Question

A grocer has two kinds of candy: one sells for 90 cents per pound, and the other sells for 40 cents per pound. How many pounds of each kind must he use to make 50 pounds that he can sell for 85 cents per pound?

Answer 1

To find out how many pounds of each kind of candy the grocer needs, we can define two variables:

- Let \( x \) be the number of pounds of the 90-cent candy.
- Let \( y \) be the number of pounds of the 40-cent candy.

We have two main conditions based on the problem statement:

1. The total weight of the candies must equal 50 pounds:
\[
x + y = 50
\]

2. The total cost of the candies must equal the total selling price of 50 pounds at 85 cents per pound:
\[
0.90x + 0.40y = 0.85(50)
\]

Let's calculate \( 0.85 \times 50 \):
\[
0.85 \times 50 = 42.5
\]
So, the second equation becomes:
\[
0.90x + 0.40y = 42.5
\]

Now we have the following system of equations:
1. \( x + y = 50 \)
2. \( 0.90x + 0.40y = 42.5 \)

We can solve these equations step by step. First, we can solve the first equation for \( y \):
\[
y = 50 - x
\]

Now we can substitute this expression for \( y \) into the second equation:
\[
0.90x + 0.40(50 - x) = 42.5
\]
Distributing \( 0.40 \):
\[
0.90x + 20 - 0.40x = 42.5
\]
Combining like terms:
\[
(0.90 - 0.40)x + 20 = 42.5
\]
\[
0.50x + 20 = 42.5
\]

Next, we subtract 20 from both sides:
\[
0.50x = 22.5
\]

Now, divide by 0.50:
\[
x = \frac{22.5}{0.50} = 45
\]

Now that we have \( x \), we can find \( y \):
\[
y = 50 - x = 50 - 45 = 5
\]

Thus, the grocer needs:
- **45 pounds of the 90-cent candy**, and
- **5 pounds of the 40-cent candy**.

To summarize:
\[
\text{The grocer must use } 45 \text{ pounds of the 90-cent candy and } 5 \text{ pounds of the 40-cent candy.}
\]