# Overview

Cost Price (CP) - This is the price at which an article is purchased.
Selling price (SP) - This is the price at which an article is sold.

Profit (P) - If the selling price exceeds the cost price (i.e. SP > CP), then there is profit.
Profit = SP – CP

Loss (L) - If the selling price of an article is less than its cost price (i.e. SP < CP), then there is loss.
Loss = CP – SP

## Profit/Loss percent

Percentage profit and percentage loss are always calculated on cost price (C.P.) unless otherwise stated (i.e. C.P. is the base).

### Profit percent

Profit percent = ($\frac{Profit}{CP}$) × 100

Now, SP = CP + Profit = CP + $\frac{CP × Profit \hspace{1ex} percent}{100}$ = ($\frac{100 + Profit \hspace{1ex} percent}{100}$) × CP

OR, CP = ($\frac{100}{100 + Profit \hspace{1ex} percent}$) × SP

### Loss percent

Similarly, Loss percent = ($\frac{Loss}{CP}$) × 100

Now, SP = CP - Loss = CP - $\frac{CP × Loss \hspace{1ex} percent}{100}$ = ($\frac{100 - Loss \hspace{1ex} percent}{100}$) × CP

OR, CP = ($\frac{100}{100 - Loss \hspace{1ex} percent}$) × SP

### Multiplying Factor

We saw that:
SP = ($\frac{100 + Profit \hspace{1ex} percent}{100}$) × CP OR ($\frac{100 - Loss \hspace{1ex} percent}{100}$) × CP

So, basically we have a relation SP = CP × Multiplying Factor (MF)

The MF is:

• greater than 1 in case of a profit and
• less than 1 in case of loss.

We can also write the expression as:
MF = $\frac{SP}{CP}$

From this multiplying factor, one can easily deduce the profit or loss percentage.

E.g. If CP = 16 and SP = 20, then:
MF = $\frac{SP}{CP}$ = $\frac{20}{16}$ = $\frac{5}{4}$ = 1.25, i.e. a profit percentage of 25%.

##### Q. Ajay bought a book for Rs. 50 and sold it for Rs. 60. What it his profit percentage?

Explanations :

Explanation 1: Formula Method or Percentage Method

Cost Price (C.P.) = Rs. 50 & Selling Price (S.P.) = Rs. 60
If the selling price exceeds the cost price (i.e. SP > CP), then there is a profit.

Our base → C.P.

Difference = Profit = S P - CP = 60 - 50 = Rs 10
Percentage of profit = (Profit/CP) × 100 = (10/50) × 100 = 20%

Explanation 2: Multiplying Factor Method

Multiplying Factor (M.F.) = SP/CP = 60/50 = 1.2 = 1 + 0.2
Percentage of profit = 0.2 × 100 = 20%

Explanation 3: Fraction Method

Cost Price (C.P.) = Rs. 50 & Selling Price (S.P.) = Rs. 60

S.P./C.P. = 60/50 = 6/5

Percentage of profit = (1/5) × 100 = 20%

##### Q. If a pen is sold for Rs. 24, there is a loss of 20%. What was the cost price of the pen?
(a) Rs. 30         (b) Rs. 40          (c) Rs. 36          (d) Rs. 20

Explanations :

Explanation 1: Formula Method or Percentage Method

Selling Price (S.P.) = Rs. 24
Our base → C.P. = x
So, loss = 20% of x = 0.2x

As there is a loss, the C.P. must be greater than the S.P.
Loss = CP - SP
Or 0.2x = x – 24
Or 0.8x = 24
Or x = 24/0.8 = Rs. 30

Explanation 2: Multiplying Factor Method

Multiplying Factor (M.F.) = 1 – 20% = 0.80
So, C.P. = S.P./ M.F. = 24/0.8 = Rs. 30

Explanation 3: Percentage Method

Let C.P. = Rs. 100
Loss of 20%
So, S.P. = 100 – 20% of 100 = 100 – 20 = Rs. 80

Rs. 80 ≡ Rs. 24
So, Rs. 100 ≡ (24/80) × 100 = Rs. 30