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 $2,250,000 and 112,500 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:

$\begin{array}{c}\text{Break-even}=\frac{\text{Fixed costs}}{\text{Contribution margin ratio}}\\ =\frac{\$900,000}{0.40}\\ =\$2,250,000\end{array}$

Hence, the break-even in dollars is $2,250,000.

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:

$\begin{array}{c}\text{Break-even units}=\frac{\text{Break-even}}{\text{Sales price}}\\ =\frac{\$2,250,000}{20}\\ =112,500\end{array}$

Hence, the break-even in units is 112,500.

b)

To determine

The amount of sales in dollars and in units, to gain $360,000 profit.

Answer to Problem 18PSA

The amount of sales in dollars and in units, to gain $360,000 profit are $3,150,000 and 157,500 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:

$\begin{array}{c}\text{Sales in dollar}=\frac{\left(\text{Fixed costs}+\text{Desired profit}\right)}{\text{Contribution margin ratio}}\\ =\frac{\left(\$900,000+\$360,000\right)}{0.40}\\ =\$3,150,000\end{array}$

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:

$\begin{array}{c}\text{Sales in units}=\frac{\text{Sales in dollars}}{\text{Sales price}}\\ =\frac{\$3,150,000}{20}\\ =157,500\end{array}$

Hence, the sales in units is 157,500.

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 $1,800,000 and 75,000 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:

$\begin{array}{c}\text{Variable cost}=\left[\text{Sales price}\times \left(1-\text{Contribution margin ratio}\right)\right]\\ =\$20\times \left(1-0.40\right)\\ =\$20\times 0.60\\ =\$12\text{ per unit}\end{array}$

Hence, the variable cost is $12 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:

$\begin{array}{c}\text{Contribution margin per unit}=\text{Increase in the sale price}-\text{Variable cost}\\ =\$24-\$12\\ =\$12\end{array}$

Hence, the contribution margin per unit is $12.

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:

$\begin{array}{c}\text{New contribution margin ratio}=\frac{\text{Contribution margin per unit}}{\text{Increase in sales price}}\\ =\frac{\$12}{\$24}\\ =0.50\end{array}$

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:

$\begin{array}{c}\text{Break-even}=\frac{\text{Fixed costs}}{\text{Contribution margin ratio}}\\ =\frac{\$900,000}{0.50}\\ =\$1,800,000\end{array}$

Hence, the break-even in dollars is $1,800,000.

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:

$\begin{array}{c}\text{Break-even units}=\frac{\text{Break-even in dollars}}{\text{Increase in sales price}}\\ =\frac{\$1,800,000}{24}\\ =75,000\end{array}$

Hence, the break-even in units is 75,000.