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.
15
on odysseyware the correct answer is 15.00
Let's break down the problem step-by-step:
Step 1: Find the amount of copper in the first alloy.
The first alloy is 10% copper and weighs 25 pounds. To find the amount of copper in the first alloy, we can multiply the weight of the alloy by its copper content in decimal form:
Amount of copper in the first alloy = 10% * 25 pounds = 0.10 * 25 pounds = 2.5 pounds
Step 2: Determine the amount of the second alloy needed.
We need to find out how much of the second alloy, with 18% copper content, we need to add to the first alloy. Let's say the weight of the second alloy is x pounds.
Since the second alloy is 18% copper, the amount of copper in the second alloy would be:
Amount of copper in the second alloy = 18% * x pounds = 0.18x pounds
Step 3: Set up the equation based on the desired composition of the mixture.
We want to achieve a mixture that is 13% copper. To set up the equation, we need to consider the total weight of the mixture and the total amount of copper in the mixture.
Total weight of the mixture = weight of the first alloy + weight of the second alloy
Total weight of the mixture = 25 pounds + x pounds = 25 + x pounds
Total amount of copper in the mixture = amount of copper in the first alloy + amount of copper in the second alloy
Total amount of copper in the mixture = 2.5 pounds + 0.18x pounds
We want the composition of the mixture to be 13% copper, so the proportion of copper to the total weight should be 13%.
This can be expressed as:
(0.13) * (Total weight of the mixture) = Total amount of copper in the mixture
Step 4: Solve the equation.
Using the equation from Step 3, we can solve for x:
(0.13) * (25 + x) = 2.5 + 0.18x
Further simplifying the equation:
3.25 + 0.13x = 2.5 + 0.18x
Subtracting 0.13x and 2.5 from both sides:
0.13x - 0.18x = 2.5 - 3.25
-0.05x = -0.75
Dividing by -0.05:
x = (-0.75)/(-0.05)
x = 15 pounds
Therefore, you need to add 15 pounds of the second alloy to the first alloy to get a 13% mixture.
To solve this problem, we can set up an equation using the concept of weighted averages.
Let's assume that x represents the amount (in pounds) of the second alloy (18% copper) that needs to be added to the first alloy.
First, we need to determine the amount of copper in the first alloy.
Since the first alloy is 10% copper and weighs 25 pounds, the amount of copper in the first alloy can be calculated as:
Amount of copper in the first alloy = 10% of 25 pounds
To find what percentage of the first alloy is copper, convert 10% into a decimal by dividing it by 100:
10% = 0.10
Therefore, the amount of copper in the first alloy is:
Amount of copper in the first alloy = 0.10 * 25 pounds
Next, let's find the amount of copper in the second alloy.
Since the second alloy is 18% copper and we want to find x pounds of it, the amount of copper in the second alloy that needs to be added is:
Amount of copper in the second alloy = 18% of x pounds
Again, convert 18% into a decimal:
18% = 0.18
So, the amount of copper in the second alloy is:
Amount of copper in the second alloy = 0.18 * x pounds
Now, we can set up the equation for the desired 13% mixture of copper:
(Amount of copper in the first alloy + Amount of copper in the second alloy) / Total weight of the mixture = 13%
((0.10 * 25) + (0.18 * x)) / (25 + x) = 0.13
Now, we solve this equation to find the value of x.
Multiply both sides of the equation by (25 + x) to eliminate the denominator:
(0.10 * 25) + (0.18 * x) = 0.13 * (25 + x)
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 = 15
Therefore, to obtain a 13% mixture, you need to add approximately 15 pounds (to the nearest tenth of a pound) of the second alloy to the first alloy.