# Profit and Loss

**Introduction:** Profit and Loss. Money makes the world go round, and math is the centripetal force that enables it. Profit and Loss are the basic driving forces of the market, and here you will understand how to wield the power of mathematics and apply it to the commercial world.

## The Big Picture: What are Profit and Loss?

Let’s take the simple example of buying and selling a video game. Suppose you go to a video game store and buy a game for Rs. 1000. After playing the game, you decide to sell it for a certain amount. In case you sell it for Rs. 1100, you would be selling it for a profit of Rs. 100. On the contrary, if you sold the game at Rs. 920, you would be selling it at a loss of Rs. 80.

Here, the initial amount that you paid for the game, Rs. 1000 is the **Cost Price** (C.P.) of the game. The price that you sold it for is called the **Selling Price** (S.P.).

Considering the above two cases mathematically:

\(\begin{align}\text{C.P. = Rs.1000}\end{align}\)

**Case 1:**

\(\begin{align} & \text{When S.P.} = \text{Rs.} 1100, \text{C.P.} < \text{S.P.}\\ & \text{Profit }\left( \text{Denoted as P} \right)= \text{S.P.} - \text{C.P.} \\&\qquad \qquad \qquad \qquad \quad\;\, = 1100 - 1000 \\&\qquad \qquad \qquad \qquad \quad\;\,= \text{Rs.} 100 \end{align}\)

**Case 2:**

\(\begin{align} & \text{When S.P.} = \text{Rs.} 920, \text{C.P.} > \text{S.P.}\\ & {\text{Loss }}\left( {{\text{Denoted as L}}} \right)\; = \text{C.P.} - \text{S.P.}\\&\qquad \qquad \qquad \qquad \quad = 1000 - 920 \\&\qquad \qquad \qquad \qquad \quad = \text{Rs.80} \end{align}\)

In the above example, let us suppose that you had to send the video game from Mumbai to Delhi and had to pay Rs. 50 for the courier. Consequently, the C.P would become Rs. 1050. So:

**Case 1:**

\(\begin{align} & \text{P = S.P.} - \text{C.P. (Since C.P. < S.P.)} \\ &\;\,\,= 1100 - 1050 = \text{Rs.} 50 \end{align}\)

**Case 2:**

\(\begin{align} & \text{L = C.P.} - \text{S.P.}\\&\;\; = 1050 - 920 \\&\;\; = \text{Rs.} 130 \end{align}\)

Since the C.P increased, the profit decreased and the loss increased.

## Why Profit and Loss? Why are Profit and Loss important?

An understanding of profit and loss opens possibilities to the vast world of commercial mathematics. A field which holds immense scope for your future when it comes to understanding how business works. Once understood, the concept of profit and loss are not limited to money and commercial math but can be extended to time and time management as well.

Imagine that a restaurant spends some amount of money promoting itself. Because of its promotions it gets 200 new customers/sales. Was the promotion successful? One may jump at saying yes, because 200 new sales looks good. Here we get into profit and loss, assuming that every sale nets some profit. Does the profit from 200 sales cover the cost of the promotion? If so, by how much? If the profit from 200 sales do not cover the cost of the promotion, it would be considered a loss. This is just a simple example and an entry to a world that consists of many layers of such considerations.

### Profit & Loss Percentage

People love percentages, from your marks to discounts. They are an easy way to gauge any kind of amount, whether there is an increase or a decrease. Profits and losses are no exceptions. As a general rule of thumb, the higher the profit percentage, the better and the reverse is true for loss percentage, lower is better. As for how they are calculated, it is pretty simple:

\(\begin{align}&\text{Profit Percentage }\left( \text{Denoted as P%} \right)\\&\qquad=\left( \frac{\text{P}}{\text{C.P.}} \right) \times 100 \\&\qquad=\left( \frac{\text{S.P.} - \text{C.P.}}{\text{C.P.}} \right)\times {\rm{100}}\\&\qquad =\left( \frac{\text{S.P.}}{\text{C.P.} - 1} \right) \times {\rm{100}}\end{align}\)

\(\begin{align}&\text{Loss Percentage }\left( \text{Denoted as L% } \right)\\&\qquad=\left( \frac{\text{L}}{\text{C.P.} \times 100} \right) \\&\qquad=\left( \frac{\text{C.P.} - \text{S.P.}}{\text{C.P.}} \right) \times 100\\&\qquad=\left( 1 - \frac{\text{S.P.}}{\text{C.P.}}\right) \times 100\end{align}\)

Always remember that both percentages are calculated relative to C.P. and NOT S.P.

Don’t worry if your profit percentage exceeds the 100% mark. There are many cases where people sell goods way above the C.P of the article. Occasionally this is the case with big clothing brands. They charge exorbitant amounts for just the brand name. A simple t shirt that might just cost about Rs. 200 can be stamped with a brand name and be sold for Rs. 1000. In such a case:

\(\begin{align}{\text{Profit}}=1000-200={\text{Rs}}.{\rm{ }}800\end{align}\)

Since profit percentage is calculated with C.P. as reference:

\(\begin{align}{\text{P% }} = \left( {\frac{{800}}{{200}}} \right) \times 100 = 400{\rm{\% }}{\rm{.}}\end{align}\)

Such cases do exist, so there is no need to worry if the percentage goes above 100%.

## Tips and Tricks

- It is often the case that we buy something and spend extra money on repairing it, transporting it, etc. These
**extra expenses are called overhead expenses. Overhead expenses add to the cost price**. The repairs/upgrades etc., add to the cost price of an item. - Often
**when calculating the profit or loss percentage, students take the selling price in the denominator instead of the cost price**. Reiterate often that profit and loss percentages are always calculated over cost price (CP)

Profit % = (Profit ÷ Cost Price) × 100