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Peter is thinking of starting up a "Personal Financial Services" consultation in his spare time. The basic office equipment will cost $$2555,$ and the variable cost $($VC$)$ for each consultation is estimated to be $$13\mathrm{.}05.$ He plans to bill each client $$61\mathrm{.}00$ per consultation.

$($a$)$ What is the break$-$even quantity of consultations for Peter's "Personal Financial Services" operation?

$($b$)$ If Peter's profit goal is $$900,$ how many consultations would he need to reach his goal?

$($c$)$ Briefly explain the concept of a "Contribution Margin," and demonstrate how the Contribution Margin $($CM$)$ determined from the information above can be used to determine the same break$-$even quantity determined in part $($a$).$

Peter has to make $53$ consultations to break even.

$($Round to the nearest integer as needed.$)$

$($b$)$ If Peter's profit goal is $$900,$ how many consultations would he need to reach his goal?

To make a $$900$ profit, Peter needs to make $72$ consultations.

$($Round to the nearest integer as needed.$)$

$($c$)$ Briefly explain the concept of a "Contribution Margin," and demonstrate how the Contribution Margin $($CM$)$ determined from the information above can be used to determine the same break$-$even quantity determined in part $($a$).$

The contribution margin is the $$ per unit minus the $$

$$ per unit. The break$-$even quantity can be found by $$ the fixed cost, $$2555,$ and the contribution margin, $$47\mathrm{.}95$

$($Type integers or decimals rounded to the nearest cent as needed.$)$

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