Q$3-$ Par Putters Company sells golf balls for $$31$ per dozen. The store's overhead expenses are $48\%$ of cost and the owners require a profit of $22\%$ of cost$.$

$($a$)$ For how much does Par Putters Company buy one dozen golf balls?

$($b$)$ What is the price needed to cover all of the costs and expenses?

$($c$)$ What is the highest rate of markdown at which the store will still break even?

$($d$)$ What is the highest rate of discount that can be advertised without incurring an absolute loss?