The Congress Company has identified two methods for producing playing cards. One method involves using a machine having a fixed cost of $10,000 and variable costs of $1.00 per deck of cards. The other method would use a less expensive machine If the selling price per deck of cards will be the same under each method, at what level of output will the two methods produce the same net operating income (EBIT)?
A) 5,000 decks
B) 10,000 decks
C) 15,000 decks
D) 20,000 decks
E) 25,000 decks

Question

Quizplus · Accepted Answer

Let's assume that the selling price per deck of cards is $X.

Method 1:

Fixed costs = $10,000 
Variable cost per deck = $1.00

Therefore, total cost per deck = $10,000/$1.00 + $1.00 = $10,001 per deck

Method 2:

Let's assume that the fixed cost of the less expensive machine is $Y, and the variable cost per deck is $Z.

Therefore, total cost per deck = $Y/Z + $Z per deck

We know that the selling price per deck is $X for both methods.

Therefore, the net operating income (EBIT) for each method can be calculated as follows:

Method 1:

EBIT = Total revenue - Total cost 
EBIT = (Number of decks sold x $X) - (Number of decks sold x $10,001) 
EBIT = Number of decks sold x ($X - $10,001)

Method 2:

EBIT = Total revenue - Total cost 
EBIT = (Number of decks sold x $X) - (Number of decks sold x ($Y/Z + $Z)) 
EBIT = Number of decks sold x ($X - $Y/Z - $Z)

To find out at what level of output the two methods will produce the same EBIT, we need to set EBIT for Method 1 equal to EBIT for Method 2:

Number of decks sold x ($X - $10,001) = Number of decks sold x ($X - $Y/Z - $Z)

Simplifying this equation, we get:

$10,001 = $Y/Z + $Z

We know that the fixed cost of the less expensive machine (Method 2) is less than $10,000, since it's supposed to be less expensive than the first method. Let's assume that it's $8,000.

Therefore:

$10,001 = $8,000/Z + Z

To find the value of Z that satisfies this equation, we need to calculate the derivative of $8,000/Z + Z with respect to Z and set it equal to zero:

d($8,000/Z + Z)/dZ = -$8,000/Z^2 + 1 = 0

Solving for Z, we get:

Z = sqrt($8,000) = $89.44 per deck

Plugging this value of Z back into the equation $10,001 = $Y/Z + Z, we get:

$10,001 = $Y/89.44 + 89.44

Solving for Y, we get:

Y = $7,956.64

This means that the fixed cost of the less expensive machine is $7,956.64, and the variable cost per deck is $89.44.

Now we can calculate the break-even point (where EBIT is zero) for each method:

Method 1:

Number of decks sold x ($X - $10,001) = 0 
Number of decks sold = $10,001/$X

Method 2:

Number of decks sold x ($X - $Y/Z - $Z) = 0 
Number of decks sold = ($Y/Z + $Z)/$X

Setting the two break-even points equal to each other, we get:

$10,001/$X = ($Y/Z + $Z)/$X

Simplifying this equation, we get:

$10,001 = $Y/Z + $Z

Plugging in the values of Y and Z, we get:

$10,001 = $7,956.64/89.44 + 89.44

$10,001 = $167.93 + $89.44

$10,001 = $257.37

This means that the break-even point for both methods is 10,000 decks. Therefore, the answer is B) 10,000 decks.

The Congress Company Has Identified Two Methods for Producing Playing