Par Putters Company sells golf balls for ​$34 per dozen. The​ store's overhead expenses are 46​% of cost and the owners require a profit of 24​% 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?

Let's break down each part of the problem step by step:

We are given:
- Selling price of golf balls: $34 per dozen.
- Overhead expenses: 46% of the cost.
- Required profit: 24% of the cost.

### Part (a): For how much does Par Putters Company buy one dozen golf balls?
Let $ C $ represent the cost price of one dozen golf balls (the amount the company buys them for). 

The selling price formula, including overhead and profit, is:
\[
\text{Selling Price} = C + 0.46C + 0.24C = 1.7C
\]

Since we know the selling price is $34, we can solve for $ C $:
\[
34 = 1.7C
\]
\[
C = \frac{34}{1.7} = 20
\]

So, Par Putters Company buys one dozen golf balls for **$20**.

### Part (b): What is the price needed to cover all of the costs and expenses?
The price to cover all costs and expenses is the cost price plus the overhead expenses (without any profit). Since the overhead is 46% of the cost:
\[
\text{Price to cover costs and expenses} = C + 0.46C = 1.46C
\]
Using $ C = 20 $, we calculate:
\[
1.46 \times 20 = 29.20
\]

So, the price needed to cover all of the costs and expenses is **$29.20**.

### Part (c): What is the highest rate of markdown at which the store will still break even?
The break-even point is when the store covers all costs and expenses, i.e., sells the product at the price calculated in part (b) ($29.20).

The markdown rate is the percentage reduction from the selling price to the break-even price:
\[
\text{Markdown rate} = \frac{34 - 29.20}{34} \times 100
\]
\[
\text{Markdown rate} = \frac{4.80}{34} \times 100 \approx 14.12\%
\]

So, the highest markdown rate at which the store will still break even is **14.12%**.

### Part (d): What is the highest rate of discount that can be advertised without incurring an absolute loss?
The absolute loss occurs when the selling price drops below the cost price of $20. So, the highest rate of discount that can be offered is the percentage reduction from the original selling price ($34) to the cost price ($20):
\[
\text{Discount rate} = \frac{34 - 20}{34} \times 100
\]
\[
\text{Discount rate} = \frac{14}{34} \times 100 \approx 41.18\%
\]

Thus, the highest rate of discount that can be advertised without incurring a loss is **41.18%**.

---

Would you like more details on any part of the solution?

Here are 5 follow-up questions for deeper exploration:
1. What would happen to the selling price if overhead expenses increased by 10%?
2. If the profit margin was reduced to 15%, what would the new selling price be?
3. How would the break-even point change if the cost of goods increased by $5 per dozen?
4. If Par Putters wanted a profit of 30%, how would that affect the markdown rate?
5. How can the company reduce overhead to maximize profit without increasing prices?

**Tip:** Understanding the relationship between cost, expenses, and profit is crucial for pricing strategies that ensure profitability.

Par Putters Company sells golf balls for ​$34 per dozen. The​ store's overhead expenses are 46​% of cost and the owners require a profit of 24​% 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?

Learn how to calculate the cost price, required price to cover expenses, highest markdown rate for breaking even, and the highest discount rate for Par Putters golf balls. This detailed solution covers algebraic and percentage-based concepts and is suitable for students in Grades 9-12.

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