Profit and Loss Questions in English

(C) Let the selling price of each article be $SP = Rs. 640$.
For the first article,profit is $20\%$. Cost price $CP_1 = \frac{100}{100+20} \times 640 = \frac{100}{120} \times 640 = Rs. \frac{1600}{3}$.
For the second article,profit is $40\%$. Cost price $CP_2 = \frac{100}{100+40} \times 640 = \frac{100}{140} \times 640 = Rs. \frac{3200}{7}$.
Total Cost Price $CP = \frac{1600}{3} + \frac{3200}{7} = \frac{11200 + 9600}{21} = Rs. \frac{20800}{21}$.
Total Selling Price $SP = 640 + 640 = Rs. 1280$.
Total Profit $= 1280 - \frac{20800}{21} = \frac{26880 - 20800}{21} = Rs. \frac{6080}{21}$.
Overall Profit $\% = \left( \frac{\text{Total Profit}}{\text{Total CP}} \right) \times 100 = \left( \frac{6080/21}{20800/21} \right) \times 100 = \frac{6080}{20800} \times 100 = \frac{608}{208} \times 10 = \frac{380}{13} = 29 \frac{3}{13}\%$.

(B) Selling price of each article $(SP)$ = $Rs. 1040$.
Cost price of the first article $(CP_1)$ = $\frac{1040}{1 - 0.20} = \frac{1040}{0.80} = Rs. 1300$.
Cost price of the second article $(CP_2)$ = $\frac{1040}{1 - 0.10} = \frac{1040}{0.90} = Rs. 1155.56$ (approx) or $\frac{10400}{9}$.
Total $CP = 1300 + \frac{10400}{9} = \frac{11700 + 10400}{9} = \frac{22100}{9} \approx Rs. 2455.56$.
Total $SP = 1040 + 1040 = Rs. 2080$.
Total Loss = Total $CP - $ Total $SP = \frac{22100}{9} - 2080 = \frac{22100 - 18720}{9} = \frac{3380}{9}$.
Loss percentage = $\left( \frac{\text{Total Loss}}{\text{Total } CP} \right) \times 100 = \left( \frac{3380/9}{22100/9} \right) \times 100 = \frac{3380}{22100} \times 100 = \frac{3380}{221} = 15 \frac{5}{17} \%$.

(D) Selling price of each article = $Rs. 1040$.
Cost price of the first article $(CP_1)$ = $\frac{1040}{1 - 0.20} = \frac{1040}{0.8} = Rs. 1300$.
Cost price of the second article $(CP_2)$ = $\frac{1040}{1 - 0.10} = \frac{1040}{0.9} = Rs. 1155.56$.
Total Cost Price = $1300 + 1155.56 = Rs. 2455.56$.
Total Selling Price = $1040 + 1040 = Rs. 2080$.
Total Loss = $2455.56 - 2080 = Rs. 375.56$.
Overall percentage loss = $\left(\frac{375.56}{2455.56}\right) \times 100 \approx 15.29\%$. Since this is not among the options,the correct answer is $D$ (None of these).

(C) Let the cost price of $1000 \text{ g}$ of goods be $1000$ units.
Since the shopkeeper sells at a $10 \%$ loss,the selling price of $1000 \text{ g}$ is $1000 - 100 = 900$ units.
The shopkeeper uses $20 \%$ less weight,meaning he actually sells $1000 - 200 = 800 \text{ g}$ of goods.
The cost price of $800 \text{ g}$ of goods is $800$ units.
Now,the shopkeeper sells $800 \text{ g}$ for $900$ units.
Profit $= 900 - 800 = 100$ units.
Profit percentage $= (\text{Profit} / \text{Cost Price}) \times 100 = (100 / 800) \times 100 = 12.5 \% = 12 \frac{1}{2} \% \text{ gain}$.

(C) Let the true length of the scale be $100 \, cm$ and the cost price of $100 \, cm$ of cloth be $100$ units.
Since the merchant sells at a $10 \%$ loss,the selling price of $100 \, cm$ of cloth is $90$ units.
Let the actual length of the inaccurate scale be $y \, cm$.
Since the merchant uses a scale of length $y \, cm$ to sell the cloth,he is effectively selling $y \, cm$ of cloth for the price of $100 \, cm$ of cloth (which is $90$ units).
The cost price of $y \, cm$ of cloth is $y$ units.
Given that he gains $15 \%$,the selling price is $1.15 \times y$ units.
Equating the selling prices: $1.15 \times y = 90$.
$y = \frac{90}{1.15} \approx 78.26 \, cm$.
Using the standard formula $\frac{100+g}{100-l} = \frac{\text{True length}}{\text{False length}}$:
$\frac{100+15}{100-10} = \frac{100}{y} \Rightarrow \frac{115}{90} = \frac{100}{y}$.
$y = \frac{100 \times 90}{115} = \frac{9000}{115} \approx 78.26 \, cm$.

(A) Let the cost price of $1 \, cm$ of cloth be $1$ unit.
The dealer charges for $100 \, cm$ but actually gives only $80 \, cm$.
Cost price $(CP)$ for the dealer $= 80 \, \text{units}$.
Selling price $(SP)$ for the dealer $= 100 \, \text{units}$.
Profit $= SP - CP = 100 - 80 = 20 \, \text{units}$.
Gain percent $= \left( \frac{\text{Profit}}{CP} \right) \times 100 = \left( \frac{20}{80} \right) \times 100 = \frac{1}{4} \times 100 = 25 \%.$

(C) Let the cost prices of the two $CDs$ be $C_1$ and $C_2$ respectively.
Given $C_1 + C_2 = 380$.
Since both are sold at the same price,we have:
$C_1(1 - 0.22) = C_2(1 + 0.12)$
$0.78 C_1 = 1.12 C_2$
$C_1 / C_2 = 112 / 78 = 56 / 39$.
Now,$C_1 = (56 / (56 + 39)) \times 380 = (56 / 95) \times 380 = 56 \times 4 = 224$.
$C_2 = 380 - 224 = 156$.
Thus,the cost prices are $Rs. 224$ and $Rs. 156$.

(A) Marked price of the article $= Rs. 65$.
Selling price of the article $= Rs. 56.16$.
Total discount amount $= 65 - 56.16 = Rs. 8.84$.
Total discount percentage $= (8.84 / 65) \times 100 = 13.6 \%$.
Let the two successive discounts be $10 \%$ and $m \%$.
The formula for the equivalent single discount is $D_{eq} = (d_1 + d_2 - (d_1 \times d_2) / 100) \%$.
Substituting the values: $13.6 = 10 + m - (10 \times m) / 100$.
$13.6 = 10 + m - 0.1m$.
$3.6 = 0.9m$.
$m = 3.6 / 0.9 = 4$.
Therefore,the second discount is $4 \%$.

(B) Total marked price of $250$ chairs $= 250 \times 50 = Rs. 12500$.
Successive discounts are $20 \%$,$15 \%$,and $5 \%$.
The effective price after discounts is calculated as:
Price $= 12500 \times (1 - 0.20) \times (1 - 0.15) \times (1 - 0.05)$
Price $= 12500 \times 0.80 \times 0.85 \times 0.95$
Price $= 12500 \times 0.646 = Rs. 8075$.

(A) Let the marked price be $MP$ and the cost price be $CP$.
Given that the selling price $SP = Rs. 1134$ after a discount of $19 \%$.
$SP = MP \times (1 - 0.19) = 0.81 \times MP = 1134$.
$MP = \frac{1134}{0.81} = 1400$.
If no discount were given,the selling price would be equal to the marked price,i.e.,$SP' = MP = 1400$.
In this case,the profit is $40 \%$ on the cost price.
$SP' = CP \times (1 + 0.40) = 1.40 \times CP$.
$1400 = 1.40 \times CP$.
$CP = \frac{1400}{1.40} = 1000$.
Thus,the cost price of each taperecorder is $Rs. 1000$.

(B) Arun's Cost Price $(C.P)$ = $Rs. 120$.
Arun sells it to Swati at a profit of $25\%$.
Swati's $C.P$ = $120 + (25\% \text{ of } 120) = 120 + 30 = Rs. 150$.
Divya sells the apples for $Rs. 198$ at a profit of $10\%$.
Divya's $C.P$ = $\frac{198}{1 + 0.10} = \frac{198}{1.1} = Rs. 180$.
Since Divya bought the apples from Swati,Swati's Selling Price $(S.P)$ = Divya's $C.P$ = $Rs. 180$.
Swati's profit = $S.P - C.P = 180 - 150 = Rs. 30$.
Swati's profit percentage = $(\frac{\text{Profit}}{C.P}) \times 100 = (\frac{30}{150}) \times 100 = 20\%$.

(C) Cost Price $(C.P)$ of $9$ bananas $= ₹ 8$.
Selling Price $(S.P)$ of $8$ bananas $= ₹ 9$.
To compare,we find the $C.P$ and $S.P$ for an equal number of bananas. Let the number of bananas be the Least Common Multiple $(LCM)$ of $9$ and $8$,which is $72$.
$C.P$ of $72$ bananas $= (8/9) \times 72 = ₹ 64$.
$S.P$ of $72$ bananas $= (9/8) \times 72 = ₹ 81$.
Since $S.P > C.P$,there is a profit.
Profit $= S.P - C.P = 81 - 64 = ₹ 17$.
Profit $\%$ $= (\text{Profit} / C.P) \times 100 = (17 / 64) \times 100 = 26.56 \%$.

(C) Let the Cost Price $(CP)$ of the pizza be $x$.
Given that selling at $Rs. 200$ results in a $20 \%$ loss,we have:
$CP - 20 \% \text{ of } CP = 200$
$0.80 \times CP = 200$
$CP = \frac{200}{0.80} = Rs. 250$.
To make a $10 \%$ profit,the new Selling Price $(SP)$ should be:
$SP = CP + 10 \% \text{ of } CP$
$SP = 250 + (0.10 \times 250)$
$SP = 250 + 25 = Rs. 275$.
Alternatively,using the ratio method:
$80 \% \text{ of } CP = 200$
$110 \% \text{ of } CP = \frac{200}{80} \times 110 = 2.5 \times 110 = Rs. 275$.

(D) Let the cost price be $CP$ and the selling price be $SP$.
The present profit percentage is given by $P = \frac{SP - CP}{CP} \times 100$.
According to the problem,if the selling price is tripled $(3SP)$ and the cost price is doubled $(2CP)$,the new profit is $65 \%$.
So,$\frac{3SP - 2CP}{2CP} = \frac{65}{100} = 0.65$.
$3SP - 2CP = 0.65 \times 2CP$.
$3SP - 2CP = 1.3CP$.
$3SP = 3.3CP$.
$SP = 1.1CP$.
Now,substitute $SP = 1.1CP$ into the profit formula:
$P = \frac{1.1CP - CP}{CP} \times 100 = \frac{0.1CP}{CP} \times 100 = 10 \%$.
Therefore,the present profit is $10 \%$.

(B) Let the Cost Price $(CP)$ of the coconut be $x$.
Given that the vendor makes a $10 \%$ loss,the Selling Price $(SP_1)$ is $90 \%$ of $CP$.
$0.90 \times x = 14.4$
$x = \frac{14.4}{0.90} = 16$.
So,the $CP$ of the coconut is $Rs. 16$.
To make a $25 \%$ profit,the new Selling Price $(SP_2)$ should be $125 \%$ of $CP$.
$SP_2 = 1.25 \times 16 = 20$.
Therefore,the vendor should sell the coconut at $Rs. 20$ to earn a $25 \%$ profit.

(B) Given that the selling price $(S.P.)$ of the chair is $Rs. 1386$ and the loss percentage is $23\%$.
We know that the relationship between cost price $(C.P.)$ and selling price in case of loss is: $S.P. = C.P. \times (1 - \text{Loss}\% / 100)$.
Substituting the values: $1386 = C.P. \times (1 - 23/100)$.
$1386 = C.P. \times (77/100)$.
$C.P. = (1386 \times 100) / 77$.
$C.P. = 18 \times 100 = 1800$.
Therefore,the cost price of the chair is $Rs. 1800$.

(C) Given: Selling Price $(SP_1)$ = $Rs. 1965$,Loss% = $25 \%$.
Since Loss% = $25 \%$,the Selling Price is $75 \%$ of the Cost Price $(CP)$.
$0.75 \times CP = 1965$
$CP = \frac{1965}{0.75} = Rs. 2620$.
Now,the new Selling Price $(SP_2)$ = $Rs. 3013$.
Profit = $SP_2 - CP = 3013 - 2620 = Rs. 393$.
Profit% = $(\frac{\text{Profit}}{CP}) \times 100 = (\frac{393}{2620}) \times 100 = 15 \%$.
Therefore,the profit percentage is $15 \%$.

(C) Let the original cost price $(CP)$ be $100$.
Since the profit is $150 \%$,the selling price $(SP)$ is $100 + 150 = 250$.
If the cost price increases by $25 \%$,the new cost price $(CP')$ becomes $100 + 25 = 125$.
The selling price remains $250$.
New profit = $SP - CP' = 250 - 125 = 125$.
New profit percentage = $\frac{\text{New Profit}}{\text{New } CP} \times 100 = \frac{125}{125} \times 100 = 100 \%$.

(B) Using the method of alligation:
Profit $1 = 20 \%$
Profit $2 = 10 \%$
Mean Profit $= 13 \%$
Difference $1 = |13 - 10| = 3$
Difference $2 = |20 - 13| = 7$
The ratio of mangoes sold at $20 \%$ profit to those sold at $10 \%$ profit is $3:7$.
Total parts $= 3 + 7 = 10$.
Mangoes sold at $20 \%$ profit $= \frac{3}{10} \times 200 = 60$ dozens.

(D) Let the cost price of the horse be $x$ and the cost price of the camel be $(51250 - x)$.
According to the problem,the selling price of both animals is the same.
Selling price of horse = $x + 25\% \text{ of } x = 1.25x$.
Selling price of camel = $(51250 - x) - 20\% \text{ of } (51250 - x) = 0.8(51250 - x)$.
Equating the two selling prices:
$1.25x = 0.8(51250 - x)$
$1.25x = 41000 - 0.8x$
$1.25x + 0.8x = 41000$
$2.05x = 41000$
$x = \frac{41000}{2.05} = 20000$.
Cost price of the horse = $Rs. 20,000$.
Cost price of the camel = $51250 - 20000 = Rs. 31,250$.
The cheaper animal is the horse,with a cost price of $Rs. 20,000$.

(D) Let the total weight of the groundnut be $100 \ kg$.
Total cost $= 100 \times (20 + 5) = Rs. 2500$.
Amount to be received after $20 \%$ profit $= 2500 \times 1.2 = Rs. 3000$.
Since $80 \%$ of the weight of groundnut is left and sold as cattle feed at $Rs. 12.5/kg$,the amount received from the waste $= 100 \times 0.8 \times 12.5 = Rs. 1000$.
Remaining amount to be received by selling the oil $= 3000 - 1000 = Rs. 2000$.
Since the weight of the oil extracted is $20 \%$ of $100 \ kg = 20 \ kg$,the selling price of the oil $= 2000 / 20 = Rs. 100$ per $kg$.

(D) Cost Price $(C.P.)$ of $10$ oranges $= 1$ rupee.
So,$C.P.$ of $1$ orange $= 1/10$ rupee.
Selling Price $(S.P.)$ of $8$ oranges $= 1$ rupee.
So,$S.P.$ of $1$ orange $= 1/8$ rupee.
Profit $= S.P. - C.P. = 1/8 - 1/10 = (5-4)/40 = 1/40$ rupee.
Profit $\% = (\text{Profit} / C.P.) \times 100 = (1/40) / (1/10) \times 100 = (1/40) \times 10 \times 100 = 100/4 = 25 \%$.

(C) Let the actual cost price of the goods be $100$ units.
When buying,the dealer cheats by $x \%$,meaning he gets $(100 + x)$ units for the price of $100$ units. Thus,the effective cost price per unit decreases.
When selling,the dealer cheats by $x \%$,meaning he sells $(100 - x)$ units for the price of $100$ units. Thus,the effective selling price per unit increases.
Alternatively,using the concept of successive percentage change for a gain of $x \%$ in both transactions:
Total Gain $\% = x + x + \frac{x \times x}{100} = \left(2 x + \frac{x^2}{100}\right) \%$.

(C) Cost Price $(CP)$ = $Rs. 1200$
Selling Price $(SP)$ = $Rs. 1500$
Profit = $SP - CP = 1500 - 1200 = Rs. 300$
Profit $\% = (\text{Profit} / CP) \times 100$
Profit $\% = (300 / 1200) \times 100 = (1 / 4) \times 100 = 25 \%$

(D) Let the cost price $(C.P.)$ of the machine be $Rs. x$ lakh.
When the machine is sold for $Rs. 39$ lakh,the loss is given by:
$\text{Loss} = C.P. - S.P. = (x - 39)$ lakh.
When the machine is sold for $Rs. 49$ lakh,the gain (profit) is given by:
$\text{Gain} = S.P. - C.P. = (49 - x)$ lakh.
According to the problem,the gain is three times the loss:
$(49 - x) = 3(x - 39)$
Expanding the equation:
$49 - x = 3x - 117$
Rearranging the terms to solve for $x$:
$49 + 117 = 3x + x$
$166 = 4x$
$x = \frac{166}{4} = 41.5$
Therefore,the cost price of the machine is $Rs. 41.5$ lakh.

(C) Total quantity of rice bought $= 8$ quintals.
Total cost price $(CP) = Rs. 3,600$.
Loss in transportation $= 10\%$ of $8$ quintals $= 0.8$ quintals.
Remaining quantity of rice $= 8 - 0.8 = 7.2$ quintals.
To earn a $15\%$ profit on the total investment,the total selling price $(SP)$ must be:
$SP = CP + (15\% \text{ of } CP) = 3600 + (0.15 \times 3600) = 3600 + 540 = Rs. 4,140$.
Now,to find the selling rate per quintal for the remaining $7.2$ quintals:
Selling rate $= \frac{\text{Total } SP}{\text{Remaining quantity}} = \frac{4140}{7.2} = Rs. 575$ per quintal.

(B) Total cost price of $16$ quintals of rice $= Rs. 5632$.
Quantity of rice lost $= 20 \% \text{ of } 16 = 0.20 \times 16 = 3.2 \text{ quintals}$.
Remaining quantity of rice $= 16 - 3.2 = 12.8 \text{ quintals}$.
To earn a $25 \%$ profit,the total selling price should be $125 \%$ of the cost price.
Total Selling Price $= 1.25 \times 5632 = Rs. 7040$.
Selling price per quintal $= \frac{\text{Total Selling Price}}{\text{Remaining Quantity}} = \frac{7040}{12.8} = Rs. 550$.

(D) Let the cost price $(C.P.)$ of the machine be $₹ x$ lakhs.
When the selling price $(S.P.)$ is $₹ 48$ lakhs,the loss is given by:
Loss $= C.P. - S.P. = (x - 48)$ lakhs.
When the selling price $(S.P.)$ is $₹ 60$ lakhs,the gain is given by:
Gain $= S.P. - C.P. = (60 - x)$ lakhs.
According to the problem,the gain is $5$ times the former loss:
$(60 - x) = 5 \times (x - 48)$
Expanding the equation:
$60 - x = 5x - 240$
Rearranging the terms to solve for $x$:
$60 + 240 = 5x + x$
$300 = 6x$
$x = 300 / 6$
$x = 50$
Therefore,the cost price of the machine is $₹ 50$ lakhs.

(A) Total Cost Price $(C.P.)$ = (Cost of paper) + (Transportation) + (Octroi) + (Porter charges)
$C.P. = (120 \times 80) + 280 + (120 \times 0.40) + 72$
$C.P. = 9600 + 280 + 48 + 72 = 10000$
Total $C.P. = Rs. 10,000$
To gain $8\%$ profit,the Total Selling Price $(S.P.)$ should be:
$S.P. = C.P. \times (1 + \frac{8}{100}) = 10000 \times 1.08 = Rs. 10,800$
Selling Price per ream = $\frac{10800}{120} = Rs. 90$

(D) The cost price $(CP)$ of $15$ mangoes is $1$ rupee.
Therefore,the $CP$ of $1$ mango is $\frac{1}{15}$ rupee.
We want a loss of $25 \%$,so the selling price $(SP)$ of $1$ mango should be $75 \%$ of the $CP$.
$SP = \frac{75}{100} \times \frac{1}{15} = \frac{3}{4} \times \frac{1}{15} = \frac{1}{20}$ rupee.
This means $1$ mango is sold for $\frac{1}{20}$ rupee.
Therefore,for $1$ rupee,the number of mangoes sold is $20$.

(B) Let the cost price $(C.P.)$ of the fan be $x$.
Given that the loss is $5 \%$,the selling price $(S.P.)$ is $95 \%$ of $C.P.$
$0.95 \times x = 1900$
$x = \frac{1900}{0.95} = 2000$
So,the cost price of the fan is $Rs. 2000$.
To gain $20 \%$,the new selling price should be $120 \%$ of $C.P.$
New $S.P. = 1.20 \times 2000 = 2400$.
Therefore,he should sell the fan for $Rs. 2400$ to gain $20 \%$.

(C) Let the Cost Price be $C.P.$ and the profit/loss amount be $x$.
When sold at $Rs. 310$,Profit $= 310 - C.P. = x$.
When sold at $Rs. 230$,Loss $= C.P. - 230 = x$.
Equating the two: $310 - C.P. = C.P. - 230$.
$2 \times C.P. = 310 + 230 = 540$.
$C.P. = 270$.
Now,if the selling price is $Rs. 180$,the loss is $270 - 180 = 90$.
Loss percentage $= (\text{Loss} / C.P.) \times 100 = (90 / 270) \times 100 = (1 / 3) \times 100 = 33 \frac{1}{3} \%$.

(B) Cost Price $(C.P.)$ of $9$ pens $= ₹ 1$.
Therefore,$C.P.$ of $1$ pen $= ₹ \frac{1}{9}$.
To gain $50 \%$,the Selling Price $(S.P.)$ of $1$ pen should be $C.P. \times (1 + \text{Profit} \%)$.
$S.P. = \frac{1}{9} \times (1 + 0.50) = \frac{1}{9} \times 1.5 = \frac{1.5}{9} = ₹ \frac{1}{6}$.
Since the $S.P.$ of $1$ pen is $₹ \frac{1}{6}$,it means $6$ pens are sold for $₹ 1$ to gain $50 \%$ profit.

(B) Let the cost price $(C.P.)$ of $1 \text{ g}$ be $₹ 1$.
Therefore,the $C.P.$ of $930 \text{ g}$ is $₹ 930$.
The shopkeeper sells $930 \text{ g}$ of goods but charges for $1000 \text{ g}$ $(1 \text{ kg})$.
Selling Price $(S.P.)$ of $930 \text{ g}$ is $₹ 1000$.
Profit = $S.P. - C.P. = 1000 - 930 = ₹ 70$.
Profit percentage = $\left( \frac{\text{Profit}}{C.P.} \right) \times 100 = \left( \frac{70}{930} \right) \times 100$.
Profit percentage = $\frac{700}{93} \approx 7.526 \% \approx 7.53 \%$.
Rounding to the nearest option,the profit percentage is $7.52 \%$.

(C) Let the Cost Price $(C.P.)$ of the article be $x$.
Given that the selling price $(S.P._1)$ is $Rs. 18450$ with a loss of $50 \%$.
$S.P._1 = C.P. - (50 \% \text{ of } C.P.) = 0.50 \times C.P.$
$18450 = 0.50 \times C.P.$
$C.P. = \frac{18450}{0.50} = Rs. 36900$.
Now,to earn a profit of $50 \%$,the new selling price $(S.P._2)$ should be:
$S.P._2 = C.P. + (50 \% \text{ of } C.P.) = 1.50 \times C.P.$
$S.P._2 = 1.50 \times 36900 = Rs. 55350$.

(D) Let the Cost Price $(C.P)$ be $₹ 100$.
Since the man gains $15 \%$,the initial Selling Price $(S.P_1)$ is $₹ 100 + ₹ 15 = ₹ 115$.
If he sells the calculator at triple the price,the new Selling Price $(S.P_2)$ will be $3 \times ₹ 115 = ₹ 345$.
The profit is calculated as $S.P_2 - C.P = ₹ 345 - ₹ 100 = ₹ 245$.
The profit percentage is calculated as $\frac{\text{Profit}}{C.P} \times 100 = \frac{245}{100} \times 100 = 245 \%$.

(D) Let the cost price $(C.P.)$ of $90$ pens be $x$.
Given that selling $90$ pens for $Rs. 80$ results in a loss of $20\%$.
Therefore,$80\%$ of $C.P. = 80$.
$0.80 \times x = 80 \implies x = 100$.
So,the cost price of $90$ pens is $Rs. 100$.
To gain a profit of $20\%$,the selling price $(S.P.)$ should be $120\%$ of the $C.P.$
$S.P. = 1.20 \times 100 = 120$.
Thus,the selling price of $90$ pens for $20\%$ profit is $Rs. 120$.

(C) Total quantity of cotton $= 500 \, kg$.
Total cost price $(C.P.) = Rs. 9,000$.
$10 \%$ of the cotton is spoiled,so the quantity of spoiled cotton $= 500 \times 0.10 = 50 \, kg$.
Remaining quantity of cotton $= 500 - 50 = 450 \, kg$.
To earn a $10 \%$ profit,the total selling price $(S.P.)$ must be $110 \%$ of the cost price.
Total $S.P. = 9,000 \times 1.10 = Rs. 9,900$.
Rate at which he should sell the remaining cotton $= \frac{\text{Total } S.P.}{\text{Remaining quantity}} = \frac{9,900}{450} = Rs. 22 \, \text{per } kg$.

(D) Using the method of alligation:
Let the quantity sold at $5 \%$ profit be $x$ and at $11 \%$ profit be $y$.
The mean profit is $7 \%$.
By alligation rule:
Ratio of quantities $= (11 - 7) : (7 - 5) = 4 : 2 = 2 : 1$.
Total quantity $= 1200 \ kg$.
Quantity sold at $5 \%$ profit $= \frac{2}{2+1} \times 1200 = \frac{2}{3} \times 1200 = 800 \ kg$.

(D) Given that the profit percentage is $20 \%$ and the difference between selling price $(SP)$ and cost price $(CP)$ is $Rs. 150$.
Profit $\% = \frac{SP - CP}{CP} \times 100$
Since $SP - CP = 150$,we substitute this into the formula:
$20 = \frac{150}{CP} \times 100$
$CP = \frac{150 \times 100}{20} = 150 \times 5 = Rs. 750$
Now,calculate the selling price:
$SP = CP + 150 = 750 + 150 = Rs. 900$

(B) Let the selling price be $S.P$.
Given that profit $P = \frac{1}{11} \times S.P$.
We know that $P = S.P - C.P$,where $C.P$ is the cost price.
Substituting the value of $P$:
$S.P - C.P = \frac{1}{11} S.P$
$S.P - \frac{1}{11} S.P = C.P$
$\frac{10}{11} S.P = C.P$
$S.P = \frac{11}{10} C.P$
Now,profit percent is calculated on the cost price:
$P \% = \left( \frac{P}{C.P} \right) \times 100$
$P = S.P - C.P = \frac{11}{10} C.P - C.P = \frac{1}{10} C.P$
$P \% = \left( \frac{\frac{1}{10} C.P}{C.P} \right) \times 100 = \frac{1}{10} \times 100 = 10 \%$.

(D) Given the ratio of cost price $(C.P.)$ to selling price $(S.P.)$ is $25:26$.
Let $C.P. = 25x$ and $S.P. = 26x$.
Profit = $S.P. - C.P. = 26x - 25x = x$.
Profit percentage is calculated as $\text{Profit} \% = \left( \frac{\text{Profit}}{C.P.} \right) \times 100$.
$\text{Profit} \% = \left( \frac{x}{25x} \right) \times 100 = \frac{1}{25} \times 100 = 4\%$.

(A) The total profit is $Rs. 144000$.
The ratio of division among Akram,Bipin,and Chintan is $3 : 2 : 7$.
The sum of the ratio parts is $3 + 2 + 7 = 12$.
Chintan's share is $\frac{7}{12}$ of the total profit.
Chintan's share $= \frac{7}{12} \times 144000 = 7 \times 12000 = Rs. 84000$.

(B) The ratio of the profit distribution is equal to the ratio of the product of investment and time period for each partner.
Ratio of investments $(P : Q : R) = (9100 \times 3) : (6825 \times 2) : (8190 \times 5)$
$= 27300 : 13650 : 40950$
Dividing by $13650$:
$= 2 : 1 : 3$
Total ratio sum $= 2 + 1 + 3 = 6$
Total profit $= Rs. 4158$
Profit share of $Q = (1 / 6) \times 4158 = Rs. 693$

(A) Let the total investment be $1$.
Raman's share = $\frac{2}{5}$.
Manan's share = $\frac{3}{8}$.
Kamal's share = $1 - (\frac{2}{5} + \frac{3}{8}) = 1 - (\frac{16+15}{40}) = 1 - \frac{31}{40} = \frac{9}{40}$.
The ratio of profits is equal to the ratio of investments.
Ratio = $\frac{2}{5} : \frac{3}{8} : \frac{9}{40}$.
To simplify,multiply each term by the least common multiple $(LCM)$ of $5, 8,$ and $40$,which is $40$.
Ratio = $(\frac{2}{5} \times 40) : (\frac{3}{8} \times 40) : (\frac{9}{40} \times 40) = 16 : 15 : 9$.

(A) Given the ratio of cost price $(CP)$ to selling price $(SP)$ is $CP : SP = 20 : 21$.
Let $CP = 20x$ and $SP = 21x$.
Profit = $SP - CP = 21x - 20x = x$.
Profit percent is calculated as: $P \% = (\text{Profit} / CP) \times 100$.
$P \% = (x / 20x) \times 100 = (1 / 20) \times 100 = 5 \%$.

(A) Let the initial cost price $(C.P)$ be $₹ 100$.
The initial selling price $(S.P)$ is $100 + 210 \% \text{ of } 100 = ₹ 310$.
If the cost price increases by $40 \%$,the new cost price $(C.P')$ becomes $100 + 40 \% \text{ of } 100 = ₹ 140$.
The selling price remains constant at $₹ 310$.
The new profit is $S.P - C.P' = 310 - 140 = ₹ 170$.
The profit as a percentage of the selling price is $\frac{170}{310} \times 100 \approx 54.83 \%$.
Rounding to the nearest whole number,we get approximately $55 \%$.

(C) Let the cost price $(CP)$ of the bucket be $x$.
Selling price at $8 \%$ gain $= x + 0.08x = 1.08x$.
Selling price at $8 \%$ loss $= x - 0.08x = 0.92x$.
The difference between the two selling prices is given as $Rs. 28$.
Therefore,$1.08x - 0.92x = 28$.
$0.16x = 28$.
$x = \frac{28}{0.16} = \frac{2800}{16} = 175$.
Thus,the cost price of the bucket is $Rs. 175$.

(B) Let the initial cost price of the article be $CP = x$.
Given that the trader sold it at a gain of $20 \%$,the initial selling price is $SP_1 = x + 0.20x = 1.2x$.
If the trader had purchased it for $40 \%$ more,the new cost price would be $CP_2 = x + 0.40x = 1.4x$.
If he sold it for $Rs. 24$ less,the new selling price would be $SP_2 = 1.2x - 24$.
According to the problem,the new transaction results in a loss of $20 \%$. Therefore,$SP_2 = CP_2 \times (1 - 0.20) = 1.4x \times 0.8 = 1.12x$.
Equating the two expressions for $SP_2$: $1.2x - 24 = 1.12x$.
Subtracting $1.12x$ from both sides: $1.2x - 1.12x = 24$.
$0.08x = 24$.
$x = 24 / 0.08 = 2400 / 8 = 300$.
Thus,the cost price of the article is $Rs. 300$.

(B) Cost Price $(C.P.)$ of each cycle $= ₹ 750$.
Total $C.P.$ of two cycles $= 750 + 750 = ₹ 1500$.
Selling Price $(S.P.)$ of the first cycle with $6 \%$ gain $= 750 \times (1 + 0.06) = 750 \times 1.06 = ₹ 795$.
Selling Price $(S.P.)$ of the second cycle with $4 \%$ loss $= 750 \times (1 - 0.04) = 750 \times 0.96 = ₹ 720$.
Total $S.P. = 795 + 720 = ₹ 1515$.
Since $S.P. > C.P.$,there is a profit.
Profit $= 1515 - 1500 = ₹ 15$.
Profit $\% = (\text{Profit} / \text{Total } C.P.) \times 100 = (15 / 1500) \times 100 = 1 \%$ gain.