For $1.80 a frocer buys a case of fruit which contains 12 dozen. She knows that two dozen will spoil before she sells them. At what price per dozel must she sell the good ones to gain half of the whole cost?

I assume you mean she wants to make 50% profit, based on a cost of $1.80/dozen. (The wording is ambiguous. If the problem was presented as you posted it, it's from a pretty sloppy text.)

10x = 1.5(1.80*12)
x = 3.24

To find the price per dozen, we need to determine the total cost and then divide it by the number of dozens.

First, let's calculate the total cost of the case of fruit:
- The price of the case is $1.80.
- There are 12 dozen in the case.
- Two dozen will spoil before they are sold, so we need to subtract them from the total.

Total cost = $1.80 - (2 dozen * price per dozen)

To gain half of the whole cost, the selling price should be half of the total cost:
Selling price = 1/2 * Total cost

Therefore, we can equate the two equations:
Selling price = Total cost / 2
$1.80 - (2 dozen * price per dozen) = (1/2) * ($1.80 - (2 dozen * price per dozen))

Let's solve this equation step-by-step:
1) Distribute the (1/2) to the terms on the right side:
$1.80 - (2 dozen * price per dozen) = $0.90 - dozen * price per dozen

2) Move the terms with price per dozen to one side and the rest to the other side:
(2 dozen * price per dozen) - price per dozen = $1.80 - $0.90

3) Combine like terms:
dozen * price per dozen = $0.90

4) Divide both sides by dozen:
price per dozen = $0.90 / dozen

Therefore, she must sell the good ones for $0.90 per dozen to gain half of the whole cost.

To solve this problem, we need to determine the cost per dozen and then calculate the selling price needed to gain half of the whole cost. Here's how we can do it:

1. Calculate the total cost of the case of fruit: Since the grocer bought a case for $1.80 and it contains 12 dozen, we can divide the total cost by the number of dozen to find the cost per dozen.
Total Cost = $1.80
Number of Dozen = 12
Cost per Dozen = Total Cost / Number of Dozen

2. Determine the number of dozen that will be sold: The problem states that two dozen will spoil before the grocer sells them. So, we subtract the number of spoiled dozen from the total number of dozen to find the number of good dozens that will be sold.
Number of Good Dozen = Total Number of Dozen - Number of Spoiled Dozen

3. Calculate the cost of the good dozens: To find the cost of the good dozens, we multiply the cost per dozen by the number of good dozens.
Cost of Good Dozen = Cost per Dozen * Number of Good Dozen

4. Determine the selling price needed to gain half of the whole cost: To gain half of the whole cost, the selling price must equal half of the total cost. So, we divide the cost of the good dozens by two and divide it by the number of good dozens to find the selling price per dozen.
Selling Price per Dozen = (Cost of Good Dozen / 2) / Number of Good Dozen

Now, let's plug in the values into the formulas:

Total Cost = $1.80
Number of Dozen = 12
Number of Spoiled Dozen = 2

1. Cost per Dozen = $1.80 / 12 = $0.15
2. Number of Good Dozen = 12 - 2 = 10
3. Cost of Good Dozen = $0.15 * 10 = $1.50
4. Selling Price per Dozen = ($1.50 / 2) / 10 = $0.075

Therefore, the grocer must sell the good dozens for $0.075 (or $0.08 rounded to the nearest cent) per dozen in order to gain half of the whole cost.