Your mother has left you in charge of the annual family yard sale. Before she leaves you to your entrepreneurial abilities, she explains that she has made the job easy for you: everything costs either $1.50 or $3.50. She asks you to keep track of how many of each type of item is sold, and you make a list, but it gets lost sometime throughout the day. Just before she’s supposed to get home, you realize that all you know is that there were 150 items to start with (your mom counted) and you have 41 items left. Also, you know that you made $227.50. Write a system of equations that you could solve to figure out how many of each type of item you sold.

A) x + y = 109
(1.5)x + 227.50 = (3.5)y
B) x + y = 109
(3.5)x + 227.50 = (1.5)y
C) x + y = 41
(1.5)x + 227.50 = (3.5)y
D) x + y = 109
(1.5)x + (3.5)y = 227.50
E) x + y = 150
(1.5)x + (3.5)y = 227.50
F) x + y = $3.50
(1.5)x + (3.5)y = 227.50

B) x + y = 109

(3.5)x + 227.50 = (1.5)y

The correct answer is option D) x + y = 109

(1.5)x + (3.5)y = 227.50

We have two unknowns, the number of items that cost $1.50 (x) and the number of items that cost $3.50 (y). We are given two pieces of information: the total number of items (150 at the start) and the total amount of money made ($227.50).

Using the first piece of information, we can write the equation x + y = 150. This represents the fact that the total number of items at the start (x + y) is equal to 150.

Using the second piece of information, we can write the equation (1.5)x + (3.5)y = 227.50. This represents the fact that the total amount of money made (1.5 times the number of $1.50 items plus 3.5 times the number of $3.50 items) is equal to $227.50.

Therefore, the correct system of equations is x + y = 109 and (1.5)x + (3.5)y = 227.50.

The correct system of equations to figure out how many of each type of item were sold is:

D) x + y = 109
(1.5)x + (3.5)y = 227.50

To understand why these equations are correct, let's break it down step by step:

1. First, we need to set up the equation based on the total number of items sold. Since there were 150 items to start with and you have 41 items left, the number of items sold can be calculated as 150 - 41 = 109.

2. Since we are considering two types of items with different prices ($1.50 and $3.50), we need to define variables to represent the number of each type. Let's use x to represent the number of $1.50 items sold and y to represent the number of $3.50 items sold.

3. The first equation, x + y = 109, represents the total number of items sold. This equation states that the sum of the number of $1.50 items sold (x) and the number of $3.50 items sold (y) should be equal to 109.

4. The second equation is based on the total amount of money earned. Since the cost of each $1.50 item is $1.50 and the cost of each $3.50 item is $3.50, we can calculate the total amount earned as (1.5)x + (3.5)y. This should be equal to the total money earned, which is $227.50.

Therefore, the correct system of equations is:

x + y = 109
(1.5)x + (3.5)y = 227.50

And the correct answer is option D.

Clearly D, once x and y are properly defined.