Question

How many quarts of peanut oil worth 19 cents per quart must be mixed with 100 quarts worth 25 cents per quart to produce a mixture that will sell for 24 cents per quart?

Answer 1

Let \( x \) be the number of quarts of peanut oil worth 19 cents per quart that needs to be mixed with 100 quarts worth 25 cents per quart.

We want to create a mixture that will be worth 24 cents per quart.

First, let's calculate the total value from each component in the mixture.

1. **Value of the peanut oil:**
\[
\text{Value of } x \text{ quarts of 19 cents} = 0.19x
\]

2. **Value of the 100 quarts of 25 cents:**
\[
\text{Value of 100 quarts of 25 cents} = 100 \times 0.25 = 25 \text{ dollars}
\]

Now we can express the total value of the mixture:
\[
\text{Total value} = 0.19x + 25
\]

Next, let's find the total number of quarts in the mixture:
\[
\text{Total quarts} = x + 100
\]

Now, the price per quart of the final mixture should equal 24 cents. Therefore, we set up the equation relating the total value to the total quarts and the desired price per quart:
\[
\frac{0.19x + 25}{x + 100} = 0.24
\]

To eliminate the fraction, we can multiply both sides by \( x + 100 \):
\[
0.19x + 25 = 0.24(x + 100)
\]

Expanding the right-hand side:
\[
0.19x + 25 = 0.24x + 24
\]

Now, we can rearrange the equation to isolate \( x \):
\[
25 - 24 = 0.24x - 0.19x
\]
\[
1 = 0.05x
\]

Dividing both sides by 0.05:
\[
x = \frac{1}{0.05} = 20
\]

Thus, the number of quarts of peanut oil worth 19 cents per quart that must be mixed is \( \boxed{20} \).