Profit and Loss Problems is related to profit, loss, selling price and cost price.

**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.

- Both profit and loss calculations are based on “cost price”.
- Always cost price is fixed and selling price is variable.

**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% = (\(\frac{50}{450} \times 100\))% = \(\frac{100}{9}\)%

**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% = (\(\frac{1.10}{27.50}\) 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% = (\(\frac{24.50}{490}\) x 100)% = 5%.

**Example 2**:

If the loss incurred in a transaction is \(\frac{3}{5}^{th}\) of the selling price, find the loss percent.

**Solution**:

- Let the selling price be \(x\). loss is \(\frac{3x}{5}\).

Loss = Cost price (C.P.) – Selling price (S.P.) (Using the Formula)

Cost price (C.P.) = Selling price (S.P.) + Loss = \(x + \frac{3x}{5}\) = \(\frac{8x}{5}\)

Loss% = \(\frac{3x}{5}\)/\(\frac{8x}{5} \times 100\) = 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 = \(x\)

C.P. of 1 article = \(\frac{x}{21}\)

S.P. of 18 articles = \(x\)

S.P. of 1 article = \(\frac{x}{18}\)

since \(\frac{x}{21}\) < \(\frac{x}{18}\), there is a gain

Gain = S.P. – C.P. = \(\frac{x}{18}\) – \(\frac{x}{21}\)

= \(\frac{(7x-6x)}{126}\)

= \(\frac{x}{126}\)

Gain % = (\(\frac{Gain \times 100}{C.P.}\))

= (\(\frac{x}{126}\) ÷ \(\frac{x}{21}\)) x 100

= (\(\frac{x}{126}\) x \(\frac{21}{x}\)) x 100

= \(\frac{100}{6}\) = \(\frac{50}{3}\) = 16\(\frac{2}{3}\)%

**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% = (\(\frac{11}{22}\) 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. (\(\frac{10}{6}\) x 12) = Rs. 20;

S.P. = Rs. (\(\frac{6}{4}\) x 12) = Rs. 18.

∴ Loss% = (\(\frac{2}{20}\) 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% = (\(\frac{10}{110}\) x 100)% = 9\(\frac{1}{11}\)%.

**Example 1**:

Find S.P. when C.P. = Rs. 56.25, Gain = 20%

**Solution**:

- S.P. = 120% of Rs. 56.25 = Rs. (\(\frac{120}{100}\) 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. (\(\frac{85}{100}\) x 80.40) = Rs. 68.34.

**Example 1**:

Find C.P. when S.P. = Rs. 40.60, Gain = 16%

**Solution**:

- C.P. = Rs. (\(\frac{100}{116}\) x 40.60) = Rs. 35.

**Example 2**:

Find C.P. when S.P. = Rs. 51.70, Loss = 12%

**Solution**:

- C.P. = Rs. (\(\frac{100}{88}\) x 51.70) = Rs. 58.75.

- Loss% = \((\frac{Common \ Loss \ and \ Gain\%}{10})^{2}\) = \((\frac{x}{10})^{2}\)

**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% = \((\frac{Common \ Loss \ and \ Gain\%}{10})^{2}\) = \((\frac{16}{10})^{2}\)% = \(\frac{64}{25}\)% = 2.56%.

- Gain% = \([\frac{Error}{(True \ Value) – (Error)} \times 100]\)%

**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% = \([\frac{Error}{(True \ Value) – (Error)} \times 100]\)% = (\(\frac{40}{960}\) x 100)% = 4\(\frac{1}{6}\)%.

2. Loss = Cost price (C.P.) – Selling price (S.P.)

3. \(Gain\%\) = \(\frac{(Gain * 100)}{Cost price}\)

4. \(Loss\%\) = \(\frac{(Loss * 100)}{Cost price}\)

5. Selling price = \(\frac{(100 + Gain\%)}{100}\) x \(Cost price\)

6. Selling price = \(\frac{(100 – Loss\%)}{100}\) x \(Cost price\)

7. Cost price = \(\frac{100}{(100 + Gain\%)}\) x \(Selling price\)

8. Cost price = \(\frac{100}{(100 – Loss\%)}\) x \(Selling price\)

9. If an object is sold at a profit of say, \(25\%\) then

Selling price = \(125\%\) of Cost price.

10. If an object is sold at a loss of say, \(25\%\) then

Selling price = \(75\%\) of Cost price.

11. When a person sells two similar items, one at a gain of

say, \(a\%\), and the other at a loss of \(a\%\), then the seller

always incurs a loss given by

\(Loss\%\) = \((\frac{common loss and gain\%}{10})^2\) = \((\frac{x}{10})^2\)

12. If a trader professes to sell his goods at cost price, but uses false weights, then

\(Gain\%\) = [\(\frac{Error}{(True value) – (Error)} * 100]\%\)

- 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% = \(\frac{Profit}{Cost price}\) x 100

⇒Profit% = \(\frac{1}{4}\) 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 \(x\) -> 1.00

By cross multiplying,

\(x\) x 0.90 = 1160 x 1

⇒\(x\) = \(\frac{1160}{0.90}\)

⇒\(x\) = \(\frac{1160 * 100}{90}\)

⇒\(x\) = 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,

\(Loss\%\) = \((\frac{common loss and gain\%}{10})^2\)

⇒\(Loss\%\) = \((\frac{8}{10})^2\%\)

⇒\(Loss\%\) = \((\frac{64}{100})\%\)

⇒\(Loss\%\) = \(0.64\%\)

**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, \(Cost price \) = \(\frac{100}{(100 + Gain\%)}\) x \(Selling price\)

⇒\(Cost price \) = \(\frac{100}{(100 + 10)}\) x \(38.50\)

⇒\(Cost price \) = 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, \(Loss\%\) = \(\frac{(Loss * 100)}{Cost price}\)

⇒\(Loss\%\) = \(\frac{(4.25 * 100)}{35}\)

⇒\(Loss\%\) = 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, \(Loss\%\) = \(\frac{(Loss * 100)}{Cost price}\)

⇒\(Loss\%\) = \(\frac{(40 * 100)}{500}\)

⇒\(Loss\%\) = 8%

So, Loss percentage = 8%