An alloy of copper is 10% copper and weighs 25 pounds. A second alloy is 18% copper. How much (to the nearest tenth lb.) of the second alloy must be added to the first alloy to get a 13% mixture.
add up the amount of copper in each part. It must total the amount of copper in the final alloy
.10*25 + .18x = .13(25+x)
now just solve for x
Let's suppose the amount of the second alloy to be added is x pounds.
Step 1: Calculate the amount of copper in the first alloy.
The first alloy is 10% copper and weighs 25 pounds, so the amount of copper in the first alloy is 10% of 25 pounds:
Amount of copper in the first alloy = (10/100) * 25 pounds = 2.5 pounds
Step 2: Calculate the amount of copper in the second alloy.
The second alloy is 18% copper, so the amount of copper in x pounds of the second alloy is:
Amount of copper in the second alloy = (18/100) * x pounds = 0.18x pounds
Step 3: Calculate the total amount of copper in the mixture.
To get a 13% mixture, the amount of copper in the mixture should be 13% of the total weight. The total weight of the mixture will be the sum of the weights of the first alloy (25 pounds) and the second alloy (x pounds).
Amount of copper in the mixture = (13/100) * (25 + x) pounds = 0.13(25 + x) pounds
Step 4: Set up an equation and solve for x.
Since the amount of copper in the mixture is the sum of the copper in the first and second alloys, we can set up the equation:
2.5 pounds + 0.18x pounds = 0.13(25 + x) pounds
Now, let's solve for x:
2.5 + 0.18x = 0.13(25 + x)
2.5 + 0.18x = 0.13 * 25 + 0.13 * x
2.5 + 0.18x = 3.25 + 0.13x
0.18x - 0.13x = 3.25 - 2.5
0.05x = 0.75
x = 0.75 / 0.05
x = 15
So, to get a 13% mixture, you need to add approximately 15 pounds (to the nearest tenth lb.) of the second alloy to the first alloy.
To solve this problem, we need to understand how to calculate the quantity of copper in each alloy and then determine the amount of the second alloy needed to achieve the desired mixture. Here's how we can approach it:
1. Let's start by finding the amount of copper in the first alloy:
- The first alloy is 10% copper, which means that for every pound of alloy, 0.10 pounds are copper.
- Therefore, the amount of copper in the first alloy is 0.10 * 25 pounds = 2.5 pounds.
2. Let's assume x represents the weight (in pounds) of the second alloy to be added.
3. Now, we need to find the amount of copper in the second alloy:
- The second alloy is 18% copper, which means that for every pound of alloy, 0.18 pounds are copper.
- So, the amount of copper in the second alloy is 0.18x pounds.
4. We want to achieve a 13% copper mixture by combining the first and second alloys. This means the amount of copper in the final mixture should be equal to 13% of the total weight of the mixture.
5. Let's set up an equation to solve for x:
- The amount of copper in the final mixture is the sum of the copper in the first alloy (2.5 pounds) and the copper in the second alloy (0.18x pounds).
- The total weight of the mixture is the sum of the weight of the first alloy (25 pounds) and the weight of the second alloy (x pounds).
- According to the 13% mixture requirement, the amount of copper in the final mixture should be 0.13 times the total weight of the mixture.
- Therefore, we have the equation: 2.5 + 0.18x = 0.13(25 + x)
6. Now, let's solve the equation to find the value of x:
- Distribute 0.13 to simplify the equation: 2.5 + 0.18x = 3.25 + 0.13x
- Subtract 0.13x from both sides: 0.05x = 0.75
- Divide both sides by 0.05: x = 0.75 / 0.05
- x is approximately 15 pounds.
Therefore, to obtain a 13% copper mixture, approximately 15 pounds of the second alloy must be added to the first alloy.