Profit (P): When selling price is greater than the cost price then the seller is said to have profit. Profit is also known as gain.
Loss (L): When selling price is less than the cost price then the seller is said to have loss.
Selling price (S.P.): The price at which a store sells the goods, is called its selling price.
Cost price (C.P.): The price at which a store owner purchases goods, is called its cost price.
Take a scenario to have detailed explanation about profit, loss, selling price, and cost price:
Consider a shopkeeper. To sell something, the person need to have something in the shop. So, for example the person bought a soap with some cost price of 5 rupees from a whole sale dealer. Now, a customer comes to buy a soap and the person had to sell it for atleast 6 rupees. One rupee extra is profit. Suppose if the person sells it for 4 rupees then loss of one rupee incurs. Therefore, if selling price is more then it is profit and if cost price is more then it is loss.

Example 1: An article is purchased for Rs. 450 and sold for Rs. 500. Find the gain percent.
Solution:

Profit = Selling price (S.P.) - Cost price (C.P.) = 500 – 450 = 50.
Gain% = ([latex]\frac{50}{450} \times 100[/latex])% = [latex]\frac{100}{9}[/latex]%

Example 2: A man buys an article for Rs. 27.50 and sells it for Rs. 28.60. Find his gain percent.
Solution:

C.P.= Rs. 27.50, S.P.= Rs. 28.60.
So, Gain = Rs. (28.60 - 27.50) = Rs. 1.10.
Therfore, Gain% = ([latex]\frac{1.10}{27.50}[/latex] x 100)% = 4%.

Example 1: If a radio is purchased for Rs. 490 and sold for Rs. 465.50, find the loss percent.
Solution:

C.P.= Rs. 490, S.P.= Rs. 465.50.
Loss = Rs. (490 - 465.50) = Rs. 24.50.
Therefore, Loss% = ([latex]\frac{24.50}{490}[/latex] x 100)% = 5%.

Example 2: If the loss incurred in a transaction is [latex]\frac{3}{5}^{th}[/latex] of the selling price, find the loss percent.
Solution:

Let the selling price be [latex]x[/latex]. loss is [latex]\frac{3x}{5}[/latex].
Loss = Cost price (C.P.) - Selling price (S.P.) (Using the Formula)
Cost price (C.P.) = Selling price (S.P.) + Loss = [latex]x + \frac{3x}{5}[/latex] = [latex]\frac{8x}{5}[/latex]
Loss% = [latex]\frac{3x}{5}[/latex]/[latex]\frac{8x}{5} \times 100[/latex] = 37.5%

Example 1: The C.P. of 21 articles is equal to S.P. of 18 articles. Find the gain or loss percent.
Solution:

C.P. of 21 articles = [latex]x[/latex]
C.P. of 1 article = [latex]\frac{x}{21}[/latex]
S.P. of 18 articles = [latex]x[/latex]
S.P. of 1 article = [latex]\frac{x}{18}[/latex]
since [latex]\frac{x}{21}[/latex] < [latex]\frac{x}{18}[/latex], there is a gain
Gain = S.P. - C.P. = [latex]\frac{x}{18}[/latex] - [latex]\frac{x}{21}[/latex]
= [latex]\frac{(7x-6x)}{126}[/latex]
= [latex]\frac{x}{126}[/latex]
Gain % = ([latex]\frac{Gain \times 100}{C.P.}[/latex])
= ([latex]\frac{x}{126}[/latex] ÷ [latex]\frac{x}{21}[/latex]) x 100
= ([latex]\frac{x}{126}[/latex] x [latex]\frac{21}{x}[/latex]) x 100
= [latex]\frac{100}{6}[/latex] = [latex]\frac{50}{3}[/latex] = 16[latex]\frac{2}{3}[/latex]%

Example 2: By selling 33 meters of cloth, one gains the selling price of 11 meters. Find the gain percent.
Solution:

(S.P. of 33 m) - (C.P. of 33 m) = Gain = S.P. pf 11m.
Therefore, S.p. of 22 m = C.P. of 33 m.
Let C.P. of each meter be Re 1. Then, C.P. of 22 m = Rs. 22, S.P. of 22 m = Rs. 33.
Therefore, Gain% = ([latex]\frac{11}{22}[/latex] x 100)% = 50%.

Example 1: A vendor bought bananas at 6 for Rs. 10 and sold them at 4 for Rs. 6. Find his gain or loss percent.
Solution:

Suppose, number of bananas bought = L.C.M. of 6 and 4 = 12.
∴ C.P. = Rs. ([latex]\frac{10}{6}[/latex] x 12) = Rs. 20;
S.P. = Rs. ([latex]\frac{6}{4}[/latex] x 12) = Rs. 18.
∴ Loss% = ([latex]\frac{2}{20}[/latex] x 100)% = 10%.

Example 2: 10% loss on selling price is what percent loss on the cost price?
Solution:

Let S.P. = Rs. 100. Then, Loss = Rs. 10, C.p. = Rs. (100 + 10) = Rs. 110.
∴ Loss% = ([latex]\frac{10}{110}[/latex] x 100)% = 9[latex]\frac{1}{11}[/latex]%.

Example 1: Find S.P. when C.P. = Rs. 56.25, Gain = 20%
Solution:

S.P. = 120% of Rs. 56.25 = Rs. ([latex]\frac{120}{100}[/latex] x 56.25) = Rs. 67.50.

Example 2: Find S.P. when C.P. = 80.40, Loss = 5 %
Solution:

S.P.= 85% of Rs. 80.40 = Rs. ([latex]\frac{85}{100}[/latex] x 80.40) = Rs. 68.34.

Example 1: Find C.P. when S.P. = Rs. 40.60, Gain = 16%
Solution:

C.P. = Rs. ([latex]\frac{100}{116}[/latex] x 40.60) = Rs. 35.

Example 2: Find C.P. when S.P. = Rs. 51.70, Loss = 12%
Solution:

C.P. = Rs. ([latex]\frac{100}{88}[/latex] x 51.70) = Rs. 58.75.

Example: A man sold two flats for Rs. 6, 75,958. On one he gains 16% while on the other he loses 16%. How much does he gain or lose in the whole transaction?
Solution:

Remember: In such a case, there is always a loss. The selling price is immaterial.
∴ Loss% = [latex](\frac{Common \ Loss \ and \ Gain\%}{10})^{2}[/latex] = [latex](\frac{16}{10})^{2}[/latex]% = [latex]\frac{64}{25}[/latex]% = 2.56%.

Example: A dishonest dealer professes to sell his goods at cost price but uses a weight of 960 gms for a kg. weight. Find his gain percent.
Solution:

Gain% = [latex][\frac{Error}{(True \ Value) - (Error)} \times 100][/latex]% = ([latex]\frac{40}{960}[/latex] x 100)% = 4[latex]\frac{1}{6}[/latex]%.

1. Profit = Selling price (S.P.) - Cost price (C.P.)
2. Loss = Cost price (C.P.) - Selling price (S.P.)
3. [latex]Gain\%[/latex] = [latex]\frac{(Gain * 100)}{Cost price}[/latex]
4. [latex]Loss\%[/latex] = [latex]\frac{(Loss * 100)}{Cost price}[/latex]
5. Selling price = [latex]\frac{(100 + Gain\%)}{100}[/latex] x [latex]Cost price[/latex]
6. Selling price = [latex]\frac{(100 - Loss\%)}{100}[/latex] x [latex]Cost price[/latex]
7. Cost price = [latex]\frac{100}{(100 + Gain\%)}[/latex] x [latex]Selling price[/latex]
8. Cost price = [latex]\frac{100}{(100 - Loss\%)}[/latex] x [latex]Selling price[/latex]
9. If an object is sold at a profit of say,  [latex]25\%[/latex] then Selling price = [latex]125\%[/latex] of Cost price.
10. If an object is sold at a loss of say, [latex]25\%[/latex] then Selling price = [latex]75\%[/latex] of Cost price.
11. When a person sells two similar items, one at a gain of say, [latex]a\%[/latex], and the other at a loss of [latex]a\%[/latex], then the seller always incurs a loss given by [latex]Loss\%[/latex] = [latex](\frac{common loss and gain\%}{10})^2[/latex] = [latex](\frac{x}{10})^2[/latex]
12. If a trader professes to sell his goods at cost price, but uses false weights, then [latex]Gain\%[/latex] = [[latex]\frac{Error}{(True value) - (Error)} * 100]\%[/latex]

1. The ratio of the cost price to the selling price is 4 : 5. Find the profit Percentage?
Solution:

Given that ratio of cost price to selling price = 4 : 5
Cost price = 4
Selling price = 5
Here selling price is more than cost price. Then,
Profit = Selling price - Cost price
P = 5 - 4
P = 1
Therefore, as profit % is calculated based on cost price
Profit% = [latex]\frac{Profit}{Cost price}[/latex] x 100
⇒Profit% = [latex]\frac{1}{4}[/latex] x 100
⇒Profit% = 0.25 x 100
⇒Profit% = 25%
Therefore, Profit percentage = 25%.

2. If a girl incurs 10% loss by selling her watch for Rs.1160. At what price should the watch be sold to earn 5% profit?
Solution:

Given that
Selling price = 1160
Loss % = 10%
As it is given loss, here selling price is less than cost price
Selling price = 90% of Cost price
If 1160 -> 0.90,
then [latex]x[/latex] -> 1.00
By cross multiplying,
[latex]x[/latex] x 0.90 = 1160 x 1
⇒[latex]x[/latex] = [latex]\frac{1160}{0.90}[/latex]
⇒[latex]x[/latex] = [latex]\frac{1160 * 100}{90}[/latex]
⇒[latex]x[/latex] = 1288.88 ≅ 1289
Now Selling price = 105% of Cost price
5% of 1289 = 64.45 ≅ 65
so add 65 to 1289 = 1289 + 65 = 1354
Hence, at Rs. 1354 should the watch sold to earn 5% profit.

3. A man sold two flats for Rs. 7,25,660 each. On one he gains 8% while on the other he loss 8%. How much does he gain or lose in the whole transaction?
Solution:

Given that
Selling price = Rs. 7,25,660
Common loss and gain percentage = 8%
Consider the formula,
[latex]Loss\%[/latex] = [latex](\frac{common loss and gain\%}{10})^2[/latex]
⇒[latex]Loss\%[/latex] = [latex](\frac{8}{10})^2\%[/latex]
⇒[latex]Loss\%[/latex] = [latex](\frac{64}{100})\%[/latex]
⇒[latex]Loss\%[/latex] = [latex]0.64\%[/latex]

4. An article was sold for Rs. 38.50 with a profit of 10%. If it were sold for Rs. 30.75then what would have been the percentage of profit or loss?
Solution:

Given that
for Selling price = Rs.38.50, Profit % = 10%
for Selling price = Rs.30.75, Profit % or loss% = ?
Now, [latex]Cost price [/latex] = [latex]\frac{100}{(100 + Gain\%)}[/latex] x [latex]Selling price[/latex]
⇒[latex]Cost price [/latex] = [latex]\frac{100}{(100 + 10)}[/latex] x [latex]38.50[/latex]
⇒[latex]Cost price [/latex] = 35Rs.
Consider, Selling price = Rs.30.75, Cost price = Rs.35 then
As selling price < cost price
⇒Loss = Cost price - Selling price
⇒Loss = 35 - 30.75 = 4.25 Rs.
Therefore, [latex]Loss\%[/latex] = [latex]\frac{(Loss * 100)}{Cost price}[/latex]
⇒[latex]Loss\%[/latex] = [latex]\frac{(4.25 * 100)}{35}[/latex]
⇒[latex]Loss\%[/latex] = 12.14%
Hence, Loss % = 12.14 %

5. If a book is purchased for Rs. 500 and sold for Rs. 460, then find the loss percent?
Solution:

Given that
Cost price = Rs. 500
Selling price = Rs. 460
Consider, loss = cost price - selling price = 500 - 460 = 40 Rs.
Therefore, [latex]Loss\%[/latex] = [latex]\frac{(Loss * 100)}{Cost price}[/latex]
⇒[latex]Loss\%[/latex] = [latex]\frac{(40 * 100)}{500}[/latex]
⇒[latex]Loss\%[/latex] = 8%
So, Loss percentage = 8%