# Calculate a Break-Even Point

In this lesson, you will learn to calculate a break-even point with some carefully chosen word problems. What is break-even point? In economy, the break even point is when you don't make a profit and you don't lose money either. In other words, your revenue or income is equal to your expenses. Say R = revenue and C = cost

R = C

Example #1:

It costs a publishing company 50,000 dollars to make books. The 50,000 is a fixed cost or a cost that cannot change. To help the publishing company sell the books, a marketing company charges 4 dollars for each book sold. If the company charges 9 dollars per book, how many books should they sell to break even?

Solution #1:

Let x be the number of books sold

Let C be the cost of producing and selling x books

Let R be the revenue made for selling x books

R = selling price of 1 book × number of books sold

R = 9x

C = fixed cost + variable cost

variable cost = fee charged for 1 book × number of books sold

variable cost = 4x

C = 50000 + 4x

R = C

9x = 50000 + 4x

9x - 4x = 50000 + 4x - 4x

5x = 50000

x = 10000 since 5 × 10000 = 50000

The break even point is to sell 10000 books.

Example #2:

It costs a man 75 dollars to buy the things that he needs to make hot dogs. The city allows him to sell his hot dogs somewhere near the city hall. However, the city hall charges him 1 dollar for each hot dog sold

Calculate the break even point if the price he charges for 1 hot dog is \$1.50

Solution #2:

Let x be the number of hot dogs sold

Let C be the cost of buying and selling x hot dogs

Let R be the revenue made for selling x hot dogs

R = selling price of 1 hot dog × number of hot dogs sold

R = 1.50x

C = fixed cost + variable cost

variable cost = fee charged for 1 hot dog × number of hot dogs sold

variable cost = 1x

C = 75 + 1x

R = C

1.50x = 75 + 1x

1.50x - 1x = 75 + 1x - 1x

0.5x = 75

x = 150 since 0.5 × 150 = 75

The break even point is to sell 150 hot dogs.

## Calculate Break even point using a formula

A break even point formula can be derived and you can just use the formula to calculate the break even point quicker

C = fixed cost + variable cost

Let x be the number of items sold and let c (lower case c ) be the fee charged for each item sold

Variable cost = fee charged for each item sold × number of items sold

Variable cost = c × x

C = fixed cost + cx

Let s be selling price of one item

R = s × x

R = C

sx = fixed cost + cx

sx - cx = fixed cost + cx - cx

sx - cx = fixed cost

x(s - c) = fixed cost

Divide both sides of the equation by s - c

x =
fixed cost / s - c

In example 2, fixed cost = \$75, s = \$1.50, and c = \$1. To calculate break even point, just put these numbers into the formula

x =
75 / 1.50 - 1

x =
75 / 0.50
= 150

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