A merchant buys cloth at $1.60 per yard. At what price per yard should he list the cloth so that he may sell it at a discount of 20% from the list price and still make a profit of 20% of the SALE PRICE?

If she makes 20% off the sale price, then 80% of the sale price is his cost.

So first find the sale price, s:
80% : $1.60
100% : s

Use cross multiplication to solve for s:
s=100*$1.60/80=$2.00

Since the sale price was after 20% discount, so again it is 80% of the list price. To calculate the list price L, we use proportions and cross multiplications:
80% : $2.00
100 : L
L=$2.00*100/80=$2.50

To determine the price per yard at which the merchant should list the cloth, we need to calculate the desired profit and the sale price.

Step 1: Calculate the desired profit:
The merchant wants to make a profit of 20% of the sale price. Let's call the sale price S.

Profit = 20% of S = 0.20S

Step 2: Calculate the sale price:
The merchant wants to sell the cloth at a 20% discount from the list price. Let's call the list price L.

Sale price = List price - Discount
S = L - 20% of L
S = L - (0.20L)
S = 0.80L

Step 3: Determine the price per yard for listing:
The merchant buys the cloth at $1.60 per yard. Let's call the price per yard for listing P.

Since the merchant wants to make a profit of 20% from the sale price, the cost of the cloth should be 80% of the sale price.

Cost = 80% of S = 0.80S

The merchant buys the cloth for $1.60 per yard, which is the same as the cost of the cloth per yard:

Cost = P

Now, we can set up an equation to solve for P:

P = 0.80S
P = 0.80(0.80L) [substituting S with 0.80L]
P = 0.64L

Therefore, the merchant should list the cloth at $0.64 per yard to sell it at a discount of 20% from the list price and still make a profit of 20% of the sale price.

To determine the price per yard at which the merchant should list the cloth, we can follow these steps:

Step 1: Establish the cost price.
Given that the merchant purchases the cloth at $1.60 per yard, the cost price is $1.60.

Step 2: Calculate the desired sale price.
To determine the desired sale price, we need to find the sale price at which the merchant wants to make a profit of 20%.

Profit percentage = 20%
Profit = (20/100) * Desired Sale Price

Since the desired sale price would be after a discount of 20%, we can calculate it as follows:
Sale Price = Cost Price + Profit

From the given information, we know that the sale price will be discounted by 20%. Thus, the sale price should be:
Sale Price = (Discounted Price / 100) * Original Price
Sale Price = (80/100) * List Price

Substituting the value of the sale price in terms of the list price into the profit equation, we get:
(20/100) * (80/100) * List Price = (20/100) * Desired Sale Price

Simplifying the equation, we obtain:
(16/100) * List Price = (20/100) * Desired Sale Price

Step 3: Find the desired sale price.
Calculate the desired sale price by dividing both sides of the equation by (20/100):
Desired Sale Price = (16/100) * List Price / (20/100)
Desired Sale Price = (4/5) * List Price

Step 4: Calculate the list price.
To determine the price per yard at which the merchant should list the cloth, we need to find the list price. We can find it by dividing both sides of the equation by the price per yard:
Desired Sale Price / List Price = (4/5)
List Price = Desired Sale Price / (4/5)

Step 5: Substitute the desired sale price and solve.
Now we can substitute the desired sale price, which is 20% less than the list price, into the equation:
List Price = (Desired Sale Price) / (4/5)
List Price = (Desired Sale Price) * (5/4)

In this case, we have:
List Price = Desired Sale Price * (5/4)
List Price = Desired Sale Price * 1.25

Hence, the merchant should list the cloth at a price per yard that is 1.25 times the desired sale price. This would allow the merchant to sell it at a discount of 20% from the list price and still make a profit of 20% of the sale price.