Chapter 3, Problem 18PSB

a)

To determine

The break-even point in dollars and in units.

Answer to Problem 18PSB

The break-even point in dollars and in units are $2,700,000 and 90,000 units respectively.

Explanation of Solution

Formula to compute the break-even in dollars:

$\text{Break-even}=\frac{\text{Fixed costs}}{\text{Contribution margin ratio}}$

Compute the break-even in dollars:

$\begin{array}{c}\text{Break-even}=\frac{\text{Fixed costs}}{\text{Contribution margin ratio}}\\ =\frac{\$540,000}{0.20}\\ =\$2,700,000\end{array}$

Hence, the break-even in dollars is $2,700,000.

Formula to compute the break-even in units:

$\text{Break-even units}=\frac{\text{Break-even in dollar}}{\text{Sales price}}$

Compute the break-even in units:

$\begin{array}{c}\text{Break-even units}=\frac{\text{Break-even in dollar}}{\text{Sales price}}\\ =\frac{\$2,700,000}{30}\\ =90,000\end{array}$

Hence, the break-even in units is 90,000.

b)

To determine

The amount of sales in dollars and in units, to gain $90,000 profit.

Answer to Problem 18PSB

The amount of sales in dollars and in units, to gain $270,000 profit are $3,150,000 and 105,000 units.

Explanation of Solution

Formula to compute the sales in dollars:

$\text{Sales in dollar}=\frac{\left(\text{Fixed costs}+\text{Desired profit}\right)}{\text{Contribution margin ratio}}$

Compute the sales in dollars:

$\begin{array}{c}\text{Sales in dollar}=\frac{\left(\text{Fixed costs}+\text{Desired profit}\right)}{\text{Contribution margin ratio}}\\ =\frac{\left(\$540,000+\$90,000\right)}{0.20}\\ =\$3,150,000\end{array}$

Hence, the amount of sales in dollar is $3,150,000.

Formula to compute the sales in units:

$\text{Sales in units}=\frac{\text{Sales in dollars}}{\text{Sales price}}$

Compute the sales in units:

$\begin{array}{c}\text{Sales in units}=\frac{\text{Sales in dollars}}{\text{Sales price}}\\ =\frac{\$3,150,000}{30}\\ =105,000\end{array}$

Hence, the sales in units is 105,000.

c)

To determine

The new break-even points in dollars and in units.

Answer to Problem 18PSB

The new break-even points in dollars and in units are $2,160,000 and 67,500 units.

Explanation of Solution

Formula to compute the variable cost:

$\text{Variable cost}=\left[\text{Sales price}\times \left(1-\text{Contribution margin ratio}\right)\right]$

Compute the variable cost:

$\begin{array}{c}\text{Variable cost}=\left[\text{Sales price}\times \left(1-\text{Contribution margin ratio}\right)\right]\\ =\$30\times \left(1-0.20\right)\\ =\$30\times 0.80\\ =\$24\text{ per unit}\end{array}$

Hence, the variable cost is $24 per unit.

Formula to compute the per unit contribution margin:

$\text{Contribution margin per unit}=\text{Increase in the sale price}-\text{Variable cost}$

Compute the per unit contribution margin:

$\begin{array}{c}\text{Contribution margin per unit}=\text{Increase in the sale price}-\text{Variable cost}\\ =\$32-\$24\\ =\$8\end{array}$

Hence, the contribution margin per unit is $8.

Formula to compute the new contribution margin ratio:

$\text{New contribution margin ratio}=\frac{\text{Contribution margin per unit}}{\text{Increase in sales price}}$

Compute the new contribution margin ratio:

$\begin{array}{c}\text{New contribution margin ratio}=\frac{\text{Contribution margin per unit}}{\text{Increase in sales price}}\\ =\frac{\$8}{\$32}\\ =0.25\end{array}$

Hence, the new contribution margin ratio is 0.25.

Formula to compute the break-even in dollars:

$\text{Break-even}=\frac{\text{Fixed costs}}{\text{Contribution margin ratio}}$

Compute the break-even in dollars:

$\begin{array}{c}\text{Break-even}=\frac{\text{Fixed costs}}{\text{Contribution margin ratio}}\\ =\frac{\$540,000}{0.25}\\ =\$2,160,000\end{array}$

Hence, the break-even in dollars is $2,160,000.

Formula to compute the break-even in units:

$\text{Break-even units}=\frac{\text{Break-even in dollars}}{\text{Sales price}}$

Compute the break-even in units:

$\begin{array}{c}\text{Break-even units}=\frac{\text{Break-even in dollars}}{\text{Increase in sales price}}\\ =\frac{\$2,160,000}{32}\\ =67,500\end{array}$

Hence, the break-even in units is 67,500.