88 - 64 = 24$.
Gain percent $= \left( \frac{\text{Gain}}{C.P.} \right) \times 100 = \left( \frac{24}{64} \right) \times 100$.
Gain percent $= \frac{3}{8} \times 100 = 37.5 \%$.

(A) Let the cost price $(CP)$ of the commodity be $100$.
Since the trader marks the price to include a $25 \%$ profit,the marked price $(MP)$ is $100 + 25 = 125$.
He allows a discount of $16 \%$ on the marked price.
Selling price $(SP) = MP - (16 \% \text{ of } MP) = 125 - (0.16 \times 125) = 125 - 20 = 105$.
Actual profit $= SP - CP = 105 - 100 = 5$.
Profit percentage $= (\text{Profit} / CP) \times 100 = (5 / 100) \times 100 = 5 \%$.

(C) Let the cost price of the saree be $Rs. x$.
Given that the selling price after a $5\%$ discount is $Rs. 266$.
Let the labelled price be $L$. Then,$L \times (1 - 0.05) = 266$.
$L \times 0.95 = 266 \implies L = \frac{266}{0.95} = Rs. 280$.
If no discount were given,the selling price would be equal to the labelled price,which is $Rs. 280$.
In this case,the profit earned is $12\%$ on the cost price $x$.
Selling Price = Cost Price + Profit = $x + 0.12x = 1.12x$.
Equating the selling price to the labelled price: $1.12x = 280$.
$x = \frac{280}{1.12} = \frac{28000}{112} = Rs. 250$.
Therefore,the cost price of each saree is $Rs. 250$.

(D) Given,selling price $(S.P.)$ after $10 \%$ discount $= Rs. 45$.
Let the marked price be $M.P.$. Since $10 \%$ discount is given,$S.P. = M.P. \times (1 - 0.10) = 0.9 \times M.P.$
$45 = 0.9 \times M.P. \implies M.P. = \frac{45}{0.9} = Rs. 50$.
If no discount is given,the new selling price $(S.P._{new})$ will be equal to the marked price,so $S.P._{new} = Rs. 50$.
Now,the shopkeeper earns a $50 \%$ profit on the original $S.P.$ of $Rs. 45$.
Cost price $(C.P.)$ $= \frac{S.P.}{1 + \text{profit } \%} = \frac{45}{1 + 0.50} = \frac{45}{1.5} = Rs. 30$.
If no discount is given,the profit percentage is calculated as:
$\text{Profit } \% = \frac{S.P._{new} - C.P.}{C.P.} \times 100$
$\text{Profit } \% = \frac{50 - 30}{30} \times 100 = \frac{20}{30} \times 100 = \frac{2}{3} \times 100 = 66 \frac{2}{3} \%$.

(D) Given: Selling Price $(S.P.)$ = $Rs. 17940$,Profit $\% = 19.6 \%$,Discount $\% = 8 \%$.
Step $1$: Calculate the Cost Price $(C.P.)$.
$C.P. = \frac{S.P.}{1 + \frac{\text{Profit } \%}{100}} = \frac{17940}{1.196} = Rs. 15000$.
Step $2$: Calculate the Marked Price $(M.P.)$.
Since $S.P. = M.P. \times (1 - \frac{\text{Discount } \%}{100})$,we have $17940 = M.P. \times 0.92$.
$M.P. = \frac{17940}{0.92} = Rs. 19500$.
Step $3$: Calculate profit percentage if no discount is offered.
If no discount is offered,$S.P. = M.P. = Rs. 19500$.
Profit = $S.P. - C.P. = 19500 - 15000 = Rs. 4500$.
Profit $\% = (\frac{\text{Profit}}{C.P.}) \times 100 = (\frac{4500}{15000}) \times 100 = 30 \%$.

(D) Let the listed price be $L$ and the cost price be $C$.
Selling price $S = L - 0.10L = 0.9L$.
Profit is given as $S - C = 70$,which implies $0.9L - C = 70$.
We have one equation with two variables ($L$ and $C$).
Since the listed price $L$ is not provided,we cannot find the unique value of the cost price $C$.
Therefore,the data provided is inadequate.

(A) Let the cost price $(C.P.)$ of the item be $100$.
Since the marked price $(M.P.)$ is $35 \%$ above the cost price,$M.P. = 100 + 35 = 135$.
To gain $8 \%$,the selling price $(S.P.)$ must be $100 + 8 = 108$.
The discount is the difference between the marked price and the selling price: $Discount = M.P. - S.P. = 135 - 108 = 27$.
The discount percentage is calculated on the marked price: $\text{Discount } \% = (\frac{Discount}{M.P.}) \times 100$.
$\text{Discount } \% = (\frac{27}{135}) \times 100 = \frac{1}{5} \times 100 = 20 \%$.

(A) Let the labelled price be $L$.
Kunal bought the suitcase at a $15 \%$ discount,so the cost price $(C.P.)$ is $C.P. = L - 0.15L = 0.85L$.
He sold the suitcase for $Rs. 2880$ with a $20 \%$ profit on the labelled price. This means the selling price $(S.P.)$ is $S.P. = L + 0.20L = 1.20L$.
Given $S.P. = 2880$,we have $1.20L = 2880$.
$L = \frac{2880}{1.20} = 2400$.
The labelled price is $Rs. 2400$.
Now,calculate the cost price: $C.P. = 0.85 \times 2400 = 2040$.
Therefore,Kunal bought the suitcase for $Rs. 2040$.

(A) Let the cost price $(C.P.)$ of the total stock be $100$ units.
Marked price $(M.P.)$ $= 100 + 20\% \text{ of } 100 = 120$ units.
He sold half the stock ($50$ units) at the marked price:
$S.P._1 = 50 \times 1.2 = 60$ units.
He sold one quarter ($25$ units) at a discount of $20\%$ on the marked price:
$S.P._2 = 25 \times (1.2 \times 0.8) = 25 \times 0.96 = 24$ units.
He sold the rest ($25$ units) at a discount of $40\%$ on the marked price:
$S.P._3 = 25 \times (1.2 \times 0.6) = 25 \times 0.72 = 18$ units.
Total Selling Price $(S.P.)$ $= 60 + 24 + 18 = 102$ units.
Total gain $= S.P. - C.P. = 102 - 100 = 2$ units.
Therefore,the total gain percentage is $2\%$.

(A) Marked price $= Rs. 30$.
Discount $= 15\%$ of $Rs. 30 = 0.15 \times 30 = Rs. 4.50$.
Selling price of the racket $= 30 - 4.50 = Rs. 25.50$.
The shopkeeper gives a shuttlecock worth $Rs. 1.50$ free,which is an additional cost to the shopkeeper,but the net revenue received from the customer is $Rs. 25.50$.
Let the cost price of the racket be $C.P$.
Total cost incurred by the shopkeeper $= C.P. + 1.50$.
Profit is $20\%$,so Selling Price $= 1.20 \times (C.P. + 1.50)$.
$25.50 = 1.20 \times (C.P. + 1.50)$.
$C.P. + 1.50 = 25.50 / 1.20 = 21.25$.
$C.P. = 21.25 - 1.50 = Rs. 19.75$.

(C) Let the cost price be $CP$ and the selling price be $SP = Rs. \,392$.
The profit percentage is $22 \frac{1}{2} \% = 22.5 \% = 0.225$.
We know that $SP = CP \times (1 + \text{Profit percentage})$.
$392 = CP \times (1 + 0.225) = CP \times 1.225$.
$CP = \frac{392}{1.225} = \frac{392000}{1225} = Rs. \,320$.
Profit $= SP - CP = 392 - 320 = Rs. \,72$.

(C) Cost Price $(C.P.)$ of the cycle $= Rs. 1400$
Loss percentage $= 15\%$
Selling Price $(S.P.)$ $= C.P. \times (1 - \frac{\text{Loss}\%}{100})$
$S.P. = 1400 \times (1 - \frac{15}{100})$
$S.P. = 1400 \times \frac{85}{100}$
$S.P. = 14 \times 85 = Rs. 1190$

(D) Let the Cost Price $(C.P.)$ of the article be $100$.
Since the gain is $20 \%$,the Selling Price $(S.P._1)$ is $100 + 20 = 120$.
If the article is sold at double the price,the new Selling Price $(S.P._2)$ will be $120 \times 2 = 240$.
The profit is calculated as $S.P._2 - C.P. = 240 - 100 = 140$.
The percentage of profit is $\frac{\text{Profit}}{C.P.} \times 100 = \frac{140}{100} \times 100 = 140 \%$.

(C) Let the cost price $(C.P.)$ of $1$ pen be $Rs. x$.
The $C.P.$ of $12$ pens is $12x$.
According to the problem,the $S.P.$ of $8$ pens is equal to the $C.P.$ of $12$ pens,so $S.P.$ of $8$ pens $= 12x$.
The $C.P.$ of $8$ pens is $8x$.
Profit $= S.P. - C.P. = 12x - 8x = 4x$.
Gain $\% = (\text{Profit} / C.P. \text{ of } 8 \text{ pens}) \times 100$.
Gain $\% = (4x / 8x) \times 100 = (1/2) \times 100 = 50\%$.

(B) Given:
Cost Price $(C.P.) = Rs. 350$
Selling Price $(S.P.) = Rs. 392$
Profit $= S.P. - C.P. = 392 - 350 = 42$
Profit Percentage $= \frac{\text{Profit} \times 100}{C.P.}$
Profit Percentage $= \frac{42 \times 100}{350} = \frac{4200}{350} = 12\%$
Therefore,the profit percentage is $12\%$.

(A) Total Cost Price $(C.P.)$ of the bicycle = Purchase Price + Repair Charges
$C.P. = 5200 + 800 = Rs. 6000$
Selling Price $(S.P.)$ = $Rs. 5500$
Since $S.P. < C.P.$,Ramesh incurred a loss.
Loss = $C.P. - S.P. = 6000 - 5500 = Rs. 500$
Loss percentage = $\frac{\text{Loss} \times 100}{C.P.}$
Loss percentage = $\frac{500 \times 100}{6000} = \frac{50000}{6000} = \frac{50}{6} = \frac{25}{3} \%$
Loss percentage = $8 \frac{1}{3} \%$

(B) Cost price of $10$ articles $(C.P.) = Rs. 8$.
Selling price of $1$ article $= Rs. 1.25$.
Selling price of $10$ articles $(S.P.) = 1.25 \times 10 = Rs. 12.50$.
Profit $= S.P. - C.P. = 12.50 - 8 = Rs. 4.50$.
Gain $\% = \frac{\text{Gain} \times 100}{C.P.} = \frac{4.50 \times 100}{8} = \frac{450}{8} = 56.25\% = 56 \frac{1}{4}\%$.
Therefore,the gain percent is $56 \frac{1}{4}\%$.

(C) Marked Price $(M.P.) = ₹ 80$.
Selling Price $(S.P.) = ₹ 68$.
Discount $= M.P. - S.P. = 80 - 68 = ₹ 12$.
Rate of discount $= (\text{Discount} / M.P.) \times 100$.
Rate of discount $= (12 / 80) \times 100 = 15\%$.

(A) Cost price of $200$ dozen oranges $= 200 \times 10 = ₹2000$.
Transportation cost $= ₹500$.
Total Cost Price $(C.P.) = 2000 + 500 = ₹2500$.
Total number of oranges $= 200 \times 12 = 2400$.
Selling Price $(S.P.) = 2400 \times 1 = ₹2400$.
Since $C.P. > S.P.$,there is a loss.
Loss $= C.P. - S.P. = 2500 - 2400 = ₹100$.
Loss $\% = \frac{\text{Loss} \times 100}{C.P.} = \frac{100 \times 100}{2500} = 4\%$.

(C) Given: Selling Price $(S.P.)$ $= Rs. 10500$,Gain percentage $= 5 \%$.
The formula for Cost Price $(C.P.)$ when gain percentage is given is:
$C.P. = \left( \frac{100}{100 + \text{Gain} \%} \right) \times S.P.$
Substituting the values:
$C.P. = \left( \frac{100}{100 + 5} \right) \times 10500$
$C.P. = \left( \frac{100}{105} \right) \times 10500$
$C.P. = 100 \times 100 = Rs. 10000$.
Thus,the cost price of the scooter is $Rs. 10000$.

(A) Given: Cost Price $(C.P.)$ = $Rs. 1800$,Loss percentage = $10 \%$.
The formula for Selling Price $(S.P.)$ when there is a loss is:
$S.P. = C.P. \times \left( \frac{100 - \text{Loss} \%}{100} \right)$
Substituting the values:
$S.P. = 1800 \times \left( \frac{100 - 10}{100} \right)$
$S.P. = 1800 \times \left( \frac{90}{100} \right)$
$S.P. = 18 \times 90 = Rs. 1620$.
Therefore,the selling price of the camera is $Rs. 1620$.

(B) Total cost price $(C.P.)$ is calculated as follows:
Cost of $120$ rims $= 120 \times 80 = Rs. 9600$
Transportation charges $= Rs. 280$
Octroi charges $= 120 \times 0.40 = Rs. 48$
Coolie charges $= Rs. 72$
Total $C.P. = 9600 + 280 + 48 + 72 = Rs. 10000$
Desired gain $= 8 \%$
Total Selling Price $(S.P.) = C.P. \times (1 + \frac{Gain \%}{100}) = 10000 \times (1 + 0.08) = Rs. 10800$
Selling price per rim $= \frac{10800}{120} = Rs. 90$

(C) In the first case,$S.P. = Rs. 31$ and $\text{loss} \% = 7 \%$.
$\therefore \text{C.P.} = \left( \frac{100}{100 - \text{Loss} \%} \right) \times S.P. = \left( \frac{100}{100 - 7} \right) \times 31$.
$= \frac{100}{93} \times 31 = \frac{100}{3} = Rs. 33.33$.
In the second case,$\text{C.P.} = Rs. \frac{100}{3}$ and $\text{gain} \% = 5 \%$.
$\therefore S.P. = \left( \frac{100 + \text{Gain} \%}{100} \right) \times \text{C.P.}$.
$= \left( \frac{105}{100} \right) \times \frac{100}{3} = \frac{105}{3} = Rs. 35$.

(A) In the first case,we have,$S.P. = Rs. 35$ and $\text{gain} \% = 40 \%$.
$\therefore \text{Cost Price } (C.P.) = \left( \frac{100}{100 + \text{Gain} \%} \right) \times S.P.$
$C.P. = \left( \frac{100}{100 + 40} \right) \times 35 = \frac{100}{140} \times 35 = Rs. 25$.
In the second case,$C.P. = Rs. 25$ and we want a $\text{gain} \% = 60 \%$.
$\therefore \text{New } S.P. = \left( \frac{100 + \text{Gain} \%}{100} \right) \times C.P.$
$S.P. = \left( \frac{100 + 60}{100} \right) \times 25 = \frac{160}{100} \times 25 = Rs. 40$.

(C) Cost Price $(CP)$ of $6$ apples = $Rs. 20$.
Cost Price of $1$ apple = $Rs. \frac{20}{6} = Rs. \frac{10}{3}$.
Selling Price $(SP)$ of $4$ apples = $Rs. 16$.
Selling Price of $1$ apple = $Rs. \frac{16}{4} = Rs. 4$.
Profit = $SP - CP = 4 - \frac{10}{3} = \frac{12 - 10}{3} = Rs. \frac{2}{3}$.
Profit $\%$ = $\left( \frac{\text{Profit}}{CP} \right) \times 100 = \left( \frac{2/3}{10/3} \right) \times 100 = \left( \frac{2}{10} \right) \times 100 = 20 \%$.

(A) Cost Price $(CP)$ of $10$ bananas = $Rs. 14$.
Cost Price of $1$ banana = $Rs. \frac{14}{10} = Rs. 1.4$.
Selling Price $(SP)$ of $12$ bananas = $Rs. 15$.
Selling Price of $1$ banana = $Rs. \frac{15}{12} = Rs. 1.25$.
Since $CP > SP$,there is a loss.
Loss = $CP - SP = 1.4 - 1.25 = Rs. 0.15$.
Loss percentage = $\left( \frac{\text{Loss}}{CP} \times 100 \right) \% = \left( \frac{0.15}{1.4} \times 100 \right) \% = \left( \frac{15}{140} \times 100 \right) \% = \left( \frac{150}{14} \right) \% = \left( \frac{75}{7} \right) \% = 10 \frac{5}{7} \%$ loss.

(B) Cost Price $(CP)$ of $12$ eggs = $Rs. 10$.
Selling Price $(SP)$ of $10$ eggs = $Rs. 12$.
To find the profit or loss percentage,we make the number of eggs equal.
$CP$ of $60$ eggs ($LCM$ of $12$ and $10$) = $\frac{10}{12} \times 60 = Rs. 50$.
$SP$ of $60$ eggs = $\frac{12}{10} \times 60 = Rs. 72$.
Since $SP > CP$,there is a profit.
Profit = $SP - CP = 72 - 50 = Rs. 22$.
Profit $\% = \left(\frac{\text{Profit}}{CP}\right) \times 100 = \left(\frac{22}{50}\right) \times 100 = 44 \%$.
Thus,there is a $44 \%$ gain.

(C) Let the cost price of $1$ watch be $CP$ and the selling price of $1$ watch be $SP$.
Given that the cost price of $21$ watches is equal to the selling price of $18$ watches.
$21 \times CP = 18 \times SP$
$\frac{SP}{CP} = \frac{21}{18} = \frac{7}{6}$
Since $SP > CP$,there is a gain.
Gain $\% = \left( \frac{SP - CP}{CP} \right) \times 100$
Gain $\% = \left( \frac{7 - 6}{6} \right) \times 100 = \frac{1}{6} \times 100 = 16 \frac{2}{3} \%$

(B) Let the cost price of $1$ metre of thread be $CP = \$1$.
Then, the cost price of $40$ metres of thread is $CP_{40} = \$40$.
The selling price of $40$ metres of thread is $SP_{40}$.
The gain is equal to the cost price of $8$ metres of thread, which is $\$8$.
We know that $\text{Gain} = SP - CP$.
So, $8 = SP_{40} - 40$, which implies $SP_{40} = \$48$.
Gain percent is calculated as $\left( \frac{\text{Gain}}{CP} \right) \times 100$.
$\text{Gain } \% = \left( \frac{8}{40} \right) \times 100 = 20\%$.

(A) Let the total quantity of milk be $x$ units and the cost price of $1$ unit of milk be $C$.
Total cost price of $x$ units of milk = $x \times C = xC$.
According to the problem,the selling price of $\frac{2}{3}x$ units of milk is equal to the cost price of $x$ units of milk.
Selling price of $\frac{2}{3}x$ units = $xC$.
Therefore,the selling price of $1$ unit of milk = $\frac{xC}{\frac{2}{3}x} = \frac{3}{2}C = 1.5C$.
Gain = Selling Price - Cost Price = $1.5C - C = 0.5C$.
Gain percent = $\left( \frac{\text{Gain}}{\text{Cost Price}} \right) \times 100 = \left( \frac{0.5C}{C} \right) \times 100 = 50\%$.

(B) Let the cost price $(CP)$ of $1$ pencil be $x$.
Then,the $CP$ of $25$ pencils $= 25x$.
The selling price $(SP)$ of $20$ pencils is equal to the $CP$ of $25$ pencils,so $SP$ of $20$ pencils $= 25x$.
$CP$ of $20$ pencils $= 20x$.
Gain $= SP - CP = 25x - 20x = 5x$.
Gain percent $= (\text{Gain} / CP) \times 100 = (5x / 20x) \times 100 = (1/4) \times 100 = 25\%$.

(C) Given,Selling Price $(S.P._1)$ = $Rs. 1230$ and loss percentage $(x)$ = $-18 \%$.
Let the required gain or loss percentage be $y \%$.
Using the relation between Selling Price and profit/loss percentage:
$\frac{S.P._1}{100 + x} = \frac{S.P._2}{100 + y}$
Substituting the values:
$\frac{1230}{100 - 18} = \frac{1600}{100 + y}$
$\frac{1230}{82} = \frac{1600}{100 + y}$
$15 = \frac{1600}{100 + y}$
$100 + y = \frac{1600}{15} = \frac{320}{3} = 106 \frac{2}{3}$
$y = 106 \frac{2}{3} - 100 = 6 \frac{2}{3} \%$
Since the value of $y$ is positive,Mohit gains $6 \frac{2}{3} \%$ by selling it for $Rs. 1600$.

(B) Let the cost price $(C.P.)$ of the article be $x$.
Case $1$: The article is sold at a gain of $10 \%$.
$S.P._1 = C.P. + 10 \% \text{ of } C.P. = 1.10x$.
Case $2$: The article is sold at a loss of $20 \%$.
$S.P._2 = C.P. - 20 \% \text{ of } C.P. = 0.80x$.
According to the problem,the difference between the two selling prices is $Rs. 180$:
$S.P._1 - S.P._2 = 180$
$1.10x - 0.80x = 180$
$0.30x = 180$
$x = \frac{180}{0.30} = 600$.
Therefore,the cost price of the article is $Rs. 600$.

(A) Let the cost price of $36$ oranges be $C.P.$
Given,Selling Price $(S.P._1)$ of $36$ oranges $= ₹ 1$.
Loss $= 4 \%$,so $S.P._1 = C.P. \times (100 - 4) / 100 = 0.96 \times C.P.$
Therefore,$C.P. = 1 / 0.96 = 100 / 96 = ₹ 25 / 24$.
To gain $8 \%$,the new Selling Price $(S.P._2)$ for $36$ oranges should be:
$S.P._2 = C.P. \times (100 + 8) / 100 = (25 / 24) \times (108 / 100) = 108 / 96 = ₹ 9 / 8$.
So,for $₹ 9 / 8$,the person sells $36$ oranges.
For $₹ 1$,the number of oranges to be sold $= 36 / (9 / 8) = 36 \times (8 / 9) = 4 \times 8 = 32$ oranges.

(C) Let the cost price $(CP)$ of the colour $TV$ be $x$.
The initial selling price $(SP_1)$ at a loss of $10\%$ is:
$SP_1 = x - 0.10x = 0.90x$.
The desired selling price $(SP_2)$ at a profit of $12 \frac{1}{2}\%$ $(12.5\%)$ is:
$SP_2 = x + 0.125x = 1.125x$.
According to the problem,the difference between the two selling prices is $Rs. 1494$:
$SP_2 - SP_1 = 1494$
$1.125x - 0.90x = 1494$
$0.225x = 1494$
Solving for $x$:
$x = \frac{1494}{0.225} = \frac{1494000}{225} = 6640$.
Therefore,the cost price of the colour $TV$ is $Rs. 6640$.

(A) Let the cost price $(C.P.)$ of the watch be $x$.
Initially,the watch is sold at a gain of $5 \%$,so the selling price $(S.P._1)$ is $x + 0.05x = 1.05x$.
If the watch is sold for $Rs. 72$ more,the new selling price $(S.P._2)$ is $1.05x + 72$.
According to the problem,the new gain is $13 \%$,so $S.P._2 = x + 0.13x = 1.13x$.
Equating the two expressions for $S.P._2$:
$1.13x = 1.05x + 72$
$1.13x - 1.05x = 72$
$0.08x = 72$
$x = \frac{72}{0.08} = \frac{7200}{8} = 900$.
Therefore,the cost price of the watch is $Rs. 900$.

(B) Let the cost price for Sita be $x$.
Sita sells it to Gita at a gain of $17 \%$,so the selling price for Sita (which is the cost price for Gita) is $x \times (1 + 0.17) = 1.17x$.
Gita sells it to Anu at a loss of $25 \%$,so the selling price for Gita (which is the cost price for Anu) is $1.17x \times (1 - 0.25) = 1.17x \times 0.75$.
Given that Anu pays $Rs. 1842.75$,we have the equation: $1.17x \times 0.75 = 1842.75$.
$0.8775x = 1842.75$.
$x = \frac{1842.75}{0.8775} = 2100$.
Thus,Sita paid $Rs. 2100$ for the calculator.

(A) Let the cost price for $A$ be $x$.
$A$ sells the article to $B$ at a profit of $10 \%$,so the selling price for $A$ (which is the cost price for $B$) is $x \times (1 + 0.10) = 1.1x$.
$B$ sells the article to $C$ at a profit of $20 \%$,so the selling price for $B$ (which is the cost price for $C$) is $1.1x \times (1 + 0.20) = 1.1x \times 1.2 = 1.32x$.
Given that $C$ pays $₹ 924$,we have the equation: $1.32x = 924$.
Solving for $x$: $x = \frac{924}{1.32} = \frac{92400}{132} = 700$.
Thus,the amount $A$ paid is $₹ 700$.

(A) Let the cost price for $A$ be $x$.
$A$ sells to $B$ at $20 \%$ gain,so $B$ buys it for $x \times (1 + \frac{20}{100}) = x \times \frac{120}{100} = 1.2x$.
$B$ sells to $C$ at $10 \%$ gain,so $C$ buys it for $1.2x \times (1 + \frac{10}{100}) = 1.2x \times 1.1 = 1.32x$.
$C$ sells to $D$ at $12 \frac{1}{2} \%$ gain,which is $12.5 \%$. So $D$ buys it for $1.32x \times (1 + \frac{12.5}{100}) = 1.32x \times 1.125 = 1.485x$.
Given that $D$ pays $Rs. 29.70$,we have $1.485x = 29.70$.
$x = \frac{29.70}{1.485} = 20$.
Therefore,the cost price for $A$ is $Rs. 20$.

(B) Let the cost price for Rajesh be $x$.
Rajesh sells it to Mihir at a loss of $10 \%$,so the selling price for Rajesh (which is the cost price for Mihir) is $x \times (1 - 0.10) = 0.9x$.
Mihir sells it to Shiv at a loss of $20 \%$,so the selling price for Mihir (which is the cost price for Shiv) is $0.9x \times (1 - 0.20) = 0.9x \times 0.8 = 0.72x$.
Given that Shiv pays $Rs. 1440$,we have the equation: $0.72x = 1440$.
Solving for $x$: $x = \frac{1440}{0.72} = \frac{144000}{72} = 2000$.
Therefore,Rajesh bought the tape recorder for $Rs. 2000$.

(A) Let the original cost price of the scooter be $CP_1$.
The man sells it at a $10\%$ loss,so the selling price for the man (which is the cost price for the friend,$CP_2$) is:
$CP_2 = CP_1 \times (1 - 0.10) = 0.90 \times CP_1$.
The friend sells the scooter for $₹ 54000$ and gains $20\%$. Therefore:
$SP_2 = CP_2 \times (1 + 0.20) = 1.20 \times CP_2$.
Substituting $CP_2$ in the equation:
$54000 = 1.20 \times (0.90 \times CP_1)$.
$54000 = 1.08 \times CP_1$.
Solving for $CP_1$:
$CP_1 = \frac{54000}{1.08} = 50000$.
Thus,the original cost price of the scooter is $₹ 50000$.

(C) Let the cost price for $A$ be $100$.
$A$ sells it to $B$ at a profit of $10 \%$,so the cost price for $B$ is $100 + 10 = 110$.
$B$ sells it to $C$ at a profit of $20 \%$. The profit for $B$ is $20 \%$ of $110$,which is $\frac{20}{100} \times 110 = 22$.
Thus,the selling price for $C$ is $110 + 22 = 132$.
The total profit is $132 - 100 = 32$.
Alternatively,using the formula for successive percentage change: $\text{Resultant profit } \% = \left(m + n + \frac{mn}{100}\right) \%$.
Here,$m = 10$ and $n = 20$.
Resultant profit $\% = \left(10 + 20 + \frac{10 \times 20}{100}\right) \% = (30 + 2) \% = 32 \%$.

(B) Let the initial cost price be $100$.
After a profit of $20 \%$,the wholesale dealer buys it for $100 + 20 = 120$.
The wholesale dealer sells it to a retail merchant at a loss of $5 \%$.
Loss $= 5 \% \text{ of } 120 = \frac{5}{100} \times 120 = 6$.
Selling price $= 120 - 6 = 114$.
Resultant profit $= 114 - 100 = 14$.
Since the value is positive,it is a $14 \%$ gain.
Alternatively,using the formula: $\text{Resultant } \% = \left(m + n + \frac{m \times n}{100}\right) \%$,where $m = 20$ and $n = -5$.
Resultant $\% = \left(20 - 5 + \frac{20 \times -5}{100}\right) = 15 - 1 = 14 \% \text{ gain}$.

(C) When two items are sold at the same selling price,one at a gain of $x \%$ and the other at a loss of $x \%$,there is always an overall loss.
The formula for the overall loss percentage is given by $\text{Loss} \% = \left( \frac{x}{10} \right)^2 \%$.
Given $x = 5$,the overall loss percentage is $\left( \frac{5}{10} \right)^2 \% = \left( \frac{1}{2} \right)^2 \% = \frac{1}{4} \%$.
Therefore,the man incurred a total loss of $\frac{1}{4} \%$.

(C) Let the selling price of each house be $SP = Rs. 1.995$ Lakhs.
For the first house,gain is $20 \%$. So,Cost Price $CP_1 = \frac{SP}{1 + 20/100} = \frac{1.995}{1.2} = Rs. 1.6625$ Lakhs.
For the second house,loss is $20 \%$. So,Cost Price $CP_2 = \frac{SP}{1 - 20/100} = \frac{1.995}{0.8} = Rs. 2.49375$ Lakhs.
Total Cost Price $CP = CP_1 + CP_2 = 1.6625 + 2.49375 = Rs. 4.15625$ Lakhs.
Total Selling Price $SP_{total} = 1.995 + 1.995 = Rs. 3.99$ Lakhs.
Since $CP > SP_{total}$,there is a loss.
Loss $= CP - SP_{total} = 4.15625 - 3.99 = 0.16625$ Lakhs.
Loss $\% = \left( \frac{\text{Loss}}{CP} \right) \times 100 = \left( \frac{0.16625}{4.15625} \right) \times 100 = 4 \%$.
Alternatively,when two items are sold at the same price with $x \%$ gain and $x \%$ loss,the overall result is always a loss of $\left( \frac{x}{10} \right)^2 \% = \left( \frac{20}{10} \right)^2 \% = 4 \%$ loss.

(A) Selling price of each bicycle = $Rs. 1500$.
Total selling price = $1500 + 1500 = Rs. 3000$.
Cost price of the first bicycle (gain of $25\%$) = $1500 \times \frac{100}{125} = Rs. 1200$.
Cost price of the second bicycle (loss of $20\%$) = $1500 \times \frac{100}{80} = Rs. 1875$.
Total cost price = $1200 + 1875 = Rs. 3075$.
Since total cost price $(Rs. 3075)$ > total selling price $(Rs. 3000)$,there is a loss.
Loss = $3075 - 3000 = Rs. 75$.
Loss percentage = $\frac{\text{Loss}}{\text{Total Cost Price}} \times 100 = \frac{75}{3075} \times 100 = \frac{1}{41} \times 100 = \frac{100}{41} \% = 2 \frac{18}{41} \% \text{ loss}$.