Chapter 3, Problem 18PSA

a)

To determine

The break-even point in dollars and in units.

Answer to Problem 18PSA

The break-even point in dollars and in units are $2,025,000 and 81,000 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:

$\begin{array}{c}\text{Break-even}=\frac{\text{Fixed costs}}{\text{Contribution margin ratio}}\\ =\frac{\$810,000}{0.40}\\ =\$2,025,000\end{array}$

Hence, the break-even in dollars is $2,025,000.

Formula to compute the break-even in units:

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

Compute the break-even in units:

$\begin{array}{c}\text{Break-even units}=\frac{\text{Break-even}}{\text{Sales price}}\\ =\frac{\$2,025,000}{25}\\ =81,000\end{array}$

Hence, the break-even in units is 81,000.

b)

To determine

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

Answer to Problem 18PSA

The amount of sales in dollars and in units, to gain $270,000 profit are $2,700,000 and 108,000 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:

$\begin{array}{c}\text{Sales in dollar}=\frac{\left(\text{Fixed costs}+\text{Desired profit}\right)}{\text{Contribution margin ratio}}\\ =\frac{\left(\$810,000+\$270,000\right)}{0.40}\\ =\$2,700,000\end{array}$

Hence, the amount of sales in dollar is $2,700,000.

Formula to compute the sales in units:

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

Compute the sales in units:

$\begin{array}{c}\text{Sales in units}=\frac{\text{Sales in dollars}}{\text{Sales price}}\\ =\frac{\$2,700,000}{25}\\ =108,000\end{array}$

Hence, the sales in units is 108,000.

c)

To determine

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

Answer to Problem 18PSA

The new break-even points in dollars and in units are $1,620,000 and 54,000 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:

$\begin{array}{c}\text{Variable cost}=\left[\text{Sales price}\times \left(1-\text{Contribution margin ratio}\right)\right]\\ =\$25\times \left(1-0.40\right)\\ =\$25\times 0.60\\ =\$15\text{ per unit}\end{array}$

Hence, the variable cost is $15 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:

$\begin{array}{c}\text{Contribution margin per unit}=\text{Increase in the sale price}-\text{Variable cost}\\ =\$30-\$15\\ =\$15\end{array}$

Hence, the contribution margin per unit is $15.

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:

$\begin{array}{c}\text{New contribution margin ratio}=\frac{\text{Contribution margin per unit}}{\text{Increase in sales price}}\\ =\frac{\$15}{\$30}\\ =0.50\end{array}$

Hence, the new contribution margin ratio is 0.50.

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:

$\begin{array}{c}\text{Break-even}=\frac{\text{Fixed costs}}{\text{Contribution margin ratio}}\\ =\frac{\$810,000}{0.50}\\ =\$1,620,000\end{array}$

Hence, the break-even in dollars is $1,620,000.

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:

$\begin{array}{c}\text{Break-even units}=\frac{\text{Break-even in dollars}}{\text{Increase in sales price}}\\ =\frac{\$1,620,000}{30}\\ =54,000\end{array}$

Hence, the break-even in units is 54,000.