# Concept: Profit & Loss

CONTENTS

INTRODUCTION

The concept of Profit and Loss is a simple application of percentage change that we learnt in the Percentage chapter.

In such questions we are concerned with the percentage change (profit/loss percentage) in the price of an article. When an article is bought at Rs. 100 and then sold for Rs. 120, the price changes from 100 to 120. Since the price has increased, we earned a profit of (120 - 100 =) Rs. 20 and the percentage change in price (i.e. profit %) = $\frac{\left(120 - 100\right)\right)}{100}$ × 100 = 20%

SOME BASIC DEFINITIONS

COST PRICE
The amount used in manufacturing an article or purchasing it is called cost price.

SELLING PRICE
The price at which an article is sold is called selling price (abbreviated as S.P.)

PROFIT or GAIN
When an article is sold for more than what it costs we say there is a profit or gain. Thus,
Profit = S.P. – C.P.

LOSS
When an article is sold for less than, what it costs we say there is loss. Thus,
Loss = C.P. – S.P.

Note: If the profit is negative it implies a loss.

SOME IMPORTANT RESULTS
• If Cost Price and Profit/Loss % is given we can calculate the selling price as:
S.P. = C.P. × $\left(1±\frac{P}{100}\right)$

Here, $\left(1±\frac{P}{100}\right)$ is simply the multiplication factor for change in price of the ariticle. Hence,
S.P. = C.P. × (multiplication factor)

• If Selling Price and Profit/Loss % is given we can calculate the cost price as:
C.P. = $\frac{S.P.}{\left(1±\frac{P}{100}\right)}$ =

• If Cost Price and Selling Price is given, we can calculate the % profit/loss as:
Profit/Loss % = $\frac{SP-CP}{CP}×100%$ = $\left(\frac{SP}{CP}-1\right)×100%$

If the % change is positive we call it profit %, else if it is negative we call it as loss %.

Example:
(a) If C.P. = 125, S.P. = 96, then loss =?
(b) If C.P. = 112, S.P. = 132, then Gain =?
(c) If C.P. = 120, S.P. = 90, then Loss % =?
(d) If C.P. = 80, S.P. = 100, then Gain % =?
(e) If C.P. = 90, Gain % = 10, then S.P. =?
(f) If C.P. = 20, loss % = 25, then S.P. =?
(g) If S.P. = 84, Gain % = 20, then C.P. =?

Solution:
(a) Loss = CP - SP = 125 - 96 = 29

(b) Gain = SP – CP = 132 – 112 = 20

(c) Loss = CP – SP = 120 – 90 = 30
Loss % = Loss/(C.P.) × 100% = 30/120 × 100 = 25%

(d) Gain % = Gain/(C.P.) × 100 = (SP - CP)/CP × 100
= 20/80 × 100 = 25%

(e) SP = CP × mf = 90$\left(1+\frac{10}{100}\right)$ = 99

(f) SP = CP × mf = 20$\left(1-\frac{25}{100}\right)$ = 15

(g) CP =  = $\frac{84}{\left(1+\frac{20}{100}\right)}$ = 70

Example: A sells a bicycle to B at a profit of 30% and B sells it to C at a loss of 20%. If C pays Rs. 520 for it, at what price did A buy?

Solution:
B sells the bicycle to C for Rs. 520 at a loss of 20%.

∴ Cost price for B = $\frac{520}{\left(1-\frac{20}{100}\right)}$ = 650

⇒ A's selling price is Rs. 650 and his profit % is 30% (given).

∴ Cost price for A = $\frac{650}{\left(1+\frac{30}{100}\right)}$ = 500

Alternate Method:
Price of the bicycle first increases by 30% and then decerases by 20% and finally becomes 520.

∴ 520 = (Initial Price) × $\left(1+\frac{30}{100}\right)\left(1-\frac{20}{100}\right)$

⇒ Initial Price = $\frac{520}{\frac{13}{10}×\frac{8}{10}}$ = Rs. 500

∴ A bought the bike for Rs. 500.

Example: A man sold a horse at a loss of 7%. Had he been able to sell it at a gain of 9%, it would have fetched Rs. 64 more than it did. What was the cost price?

Solution:
Let the cost price of hourse be Rs. 'x'

It is initially sold at a loss of 7%, i.e. SP = 0.93x

If the horse were sold for Rs. 64 more there would have been a gain of 9% i.e. SP = 1.09x

⇒ 0.93x + 64 = 1.09x

⇒ 0.16x = 64

⇒ x = Rs. 400

Alternately:
Due to Rs. 64 increase, there is a change of 16% (7% loss to 9% profit)

∴ 16% of CP = 64

⇒ CP = Rs. 400

Example: An article is sold at 20% profit. If its cost price is increased by Rs. 50 and at the same time if its selling price is also increased by Rs. 30, the percentage of profit decreases by 3$\frac{1}{3}$%. Find the cost price.

Solution:

Let the original cost price be Rs. x

∴ original selling price is Rs. 1.2x

⇒ (1.2x + 30) = (x + 50) × $\left(1+\frac{20-3\frac{1}{3}}{100}\right)$

⇒ 1.2x + 30 = (x + 50) × $\frac{7}{6}$

⇒ 7.2x + 180 = 7x + 350

∴ x = Rs. 850

Example: I sell 16 articles for the same money as I paid for 20. What is my gain per cent?

Solution:
16 × SP = 20 × CP

$\frac{SP}{CP}$

= $\frac{20}{16}$ = $\frac{5}{4}$

∴ % profit = $\left(\frac{SP}{CP}-1\right)×100%$  = $\left(\frac{5}{4}-1\right)×100%$ = 25%

Direct Formula: When selling price of 'a' articles is same as cost price of 'b' articles. % profit = $\left(\frac{b}{a}-1\right)×100%$

In previous example, % Profit = (20/16 - 1) × 100 = 25%

Example: If goods be purchased for Rs. 840, and one-fourth be sold at a loss of 20%, at what gain per cent should the remainder be sold so as to gain 20% on the whole transaction?

Solution:
Cost of goods = Rs. 840

CP of one-fourth of these goods = 1/4 × 840 = Rs. 210

SP of these one-fourth goods = 0.8 × 210 = 168

Overall gain % = 20%

∴ Total revenue = 1.2 × 840 = Rs. 1008

∴ Revenue for the remaining three-fourth goods = 1008 - 168 = 840

CP of these three-fourth of these goods = 3/4 × 840 = Rs. 630

∴ profit on remaining three-fourth goods = 840 - 630 = Rs. 210

⇒ % Profit = $\frac{210}{630}$ × 100 = 33$\frac{1}{3}$%

Important
The gain or loss percent is always calculated on cost price.

In some questions profit or loss percentage would be given on selling price and you would be required to calculate actual profit or loss percentage.

Example: If profit is 20% on selling price, calculate actual profit%?

Solution:
Let the sellign price be Rs. 100

∴ Profit = 20% of 100 = Rs. 20

∴ Cost price = 100 - 20 = Rs. 80

⇒ Actual profit % = 20/80 × 100 = 25%

Example: Three articles are bought at same cost price and sold at 40% profit, 30% profit and 10% loss respectively. Calcular the over profit/loss percentage?

Solution:
Let the cost price of each article be Rs. 100

Profit on first article = 40% of 100 = Rs. 40

Profit on second article = 30% of 100 = Rs. 30

Loss on third article = 10% of 100 = Rs. 10

Overall Profit on these three articles = 40 + 30 - 10 = 60

Total cost price of these three articles = 3 × 100 = Rs. 300

∴ overall profit % = 60/300 × 100 = 20%

Example: Two articles are sold at same price, such that one is sold at 20% profit and the other at 20% loss. Calcular the over profit/loss percentage?

Solution:
Let the selling price of each article be Rs. 960

Cost price of first article = 960/1.2 = Rs. 800

Cost price of second article = 960/0.8 = Rs. 1200

Total cost price of these articles = 800 + 1200 = Rs. 2000

Total selling price of these articles = 2 × 960 = Rs. 1920

Since total cost price is more than total selling price, there is a loss of 2000- 1920 = Rs. 80

∴ overall loss % = 80/2000 × 100 = 4%

SOME IMPORTANT RESULTS
• When 2 or more articles are bought at same price and sold at a%, b%, ... profit respectively, then

Overall % profit = $\frac{\left(a + b + ...\right)}{\left(no. of articles\right)}$%

i.e. Overall % profit = Simple average of individual profit/loss %.

• When exactly 2 articles are sold at same price such that one is sold for P% profit and the other at P% loss, then

Overall % loss = $\frac{{P}^{2}}{100}$%

there will always be a loss in such cases.

MARK UP AND DISCOUNT

In such questions a shopkeeper would buy an item and then fix its price by increasing it by certain percetage. And then while selling he/she would give a certain % discount on this increased price.

SOME BASIC DEFINITIONS

MARKED PRICE (MP)
List price or the price printed on the article is known as Marked Price (abbreviated as M.P.)

MARKUP PERCETNAGE (m %)
Percentage by which the shopkeeper increases the price.

Remember: Markup percentage is always calculated with cost price as the base.

DISCOUNT PERCENTAGE (d %)
While selling the shopkeeper may give a certain percentage discount on the marked price of an article, this is called discount percetage.

Remember: Discount percentage is always calculated with marked price as the base.

Suppopse a shopkeeper buys an article for Rs. 200 (Cost Price). He then marks it up by 50% (mark up %). Now the price of the article becomes Rs. 300 (Marked Price). While selling the shopkeer gives a discount of 20% (discount %) and sells it for Rs. 240 (Selling Price).

The process of marking up the price and then giving discount can be solved using the concept of successive percentage change.

Price of the article is first increased by m% and then decreased by d%. Hence, the overall percentage change = m - d - $\frac{m × d}{100}$

Example: A shopkeeper marks up his goods by 25% and then sells them at a discount of 20%. Find his profit/loss percetange?

Solution:
Method 1:
Let the cost price of each article be Rs. 100

The articles is marked up by 25%, hence its marked price = 100 × 1.25 = Rs. 125

Now the article is sold by giving a discount of 20%, hence the selling price = 125 × 0.8 = Rs. 100

Since cost price = selling price = Rs. 100, the shopkeeper earns no profit no loss

∴ % profit = 0%

Alternately:
Price of the article is first increased by 25% and then decreased by 20%.

⇒ overall % change in price = 25 - 20 - $\frac{25 × 20}{100}$ = 5 - 5 = 0%

There overall no profit no loss

Alternately:
Price of the article is first increased by 25% and then decreased by 20%.

Overall multiplication factor = $\left(1+\frac{25}{100}\right)\left(1-\frac{20}{100}\right)$ = $\frac{5}{4}$ × $\frac{4}{5}$ = 1

∴ overall percentage change = (1 - 1) × 100 = 0% i.e. no profit no loss.

DISHONEST MERCHANT

In such questions the shopkeeper cheats the customer by giving him less than the quoted quantity. Let us understand this with an example.

Example: A shopkeeper claims to sell goods at cost price but gives 900 gms for a kg. Find his profit percentage?

Solution:
Let the cost price = selling price = Rs. 1/gm

For a kg (1000 gms) he actually gives only 900 gm.

His cost price for 900 gms = 900 × 1 = Rs. 900

His selling price for 1000 gms = 1000 × 1 = Rs. 1000

∴ he earns a profit of = 1000 - 900 = Rs. 100

⇒ profit % = 100/900 × 100 = 11.11%

Example: A shopkeeper sells goods at 20% more than his cost price but gives only 800 gms for a kg. Find his profit percentage?

Solution:
Let the cost price = 10/gm
selling price = Rs. 12/gm

For a kg (1000 gms) he actually gives only 800 gm.

His cost price for 800 gms = 800 × 10 = Rs. 8000

His selling price for 1000 gms = 1000 × 12 = Rs. 12000

∴ he earns a profit of = 120000 - 8000 = Rs. 4000

⇒ profit % = 4000/8000 × 100 = 50%

Alternately:
Same question can also be solved using the concept of successive percentage change.

We know, total price = (price/gm) × (quantity)

multiplifation factor (total price) = multiplication factor (price) × multiplication factor (quantity)

[Using the concpet of successive % change]

Multiplication factor for price (20% increase) = (1 + 20/100) = 6/5

An initial quantity of 800 gms is quoted to be as 1000 gms, hence multiplication factor for quantity = 1000/800 = 5/4

∴ multiplication factor for total price = 6/5 × 5/4 = 3.2 = 1.5

⇒ % change in price = (1.5 - 1) × 100 = 50%

∴ percentage profit = 50%

Example: A shopkeeper sells goods at 10% less than his cost price but gives only 800 gms for a kg. Find his profit percentage?

Solution:
We know, total price = (price/gm) × (quantity)

multiplifation factor (total price) = multiplication factor (price) × multiplication factor (quantity)

Multiplication factor for price (10% decrease) = 9/10

Multiplication factor for quantity = 1000/800 = 5/4

∴ multiplication factor for total price = 9/10 × 5/4 = 9/8 = 1.125

⇒ % change in price = (1.125 - 1) × 100 = 12.5%

∴ percentage profit = 12.5%

Example: A shopkeeper sells goods at 20% less than his cost price but to the customer he gives quantity less than the amount he charges for. Find how much quantity he actually gives for a kg if he earns 60% profit?

Solution:
Let the shopkeeper gives x gms for a kg.

We know, total price = (price/gm) × (quantity)

multiplifation factor (total price) = multiplication factor (price) × multiplication factor (quantity)

Multiplication factor for price (20% decrease) = 8/10 = 4/5

Multiplication factor for quantity = 1000/x

∴ multiplication factor for total price = 1 + 60/100 = 8/5

$\frac{8}{5}$ = $\frac{4}{5}$ × $\frac{1000}{x}$

⇒ x = $\frac{1000 × 4}{8}$ = 500

∴ For every kg, the shopkeeper actually gives only 500 gms.

Example: A shopkeeper sells goods at his cost but cheats by 20% while buying as well as while selling. Find his profit %?

Solution:
We know, total price = (price/gm) × (quantity)

multiplifation factor (total price) = multiplication factor (price) × multiplication factor (quantity)

Multiplication factor for price (no change) = 1

while buying the shopkeer gets 1200 gms for every 1000 gms he pays for, hence multiplication factor for quantity while buying = 1200/1000 = 6/5

while selling for every 800 gms he actually gives the shopkeer charges for 1000 gms, hence multiplication factor for quantity while selling = 1000/800 = 5/4

∴ Multiplication factor for quantity = 6/5 × 5/4 = 3/2

∴ multiplication factor for total price = 1 × 3/2 = 3/2 = 1.5

⇒ % change in total price = (1.5 - 1) × 100 = 50%

∴ percentage profit for the shopkeeper = 50%

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