Aptitude Profit and Loss Theory

Basic Definitions and Formulas :

Cost price (C.P.):

This is the price at which an article is purchased.

Selling price (S.P.):

This is the price at which an article is sold.

Profit or Gain:

If the selling price is more than the cost price, the difference between them is the profit incurred.
Formula:
Profit or Gain = S.P. – C.P.

Loss:

If the selling price is less than the cost price, the difference between them is the loss incurred.
Formula:
Loss = Cost price (C.P.) – Selling Price (S.P.)

Profit or Loss is always calculated on the cost price.

Marked price:

This is the price marked as the selling price on an article, also known as the listed price.

Discount or Rebate:

This is the reduction in price offered on the marked or listed price.

Below is the list of some basic formulas used in solving questions on profit and loss:

Gain % = (Gain / CP) * 100
Loss % = (Loss / CP) * 100
SP = [(100 + Gain%) / 100] * CP
SP = [(100 – Loss %) / 100]*CP

The above two formulas can be stated as,
If an article is sold at a gain of 10%, then SP = 110% of CP.
If an article is sold at a loss of 10%, then SP = 90% of CP.

CP = [100 / (100 + Gain%)] * SP
CP = [100 / (100 – Loss%)] * SP

7 Types Of Profit And Loss Problems :

Type I: Direct Formula Based Profit And Loss Percentages :

This type is very straightforward and is formula based.
This is very easy because, you have to remember just 4 very simple formulas to solve this type.

Let CP be cost price of an item and SP be its selling price.

(i) If SP is greater than CP, then there is profit in the transaction. Profit value and percentage can be calculated using below two formulas:

(ii) If SP is lesser than CP, then there will be loss in the transaction. Loss value and percentage can be calculated using the below formulas.

Solved Examples - Easy

Ram buys a book for Rs.100 and sells it for Rs.150. Find his gain or loss percentage.

Solution:
Cost Price CP =Rs.100
Selling Price SP = Rs.150
Here, SP is greater than CP. Therefore, there is profit in the transaction.
Based on formula, you know that Profit = SP – CP = 150 – 100 = Rs. 50
You also know the formula that ​Profit Percentage = (Profit / CP) x 100%
Therefore, Profit Percentage = (50 / 100) x 100% = (1/2)x100 % = 50%

Exercise:

Type II: Profit And Loss When Selling Different Varieties Of Same Item

In this type, a seller will buy two (or more) varieties of an item at two different cost prices. Then he will sell them together (by mixing them) at common selling price.

Solved Examples - Easy

Uma bought a number of roses at 4 for a rupee and an equal number at 2 for a rupee. At what price per dozen should she sell them to make a profit of 25%?

Solution:
Uma buys two varieties of roses. Type I at 4 roses per rupee and type II at 2 roses per rupee.
CP of 4 roses of type I = 1 and
CP of 2 roses of type II = 1

Therefore, CP of 1 rose of type I = ¼ and
CP of 1 rose of type II = 1/2

Now assume that Uma had bought 1 dozen (12) roses of each variety.
Therefore, CP of 1 dozen roses of type I = ¼ x12 = 3 and
CP of 1 dozen roses of type II = 1/2 x 12 = 6

If Uma mixes 1 dozen of type I and 1 dozen of type II together,
CP of 2 dozen mixed roses = CP of 1 dozen roses of type I + CP of 1 dozen roses of type II
= 3 + 6 = Rs. 9
So, CP of 1 dozen mixed roses = 9/2 = Rs. 4.5

Let SP of 1 dozen mixed roses be X
You know that the Profit = SP – CP = X – 4.5
And Profit Percentage = Profit / CP x 100%
= (X – 4.5) / 4.5 x 100%

To answer the question, you have to find X value when profit percentage is 25. Therefore,
(X – 4.5) / 4.5 x 100 = 25
Or X – 4.5 = 25 x 4.5 / 100
Or X – 4.5 = 1.125
Or X = 5.625
Therefore, to make a profit of 25%, Uma has to sell the mixture at Rs. 5.625 per dozen

Exercise:

Type III: Same Selling Prize, Equal Profit And Loss Percentages

Assume that a vendor sells 2 items at same selling price. Also assume he makes profit in one transaction and loss in the other. Let the profit percentage in the first transaction be equal to the loss percentage in second transaction. In such case, overall there will be a loss.

Solved Examples - Easy

A man sold two bicycles at Rs.1500 each. He sold one at a loss of 23% and other at a profit of 23%. Find his profit or loss percentage.

Solution:
Whenever you see such problems where one is sold at x% loss and another at an equal x% profit, you can be sure that there will always be loss. To calculate loss %, you can use the below shortcut formula:
If one item is sold at X% profit and other at X% loss and selling prices in both the transactions are equal, then
Loss % = (X/10)2
In our example, the value of X is 23
Therefore, Loss percentage = (23/10)2 = 2.3 x 2.3 = 5.29

Exercise:

Type IV: Profit When Seller Is Not Honest And Uses False Weighing Stone Or Scale

If a seller (e.g., vegetable seller) uses false weighing stone (for example, 750 gram instead of 1 kilogram weighing stone), he will make higher profit compared to an honest seller, right?

Type IV is all about such dishonest sellers.
(Like type III, you can solve type IV questions using simple formula.)

Solved Examples - Easy

A seller uses a weighing stone of 900gms instead of 1 Kg. Find his real profit percent.

Solution:
You have to use below formula in such problems:

In question, you will see that the seller uses 900g weight instead of 1000g or 1Kg weight.
Therefore, Error = 1000 – 900 = 100
But a true weighing stone will be 1 Kg or 1000g.
Therefore, True value = 1000
If you apply above values in our Real Profit % formula, you will get
Real Profit % = 100 / (1000 – 100) x 100
= 100/900 x 100 = 11.11%

Exercise:

Type V: Multiple Transactions Based Profit And Loss Problems

In all the above types, you saw only one transaction. In the below example, you will find two or more continuous transactions.

Solved Examples - Easy

Rahul sells a bicycle to Banu at a profit of 15%. Banu sells it to Sona at a profit of 20%. If sona pays Rs.3000 for it, then the cost price of the bicycle for Rahul is.

Solution:
First, assume CP of the bicycle when Rahul bought be Rs.X.
He sells it to Banu at profit of 15%. In other words, Banu buys the bicycle from Rahul by giving 15% more than Rahul’s CP. Therefore, CP of bicycle to Banu = 15% more than CP of bicycle to Rahul

CP of bicycle to Banu = CP of bicycle to Rahul + 15/100 x CP of bicycle to Rahul
= X + 15/100 x X
= X x (115/100) …equation 1

Banu sells it to Sona at a profit of 20%. In other words, Sona buys the bicycle from Banu by giving 20% more than Banu’s CP.
Therefore, CP of bicycle to Sona = 20% more than CP of bicycle to Banu
= CP of bicycle to Banu + 20/100 x CP of bicycle to Banu
= CP of bicycle to Banu x (120/100)

But you know from equation 1 that CP of bicycle to Banu = X x (115/100). If you substitute this in above equation, you will get:
CP of bicycle to Sona = X x (115/100) x (120/100) … equation 2

In question, you can see that Sona pays Rs 3000 for the bicycle.
Or, CP of bicycle to Sona = 3000
If you substitute above value in equation 2, you will get,
3000 = 120/100 x 115/100 x X
Or X = 3000 x 100/120 x 100/115
Or X = Rs. 2173.91
Therefore, CP of bicycle to Rahul is Rs. Rs. 2173.91

Exercise:

Type VI: Marked Price And Discounts

In shops, you can see products with price mentioned on labels. This is called marked price (or printed price). If a seller gives discount on marked price, you will get this type VI problems.

Solved Examples - Easy

A vendor buys 30 pencils at the marked price of 25 pencils from a wholesaler. If he sells these pencils giving a discount of 2%, then what is his profit percentage ?

Solution:
First, let us assume that the marked price of each pencil be Rs.1
In question, you can see that the Vendor buys 30 pencils at the marked price of 25 pencils.
Therefore, CP of 30 pencils = Marked price of 25 pencils = 25 x 1 = Rs.25

Without discount, SP of 30 pencils = Marked price of 30 pencils = 30 x 1 = Rs. 30
But, the vendor sells these pencils at a discount of 2%.

Therefore, SP of 30 pencils = Marked price of 30 pencils – (2/100) of Marked price of 30 pencils
= Marked price of 30 pencils (1 – 2/100)
= Marked price of 30 pencils x 98/100
= 30 x 98/100
= Rs. 29.40

Therefore, his Profit = SP – CP
= 29.40 – 25 = Rs. 4.40
Profit % = Profit /CP x 100
= 4.40/25 x 100
= 17.6%

Exercise:

Type VII: Profit And Loss Problems With Ratio Calculations

If any of the above types is combined with ratio calculation, you will get this type VII.

Solved Examples - Easy

A vendor earns a profit of 10% on selling a book at 15% discount on the printed price (marked price). The ratio of the cost price to the printed price of the book is.

Solution:
Let the CP be Rs.100.
Vendor earns a profit of 10%. Therefore his SP will be CP + 10% of CP
SP = 100 + 10% of 100
Or SP = Rs. 110 … equation 1

Let the printed price be Rs.X
From the question, you know that SP is printed price with 15% discount.
Or SP = Printed Price – 15% Printed Price
Or SP = X – (15/100) x X
Or SP = .85X … equation 2

From equations 1 and 2, you can write,
.85X = 110
or X = 110/.85 = 11000/85 = 2200/17

Based on our assumption that CP is 100, we have found that printed price will be 2200/17
Therefore, required ratio = CP : Printed price
=100 : 2200/17
To simplify the above ratio, you can multiply both the terms by 17. So the above ratio becomes,
1700 : 2200
=17 : 22

Exercise:

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