- A company achieves its break-even point when it sells 5,000 units per month. The company's fixed costs are $200,000 per month. If the company sells 5,800 units in a given month, what is its profit for that month?
Answer: $32,000
The break-even point formula is x=F/(p−v), where F is fixed costs and (p−v) is the contribution margin per unit. We are given x=5,000 and $F=$200,000. We can find the contribution margin: $5000 = 200000 / (p - v) \Rightarrow (p - v) = 200000 / 5000 = $40. Profit is only generated on units sold above the break-even point. The number of profit-generating units is 5800−5000=800. The total profit is the number of extra units multiplied by the contribution margin per unit: $Profit = 800 \times $40 = 32,000. - A firm's break-even quantity is determined by its fixed costs F, its selling price per unit p, and its variable cost per unit v, where p>v. If both the selling price p and the variable cost v increase by the same positive dollar amount, k, what is the effect on the break-even quantity?
Answer: The break-even quantity remains unchanged.
The formula for the break-even quantity is xBE=F/(p−v). The term (p−v) is the contribution margin. The new price is p′=p+k, and the new variable cost is v′=v+k. The new contribution margin is (p′−v′)=(p+k)−(v+k)=p−v. Since the fixed costs F are unchanged and the contribution margin (p−v) is also unchanged, the new break-even quantity, xBE′=F/(p′−v′), is identical to the original break-even quantity. - A company's monthly cost function is C(x)=8000+25x+0.02x2 and revenue function is R(x)=75x−0.01x2, where x is the number of units produced and sold. The company is considering a process improvement that would reduce the coefficient of x2 in the cost function from 0.02 to 0.015, but would increase fixed costs to $9,200. At what production level would the profit improvement from this change be exactly $500?
Answer: 1,414 units with profit improvement of exactly $500
Original profit: P₁(x) = 75x - 0.01x² - 8000 - 25x - 0.02x² = 50x - 0.03x² - 8000. New profit: P₂(x) = 75x - 0.01x² - 9200 - 25x - 0.015x² = 50x - 0.025x² - 9200. Improvement: P₂(x) - P₁(x) = -0.025x² + 0.03x² - 9200 + 8000 = 0.005x² - 1200. Setting equal to 500: 0.005x² - 1200 = 500, so 0.005x² = 1700, thus x² = 340,000, and x = 1,414 units. Other choices result from computational errors in the quadratic terms.
- A concert promoter has fixed costs of $180,000 for a venue. The variable cost per attendee is $25. The ticket price is set at $75. A new city ordinance imposes a tax of 10% on the ticket price, which the promoter must pay from the proceeds of each ticket. How does the imposition of this tax affect the number of tickets that must be sold to break even?
Answer: It increases by approximately 636 tickets.
First, calculate the original break-even point. The contribution margin is $CM = $75 - $25 = $50. The original break-even point is xBE=F/CM=180000/50=3600 tickets. Next, calculate the effect of the tax. The tax is 10% of the $75 ticket price, which is $0.10 \times 75 = $7.50. This tax reduces the revenue per ticket, so the new effective contribution margin is $CM' = ($75 - $7.50) - $25 = $67.50 - $25 = $42.50. The new break-even point is xBE′=180000/42.50≈4235.29, which means 4,236 tickets must be sold. The change is 4236−3600=636 tickets. - A company has a linear cost function and a linear revenue function. At a production level of 1,000 units, the total cost is $80,000 and the profit is $10,000. At a production level of 1,500 units, the total cost is $110,000. What is the company's fixed cost?
Answer: $20,000
The cost function is linear, C(x)=vx+F. We have two points on the cost function: (1000, 80000) and (1500, 110000). The variable cost per unit, v, is the slope of the cost function: $v = (110000 - 80000) / (1500 - 1000) = 30000 / 500 = $60. Now we can use one of the points to find the fixed cost, F. Using the first point: C(1000)=60(1000)+F=80000. This simplifies to 60000+F=80000, so $F = $20,000. The information about profit is consistent but not needed to find the fixed cost. - A software company sells a subscription for $50 per month. The variable cost associated with each user is $10 per month, and the total monthly fixed costs are $80,000. The company is currently operating at a monthly profit of $40,000. Management is considering a new advertising campaign that will increase fixed costs by 20%. To justify this expense, the campaign must increase the company's monthly profit by 10%. What is the minimum number of additional monthly subscriptions the company must sell to achieve this goal?
Answer: 500
First, determine the current number of subscriptions sold. The profit function is P(x)=R(x)−C(x). The contribution margin per unit is Price - Variable Cost = $50 - $10 = $40. So, P(x)=40x−80000. Given the current profit is $40,000, we can solve for x: 40000=40x−80000⇒120000=40x⇒x=3000 subscriptions. Next, determine the new fixed costs and target profit. New fixed costs are $80000 \times 1.20 = $96,000. New target profit is $40000 \times 1.10 = 44,000.Now,findthenumberofsubscriptions(x'$) needed to meet the new target: 44000=40x′−96000⇒140000=40x′⇒x′=3500 subscriptions. The question asks for the additional subscriptions needed, which is 3500−3000=500. - A manufacturing company produces widgets with fixed costs of $15,000 monthly and variable costs of $6 per widget. The selling price follows the demand equation p=30−0.001q, where p is price in dollars and q is quantity. What quantity should be produced to maximize monthly profit?
Answer: 12,000 widgets with maximum profit of $129,000
Revenue = q(30 - 0.001q) = 30q - 0.001q². Profit = 30q - 0.001q² - 6q - 15,000 = 24q - 0.001q² - 15,000. Taking derivative: 24 - 0.002q = 0, so q = 12,000. Maximum profit = 24(12,000) - 0.001(12,000)² - 15,000 = 288,000 - 144,000 - 15,000 = $129,000. Choice B uses incorrect profit calculation, C and D have wrong quantities.
- A company manufactures an item with a variable cost of $80 per unit and fixed costs of $300,000. The company aims for a target profit of $150,000 by selling 5,000 units. Due to a supply chain issue, the variable cost per unit is projected to increase by 15%. To maintain the same target profit ($150,000) at the same sales volume (5,000 units), by what amount must the selling price per unit be increased?
Answer: $12.00
The increase in variable cost per unit is $80 \times 0.15 = $12. For the profit to remain unchanged at a constant sales volume, the total revenue must increase by the same amount as the total cost. The total cost increases by (increase in variable cost per unit) × (number of units), which is $12 \times 5000 = $60,000. The increase in total revenue is (increase in price per unit) × (number of units). Let Δp be the price increase. Then $\Delta p \times 5000 = $60,000. Solving for Δp gives $\Delta p = $60,000 / 5000 = $12. Therefore, the price must be increased by $12.00. - Two companies are considering a merger. Company A has monthly fixed costs of $50,000 and variable costs of $15 per unit, while Company B has monthly fixed costs of $30,000 and variable costs of $20 per unit. Both sell identical products for $35 per unit. After the merger, fixed costs would be $65,000 monthly (due to elimination of duplicate overhead), and variable costs would be $17 per unit (average of both companies). If both companies currently sell the same number of units individually, at what total monthly sales volume would the merged company achieve the same total profit as both companies operating separately?
Answer: 5,556 units per month for the merged company
Let x = units sold by each company separately, so total separate volume is 2x. Company A profit: (35-15)x - 50,000 = 20x - 50,000. Company B profit: (35-20)x - 30,000 = 15x - 30,000. Combined separate profit: 35x - 80,000. For the merged company selling volume y: (35-17)y - 65,000 = 18y - 65,000. Setting merged profit equal to combined separate profit at equal total volume (y = 2x): 18y - 65,000 = 35(y/2) - 80,000. Solving: 18y - 65,000 = 17.5y - 80,000, so 0.5y = -15,000, giving y = 5,556 units.
- Two competing pizza shops have the following cost structures: Shop A has fixed costs of $2,400 per month with variable costs of $3 per pizza, while Shop B has fixed costs of $1,800 per month with variable costs of $4 per pizza. Both shops sell pizzas for $12 each. At what monthly sales volume will both shops have identical total profits?
Answer: 600 pizzas per month for each shop individually
Shop A profit: $12x - $3x - $2,400 = $9x - $2,400. Shop B profit: $12x - $4x - $1,800 = $8x - $1,800. Setting equal: $9x - $2,400 = $8x - $1,800, so x = 600 pizzas per shop. Choice A incorrectly assumes combined volume, while C and D result from computational errors.
- A company's cost function is C(x)=50x+10000 and its revenue function is R(x)=75x. Management implements changes that increase fixed costs by 10% and decrease the variable cost per unit by $5. How does the break-even quantity change?
Answer: It decreases by 33 units.
First, calculate the original break-even point by setting R(x)=C(x): 75x=50x+10000⇒25x=10000⇒xold=400 units. Next, determine the new cost function parameters. The new fixed cost is $10000 \times 1.10 = $11,000. The new variable cost per unit is $50 - 5 = $45. The new cost function is C′(x)=45x+11000. Now, calculate the new break-even point: 75x=45x+11000⇒30x=11000⇒xnew=11000/30≈366.67 units. The change is xnew−xold=366.67−400=−33.33. The break-even quantity decreases by approximately 33 units. - A company produces two products, Alpha and Beta. Product Alpha has a contribution margin of $10 per unit and Product Beta has a contribution margin of $15 per unit. The company's sales data shows a consistent sales mix of 3 units of Alpha for every 2 units of Beta. If the total fixed costs for the company are $240,000, how many units of Product Beta are sold at the break-even point?
Answer: 8,000
To solve a multi-product break-even problem, create a 'bundle' based on the sales mix. One bundle consists of 3 units of Alpha and 2 units of Beta. Calculate the contribution margin of one bundle: $CMbundle = (3 × CMAlpha) + (2 × CMBeta) = (3 \times $10) + (2 \times $15) = $30 + $30 = $60. Next, find the number of bundles needed to break even: Bundles to break even = Total Fixed Costs / $CMbundle = $240,000 / $60 = 4,000 bundles. The question asks for the number of units of Product Beta. Since each bundle contains 2 units of Beta, the total number of Beta units is 4,000 bundles×2 units/bundle=8,000 units. - A company has a revenue function R(x)=150x and a cost function C(x)=90x+48000. Management is considering a price reduction of 10% to boost sales. They estimate that this will increase the number of units sold by 25%. If the company is currently operating exactly at its break-even point, what will be the net effect on profit from implementing these changes?
Answer: A decrease in profit of $3,000
First, find the current break-even point: 150x=90x+48000⇒60x=48000⇒x=800 units. At this point, the current profit is $0. Next, evaluate the proposed changes. The new price is $p' = 150 \times (1 - 0.10) = $135. The new sales volume is x′=800×(1+0.25)=1000 units. The cost function parameters (v=90,F=48000) are unchanged. Calculate the new profit: $P' = R'(x') - C(x') = (135×1000) - (90×1000+48000) = 135000 - (90000 + 48000) = 135000 - 138000 = -$3,000. The net effect is the new profit minus the old profit, which is -$3,000 - $0 = -$3,000. This is a decrease in profit of $3,000. - A company sells a product for $120. The variable cost per unit is 60% of the selling price. The company's fixed costs are $96,000. The company projects a loss of $12,000 for the upcoming period. How many additional units must be sold to transform this projected loss into a target profit of $24,000?
Answer: 750
First, determine the contribution margin per unit. The variable cost is $v = 0.60 \times $120 = $72. The contribution margin is $p - v = $120 - $72 = $48. The total change in profit required is from a loss of $12,000 to a profit of $24,000, which is a total swing of $24,000 - (-$12,000) = $36,000. Each additional unit sold contributes $48 to covering fixed costs and generating profit. To find the number of additional units needed, divide the required profit change by the contribution margin per unit: Additional Units = $36,000 / $48 = 750 units.
- A manufacturing firm is choosing between two production technologies for a new product that will sell for $70 per unit. Technology A has fixed costs of $150,000 and a variable cost of $25 per unit. Technology B has fixed costs of $90,000 and a variable cost of $40 per unit. At what minimum sales volume does the profit from Technology A first exceed the profit from Technology B?
Answer: 4,001 units
Let x be the number of units sold. The profit function for Technology A is PA(x)=(70−25)x−150000=45x−150000. The profit function for Technology B is PB(x)=(70−40)x−90000=30x−90000. To find where the profit from A exceeds the profit from B, we set up the inequality PA(x)>PB(x): 45x−150000>30x−90000. Subtracting 30x from both sides gives 15x−150000>−90000. Adding 150,000 to both sides gives 15x>60000. Dividing by 15 gives x>4000. The smallest integer number of units greater than 4,000 is 4,001.
