10. 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.

Question

10. 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.

Answer 1

Let's use the variables x and y to represent the number of items sold at $1.50 and $3.50, respectively.

We know that:

x + y = 150 (because there were 150 items to start with)
x - 41 = number of $1.50 items left
y - 41 = number of $3.50 items left
1.5x + 3.5y = 227.5 (because you made $227.50)

Using substitution, we can solve for one of the variables in one equation and substitute it into the other equation:

x + y = 150

y = 150 - x

1.5x + 3.5y = 227.5

1.5x + 3.5(150 - x) = 227.5

1.5x + 525 - 3.5x = 227.5

-2x = -297.5

x = 148.75

Now that we know x, we can solve for y:

y = 150 - x

y = 1.25

So we sold 148.75 items at $1.50 and 1.25 items at $3.50. Of course, this is not possible because we can't sell 0.75 of an item. We made a mistake somewhere in our calculations. One possibility is that we made a mistake when calculating the number of $3.50 items sold (y). We can check by solving for y using the other equation:

y - 41 = number of $3.50 items left

y = number of $3.50 items left + 41

y = (150 - x) + 41

y = 191 - x

Now we can substitute this expression for y in the second equation:

1.5x + 3.5y = 227.5

1.5x + 3.5(191 - x) = 227.5

1.5x + 668.5 - 3.5x = 227.5

-2x = -441

x = 220.5

This doesn't make sense either because we can't sell 70.5 more items than we started with. So there must be some other mistake in our calculations. We should double-check our work and make sure we didn't make any typos or arithmetic errors.