# Simple Examples That Show How to Solve the Equation for Profit

Prachi Patkar May 4, 2019
Making profits being the main goal of any business, assigns it a vital position in the field of economics. No business will run successfully without profit. Here are some examples that show how to solve the profit equation.

### Did You Know?

Every business enterprise invests some capital in order to succeed. Profit is thus termed as 'ROI' or 'return on investment'. Conjugally, higher the ROI, higher is the investment.
The success of any business or organization depends solely on its profits. The higher the profits, the greater the efficiency of the firm. Profit is a reward for any commodity sold or any transaction done in business. These commodities could be any product or service.
The price at which a commodity is bought by the business is the cost price. The price at which a commodity is sold in the market is the selling price. Profit mainly depends on the cost price and the selling price.
For any business or entity to make profit, the selling price must be greater than the cost price, i.e., the price at which a commodity is sold must be greater than the price at which it was bought.

## Profit Equation in Economics

In order to understand the profit equation, let us first understand what a function is. A function f(x) is a mathematical relation which takes x as the input and generates an output which is a discrete value.

Consider the equation given:
f(x) = 7x +1

Solving this equation for x=1
f(1) = (7 * 1) + 1
f(1) = 8

At every value of x, the value of f(x) will change.
Now that we know what a mathematical function is, let us consider CP as the cost price, SP as the selling price, and P as the profit incurred.

The profit equation can be given in terms of functions as represented:

P(x) = SP(x) - CP(x) ---------- Equation I
The profit function equation can also be expressed as follows:

Profit(x) = Selling Price(x) - Cost Price(x)
Profit(x) = Revenue(x) - Expenditure(x)

Selling price can alternatively be termed as the revenue, and cost price can also be termed as the expenditure.
The equation containing direct selling and cost price is called 'gross profit equation'.

Gross Profit = Direct Revenue - Direct Expenditure

The equation containing indirect expenditures is called 'net profit equation'.

Net Profit = Gross Profit + Indirect Incomes - Indirect Expenditures
CP(x) from Equation I can also be expressed as the sum of the variable cost and fixed cost.
CP(x) = Variable costs + Fixed Costs

P(x) = SP(x) - (Variable Cost + Fixed Cost) ---------- Equation II

Equation I and II are two formulas for solving profit equations.
Here are three possible combinations of selling and cost price in a profit equation.

Case I SP > CP Value of P is positive
Positive value of P indicates that profit is incurred. Profit is maximum when the difference between SP and CP is maximum.
Case II SP < CP Value of P is negative
Negative value of P indicates that loss has occurred. Loss is the highest when the difference between SP and CP is maximum.

Case III SP = CP Value of P is zero
This indicates a no-gain and no-loss scenario.

### Examples

Problem I: (Based on equation I) A book vendor buys three books of USD 5 each. He sells each of the book for USD 10. What is the profit?

Solution: Let us recall the formula for profit.

Profit = Selling Price - Cost Price
For one book: Profit = 10 - 5 = USD 5
For three books: Profit = USD 5 X 3 = USD 15 or Profit = (10 X 3) - (5 X 3) = 30 - 15 = 15
Problem II: (Based on equation II) Consider an ice cream stall which sells each ice cream at USD 8. The cost price of each ice cream is USD 6. The rent of the stall for a day is USD 10. Calculate the profit made if the stall sells 10 ice creams a day.
Solution: Writing the equations for selling and cost price functions.

SP(x) = (8 X 10) = 80
CP(x) = Fixed Cost + Variable Cost = 10 + (6 X 10) = 70

Substituting the values of SP(x) and CP(x) in equation II we get:
P(x) = SP(x) - CP(x) = 80 - 70 = 10

On sale of 10 ice creams, the profit made by the ice cream stall is USD 10.
Profit defines how well the business is doing in terms of its growth. From these examples, it is clear that calculation of profit is quite simple. Isn't it? This computation will come in handy for each and every business.