# Profit – Loss Problems

#### Chapter 10

5 Steps - 3 Clicks

# Profit – Loss Problems

### Introduction

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

### Methods

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

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.

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%.

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$$\frac{1}{6}$$%.

### Formulae

1. Profit = Selling price (S.P.) – Cost price (C.P.)

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

### Samples

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% = $$\frac{Profit}{Cost price}$$ 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%