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A producer of felt-tip pens has received a forecast of demand of 31,000 pens for the coming month from its marketing department. Fixed costs of $21,000 per month are allocated to the felt-tip operation, and variable costs are 25 cents per pen. a. Find the break-even quantity if pens sell for $1 each. (Round your answer to the next whole number.) QBEP units b. At what price must pens be sold to obtain a monthly profit of $18,000, assuming that estimated demand materializes

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Edufirst

Answer:

  • a. Break-even quantity: 28,000 pens

  • b. Price: $1.51 per pen

Explanation:

1. Break-even quantity

a) Revenue, R(x)

The  monthly revenue is the product of the price by the number of units sold in the month.

Naming x the number of pens sold in the month:

  • R(x) = $1 × x = x

b) Cost, C(x)

The monthly cost is the sum of the fixed cost per month plus the variable costs:

  • C(x) = $21,000 + 0.25 × x = 21,000 + 0.25x

c) Break-even

Break-even is the point when the revenue and the total costs are equal, this is, when the profit is zero. Write the equation and solve:

  • x = 21,000 + 0.25x
  • x - 0.25x = 21,000
  • 0.75x = 21,000
  • x = 21,000 / 0.75
  • x = 28,000

Hence, the break-even quantity is 28,000 pens.

2. Price pens must be sold to obtain a monthly profit of $18,000

Profit = Revenue - Total cost

  • P(x) = R(x) - C(x)

  • P(x) = x.p - [ 0.25x + 21,000]

Where p is the price.

  • P(x) = x.p - 0.25x - 21,000

Substitute the quantity demanded, x, with 31,000, and the profit, P(x) with 18,000:

  • 18,000 = 31,000p - 0.25(31,000) - 21,000

Solve for p and compute:

  • 31,000p = 18,000 + 7,750 + 21,000

  • 31,000 p = 46,750

  • p = 1.51

That is $1.51 per pen.

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