Certainly! To find the amount of the 60% mixture used to create the final 65% mixture, we need to set up the equation based on the information provided.
Let's break it down step-by-step.
1. Define the variables:
- Let [tex]\( x \)[/tex] be the amount of the 60% copper mixture (in pounds).
- Since the total weight of the final mixture is 100 pounds, the amount of the 80% copper mixture will be [tex]\( 100 - x \)[/tex] pounds.
2. Set up the equation:
The total amount of copper contributed by each mixture should equal the amount of copper in the final mixture.
- From the 80% copper mixture:
[tex]\[
0.8 \times (100 - x)
\][/tex]
Here, [tex]\( 0.8 \)[/tex] represents the 80% copper content, and [tex]\( 100 - x \)[/tex] is the amount of the 80% mixture.
- From the 60% copper mixture:
[tex]\[
0.6 \times x
\][/tex]
Here, [tex]\( 0.6 \)[/tex] represents the 60% copper content, and [tex]\( x \)[/tex] is the amount of the 60% mixture.
- The amount of copper in the final 65% mixture:
[tex]\[
0.65 \times 100
\][/tex]
Since the final mixture weighs 100 pounds with 65% copper content.
3. Combine these into a single equation:
[tex]\[
0.8(100 - x) + 0.6x = 100 \times 0.65
\][/tex]
So the equation properly set up to find [tex]\( x \)[/tex] is:
[tex]\[
0.8(100 - x) + 0.6x = 100 \times 0.65
\][/tex]
You can solve this equation to find the value of [tex]\( x \)[/tex], the amount of the 60% copper mixture used.
