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 $1,800,000 and 45,000 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:

$\begin{array}{c}\text{Break-even}=\frac{\text{Fixed costs}}{\text{Contribution margin ratio}}\\ =\frac{\$540,000}{0.30}\\ =\$1,800,000\end{array}$

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

Formula to compute the break-even in units:

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

Compute the break-even in units:

$\begin{array}{c}\text{Break-even units}=\frac{\text{Break-even in dollar}}{\text{Sales price}}\\ =\frac{\$1,800,000}{40}\\ =45,000\end{array}$

Hence, the break-even in units is 45,000.

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 $2,100,000 and 52,500 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:

$\begin{array}{c}\text{Sales in dollar}=\frac{\left(\text{Fixed costs}+\text{Desired profit}\right)}{\text{Contribution margin ratio}}\\ =\frac{\left(\$540,000+\$90,000\right)}{0.30}\\ =\$2,100,000\end{array}$

Hence, the amount of sales in dollar is $2,100,000.

Formula to compute the sales in units:

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

Compute the sales in units:

$\begin{array}{c}\text{Sales in units}=\frac{\text{Sales in dollars}}{\text{Sales price}}\\ =\frac{\$2,100,000}{40}\\ =52,500\end{array}$

Hence, the sales in units is 52,500.

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 $1,227,273 and 24,545 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:

$\begin{array}{c}\text{Variable cost}=\left[\text{Sales price}\times \left(1-\text{Contribution margin ratio}\right)\right]\\ =\$40\times \left(1-0.30\right)\\ =\$40\times 0.70\\ =\$28\text{ per unit}\end{array}$

Hence, the variable cost is $28 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:

$\begin{array}{c}\text{Contribution margin per unit}=\text{Increase in the sale price}-\text{Variable cost}\\ =\$50-\$28\\ =\$22\end{array}$

Hence, the contribution margin per unit is $22.

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:

$\begin{array}{c}\text{New contribution margin ratio}=\frac{\text{Contribution margin per unit}}{\text{Increase in sales price}}\\ =\frac{\$22}{\$50}\\ =0.44\end{array}$

Hence, the new contribution margin ratio is 0.44.

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:

$\begin{array}{c}\text{Break-even}=\frac{\text{Fixed costs}}{\text{Contribution margin ratio}}\\ =\frac{\$540,000}{0.44}\\ =\$1,227,273\left(\text{rounded}\right)\end{array}$

Hence, the break-even in dollars is $1,227,273.

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:

$\begin{array}{c}\text{Break-even units}=\frac{\text{Break-even in dollars}}{\text{Increase in sales price}}\\ =\frac{\$1,227,273}{50}\\ =24,545\left(\text{rounded}\right)\end{array}$

Hence, the break-even in units is 24,545.