# CHAPTER 9 BREAK-EVEN POINT AND COST-VOLUME-PROFIT ANALYSIS

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1 CHAPTER 9 BREAK-EVEN POINT AND COST-VOLUME-PROFIT ANALYSIS 11. a. Break-even in units = \$90,000 (\$70 \$40) = 3,000 units b. In dollars break-even = 3,000 \$70 = \$210, a. Break-even point in rings = \$345,000 (\$600 \$300) = 1,150 b. Break-even point in sales dollars = 1,150 \$600 = \$690,000 c. Break-even point \$345,000 (\$600 \$306) = 1,174 rings (rounded) d. Break-even point would be \$339,000 (\$600 \$300) = 1,130 rings 14. a. Break-even in units is \$260,000 (\$1,800 \$1,000) = 325 garden sheds. b. To earn a pre-tax profit of \$200,000 = (\$260,000 + \$200,000) \$800 = 575 garden sheds c. To earn a pre-tax profit of \$280,000 = (\$260,000 + \$280,000) \$800 = 675 garden sheds 15. a. Contribution margin per unit = Sales less variable costs \$180 (\$30 + \$25 + \$17) = \$108 b. Contribution margin ratio = Contribution margin Sales \$108 \$180 = 60% c. Break-even in units is fixed costs Contribution margin per unit \$62,640 \$108 = 580 units d. Break-even in dollars is fixed costs Contribution margin ratio \$62, = \$104,400 e. To earn \$51,840 in pre-tax profit, Austin Automotive must sell: (\$62,640 + \$51,840) \$108 = 1,060 units 16. a. Contribution margin per unit = Sales less variable costs \$180 (\$30 + \$25 + \$17) = \$108 b. Contribution margin ratio = Contribution margin Sales \$108 \$180 = 60% c. Break-even in units is fixed costs Contribution margin per unit \$62,640 \$108 = 580 units f. Break-even in dollars is fixed costs Contribution margin ratio \$62, = \$104,

2 262 Chapter 9 g. To earn \$51,840 in pre-tax profit, Austin Automotive must sell: (\$62,640 + \$51,840) \$108 = 1,060 units 17. a. Convert after-tax to pre-tax profit: \$182,000 (1 0.35) = \$280,000 The number of garden sheds that must be sold to generate \$280,000 = (\$260,000 + \$280,000) \$800 = 675 garden sheds. b. Let R = revenue; then 0.08R = After-tax income desired Before-tax income = 0.08R (1 0.35) = 0.123R Revenue Variable costs Fixed costs = Income before tax Let X = Units sold SP(X) VC(X) FC = Income before tax \$1,800X \$1,000X \$260,000 = 0.123(\$1,800)X \$800X \$260,000 = \$221.4X \$578.6X = \$260,000 X = 450 units (rounded) sold to earn 8 percent of revenue after tax Amount of revenue = 450 \$1,800 = \$810,000 Check: \$810, = \$64,800 after-tax income needed (round to \$65,000) \$64, = \$99,692 before-tax income (round to \$100,000) \$1,800(450) \$1,000(450) \$260,000 = \$100,000 (before-tax income) \$100, (\$100,000) = \$100,000 \$35,000 = \$65,000 \$65,000 \$810,000 = 8% 18. a. Convert the after-tax income to pre-tax desired income: \$135,800 (1 0.30) = \$194,000 The number of units required to earn an after-tax profit of \$135,800: (\$62,640 + \$194,000) \$108 = 2,376.3 or 2,376 units b. Convert the after-tax to pre-tax profit: \$7.20 \$180 = 0.04, or 4%; 0.04 (1 0.30) = 5.7% of sales A pre-tax return on sales of 5.7 percent is required to generate an after-tax profit of \$7.20 per unit Let R = the Level of revenue that generates a pre-tax return of 5.7%: Variable costs = (\$30 + \$ ) \$180 = 0.4, or 0.4R R \$62, R = 0.057R 0.543R = \$62,640 R = \$115,359 \$115,359 \$180 = or 641 units (rounded) 19. Let Y = Level of sales generating income equal to 30% of sales, then: Y 0.60Y (\$25,000 per month 12 months) = 0.30Y 0.10Y = \$300,000 Y = \$3,000,000 Since existing sales are \$2,250,000, sales would need to increase by \$3,000,000 \$2,250,000 = \$750,000.

3 Chapter a. First, convert the desired after-tax income to a pre-tax desired income: \$600,000 (1 0.40) = \$1,000,000 Note that total variable costs per unit = \$3,000, and total fixed costs = \$370,000. Next, let P represent the number of golf carts that must be sold to generate \$1,000,000 in pre-tax income: \$5,000P \$3,000P \$370,000 = \$1,000,000 \$2,000P = \$1,370,000 = 685 golf carts b. Find after-tax equivalent of 20%: 20% (1 0.40) = 33.33% Variable costs as a percentage of sales: \$3,000 \$5,000 = 60% Let R = Level of revenue that generates a pre-tax return of 33.33%: R 0.6R \$370,000 = R R = \$370,000 R = \$5,547,226 Proof: Sales \$ 5,547,226 Variable costs (60%) (3,328,336) Contribution margin \$ 2,218,890 Fixed costs (370,000) Income before tax \$ 1,848,890 Income tax (40%) (739,556) Net income \$ 1,109,334 \$1,109,334 \$5,547,226 = 20% 22. a. \$1,450 \$0.50 = 2,900 passengers per day i. Break-even: \$2,000 2,900 = \$0.69 (rounded) per passenger Earn \$250: (\$2,000 + \$250) 2,900 = \$0.78 (rounded) ii. Total variable cost = \$2,000 (\$2, ) = \$400 Variable cost per passenger = \$400 2,900 = \$0.14 (rounded) Profit if fare is \$0.60 = (2, \$0.60) (2, \$0.14) \$1,600 = \$(399.40) Current loss = \$1,450 \$2,000 = \$(550) County will be better off by \$(399.40) (\$550) = \$ iii. At a fare of \$0.70: (2,900 \$ ) (2,900 \$ ) \$1,600 = \$(57.20) The county would incur a slight loss at a fare of \$0.70. At a fare of \$0.90: (2,900 \$ ) (2,900 \$ ) \$1,600 = \$ The company would first make a profit when the fare is set at \$0.90. iv. Increasing volume will help improve profitability only if the volume change increases total contribution margin. Because an increase in volume can often

4 264 Chapter 9 be achieved only with a decrease in price, the change in contribution margin may be negative rather than positive. 23. a. Current sales volume for both companies = \$2,000,000 \$40 = 50,000 New selling price \$40 (0.3 \$40) = \$28; Variable costs = \$1,400,000 50,000 = \$28 Ainsley: (50, \$28) (50, \$28) \$0 = \$0 Bard: (50, \$28) (50, \$0) \$1,400,000 = \$840,000 This strategy is best used by Bard. b. New selling price: \$ = \$52 Ainsley: (50, \$52) (50, \$28) \$0 = \$1,020,000 Bard: (50, \$52) (50, \$0) \$1,400,000 = \$810,000 This strategy is best used by Ainsley. c. Ainsley: (65,000 \$40) (65,000 \$28) \$200,000 = \$580,000 Bard: (65,000 \$40) (65,000 \$0) \$1,600,000 = \$1,000,000 This strategy is best used by Bard. 24. a. CM per unit of sales mix = (\$3 8) + (1 \$6) = \$30 Break-even = \$180,000 \$30 = 6,000 units of sales mix, or 18,000 wallets and 6,000 money clips Total revenue = (18,000 \$30) + (6,000 \$15) = \$630,000 b. Sales mix units = (\$180,000 + \$150,000) \$30 = 11,000 = 33,000 wallets and 11,000 money clips Total revenue = (33,000 \$30) + (11,000 \$15) = \$1,155,000 c. Equivalent pre-tax profit = \$150,000 (1 0.40) = \$250,000 Sales mix units = (\$180,000 + \$250,000) \$30 = 14, = 43,000 wallets and 14,333 money clips Total revenue = (43,000 \$30) + (14,333 \$15) = \$1,504,995 d. Units of sales mix = \$1,155,000 [(5 \$30) + (2 \$15)] = 6,417 (rounded) = 32,085 wallets and 12,834 money clips Income = (32,085 \$8) + (12,834 \$6) \$180,000 = \$153,684 The sales mix shifted such that the ratio of wallets to money clips declined, and the break-even point was reduced because money clips have a higher contribution margin ratio than money clips. Hence, at a sales level of \$1,155,000, more contribution margin is generated at the actual sales mix than at the planned sales mix. 25. a. Fixed costs Contribution margin = Break-even point in units \$1,080,000,000 [(3 \$300) + (5 \$700) + (2 \$1,000)] = \$1,080,000,000 \$6,400 = 168,750 bags Mod = 3 168,750 = 506,250 units \$2,200 = \$1,113,750,000 Rad = 5 168,750 = 843,750 units \$3,700 = 3,121,875,000 X-treme = 2 168,750 = 337,500 units \$6,000 = 2,025,000,000 Revenue to break-even \$6,260,625,000

5 Chapter b. Convert after-tax to pre-tax income. \$1,000,000,000 (1 0.5) = \$2,000,000,000 (\$2,000,000,000 + \$1,080,000,000) \$6,400 = 481,250 bags Mod = 3 481,250 = 1,443,750 units \$2,200 = \$ 3,176,250,000 Rad = 5 481,250 = 2,406,250 units \$3,700 = 8,903,125,000 X-treme = 2 481,250 = 962,500 units \$6,000 = 5,775,000,000 Total revenue needed \$17,854,375,000 c. This change will increase the number of units required to break even because fewer units of Rad and X-treme, which have the greatest contribution margin, are being sold and more units of Mod, which has the lowest contribution margin, are being sold. Scooter Contribution Margin Mod 5 \$300 = \$1,500 Rad 4 \$700 = 2,800 X-treme 1 \$1,000 = 1,000 Total \$5,300 Now the contribution margin is \$5,300 per bag, which is less than the contribution margin per bag of \$6,400 in (a) above. d. If Green Rider sells more of its scooters with the greatest contribution margin (X-treme) and fewer of the scooters with the lowest contribution margin (Mod), then fewer scooters would be needed to be sold to break even. 26. a. Break-even is \$264,000 (\$9.60 \$7.60) = 132,000 bushels 132,000 bushels \$9.60 = \$1,267,200 Bushels per acre = 132,000 1,200 = 110 bushels per acre b. Bushels sold Break-even bushels = Margin of safety 174, ,000 = 42,000 bushels (174,000 \$9.60) \$1,267,200 = \$403,200 \$403,200 \$1,670,400 = 24.1% 31. a. Each bag contains one unit of liquid and two units of spray. Thus, each bag generates contribution margin of: (1 \$10) + (2 \$5) = \$20. The break-even point would be: \$100,000 \$20 = 5,000 bags. Since each bag contains two units of spray, at the break-even point 5,000 2 or 10,000 units of spray must be sold. i. At the break-even point, Total CM = Total FC; and the CM per unit would be \$1,600 4,000 = \$0.40. If one unit is sold beyond the break-even point, net income would rise by \$0.40. ii. iii. \$10X 0.40(\$10X) \$216,000 = 0.25(\$10X) \$3.50X = \$216,000 X = 61,715 units (rounded) In units: 3,200 2,800 = 400 units In dollars: 400 units \$65 per unit = \$26,000

6 266 Chapter 9 Percentage: \$26,000 (\$65 3,200) = 12.5% 38. a. Total variable cost = \$28 + \$12 + \$8 = \$48 Contribution margin per unit = \$70 \$48 = \$22 per unit Contribution margin ratio = \$22 \$70 = 31.4% (rounded) Total fixed costs = \$10,000 + \$24,000 = \$34,000 Break-even point in units = \$34,000 \$22 per unit = 1,545 units (rounded) Break-even point in dollars = \$34, = \$108,280 (rounded) b. (\$40,000 + \$34,000) = \$235,669 (rounded) (\$235,669 \$70) = 3,367 units (rounded) c. Convert after-tax earnings to pre-tax earnings: \$40,000 (1 0.40) = \$66,667 Required sales = (\$66,667 + \$34,000) = \$320,596 (rounded) \$320,596 \$70 = 4,580 units (rounded) d. Convert the after-tax rate of earnings to a pre-tax rate of earnings: [20% (1 0.40)] = 33.33% Because the CM% is only 31.4%, no level of sales would generate net income equal to, on a pre-tax basis, 33.33% of sales. e. Variable cost savings (5,000 \$6.00) \$ 30,000 Additional fixed costs (40,000) Decrease in profit \$(10,000) The company should not buy the new sewing machine. f. Existing CM per unit = \$22 CM under proposal = (\$ ) \$48 = \$15 Total CM under proposal (3, \$15) \$ 58,500 Existing CM (3,000 \$22) (66,000) Change in CM \$ (7,500) Change in fixed costs (10,000) Change in net earnings before taxes \$ (17,500) No, these two changes should not be made because they would lower pre-tax profits by \$17,500 relative to existing levels. 41. a. Total sales price per bag: Commercial (\$5,600 1) \$5,600 Residential (\$1,800 3) 5,400 \$11,000 Total variable costs per bag: Commercial (\$3,800 1) \$3,800 Residential (\$1,000 3) 3,000 (6,800) Total contribution margin \$ 4,200 Break-even point in units = \$8,400,000 \$4,200 = 2,000 bags Commercial: 2,000 1 = 2,000 mowers Residential: 2,000 3 = 6,000 mowers b. (\$8,400,000 + \$1,260,000) \$4,200 = 2,300 bags

7 Chapter Commercial: 2,300 1 = 2,300 mowers Residential: 2,300 3 = 6,900 mowers c. Pre-tax equivalent of \$1,008,000 after-tax = \$1,008,000 (1 0.40) = \$1,680,000 (\$8,400,000 + \$1,680,000) \$4,200 = 2,400 bags Commercial: 2,400 1 = 2,400 mowers Residential: 2,400 3 = 7,200 mowers d. Let X = number of bags that must be sold to produce pre-tax earnings equaling 12 percent of sales revenue, then: \$4,200X \$8,400,000 = 0.12(\$11,000X) X = 2,917 bags (rounded) Commercial: 2,917 1 = 2,917 mowers Residential: 2,917 3 = 8,751 mowers e. Convert the after-tax return to a pre-tax rate of return: 0.08 (1 0.40) = 13% (rounded) \$4,200X \$8,400,000 = 0.13(\$11,000X) X = 3,032 bags (rounded) Commercial: 3,032 1 = 3,032 mowers Residential: 3,032 3 = 9,096 mowers 42. a. Ducks Ducklings Sales \$ \$12.00 Variable costs (12.00) (8.00) Contribution margin \$ \$ 4.00 Mix 1 5 Total contribution margin \$ \$20.00 The average contribution margin ratio is \$32 \$84 = 38.1% (rounded) b. Break-even point = \$288,000 \$32 = 9,000 bags per year or 750 bags a month Ducks: = 750 per month Ducklings: = 3,750 per month c. Target profit is \$96, = \$1,152,000 (\$288,000 + \$1,152,000) \$32 = 45,000 bags per year or 3,750 bags a month. Ducks: 3,750 1 = 3,750 per month Ducklings: 3,750 5 = 18,750 per month d. Ducks Ducklings Sales \$ \$12.00 Variable costs (12.00) (8.00) Contribution margin \$ \$ 4.00 Mix 1 9 Total contribution margin \$ \$36.00 Target profit after tax is \$31,680. Pre-tax profit is \$31,680 (1 0.40) = \$52,800 monthly or \$633,600 per year. Break-even = (\$633,600 + \$288,000) \$48 = 19,200 bags per year, or 1,600 per month Units Revenue

8 268 Chapter 9 Ducks (19,200 \$24) 19,200 \$ 460,800 Ducklings (19,200 9 \$12) 172,800 2,073,600 Total \$2,534,400 e. [\$288,000 + (\$8,500 12)] [\$12 + (\$8 5)] (\$288,000 + \$102,000) \$52 = 7,500 Yes, the company would want to make the change because the break-even point is reduced from 9,000 mix units to 7,500 mix units.

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