A division of a company produces "Personal Income Tax" diaries. Each diary sells for $6. The monthly fixed costs incurred by the division are $30,000, and the variable cost of producing each diary is $1.
(a) Find the break-even point for the division.
(x, y) = (6000, 36000)
(b) What should be the level of sales in order for the division to realize a 10% profit over the cost of making the diaries? (Round your answer up to the nearest whole number.)
_______ diaries
To find the break-even point for the division, we need to understand the concept of break-even analysis. This analysis helps determine the quantity of units a company needs to sell to cover its fixed and variable costs.
Let's define the variables:
- Quantity of diaries sold (x)
- Selling price per diary ($6)
- Monthly fixed costs ($30,000)
- Variable cost per diary ($1)
First, let's calculate the total cost per diary. This is the sum of the fixed cost and the variable cost:
Total cost per diary = Fixed cost + Variable cost
= $30,000 + $1
= $30,001
Next, we calculate the revenue obtained from selling x diaries:
Revenue = Selling price per diary * Quantity of diaries sold
= $6 * x
= $6x
To find the break-even point, we set the revenue equal to the total cost:
$6x = $30,001
Now, let's solve for x:
x = $30,001 / $6
x ≈ 5000
Therefore, the break-even point for the division is 5000 diaries. This means the division needs to sell at least 5000 diaries to cover its costs.
To find the level of sales required for the division to realize a 10% profit over the cost of making the diaries, we can modify the revenue equation:
Revenue = (Selling price per diary + 10% of cost per diary) * Quantity of diaries sold
= ($6 + 0.10*$1) * x
= $6.10 * x
Setting this equation equal to the total cost:
$6.10 * x = $30,001
Now, let's solve for x:
x = $30,001 / $6.10
x ≈ 4918
Therefore, the division would need to sell approximately 4918 diaries to realize a 10% profit over the cost of making the diaries.
Rounding up to the nearest whole number, the level of sales required would be 4919 diaries.