An alloy of metals is 25% copper. Another alloy is 50% copper. How much of each alloy should be used to make 1000 grams of an alloy that is 45% copper
To determine how much of each alloy should be used, we can set up a system of equations.
Let's assume we use x grams of the first alloy (25% copper) and y grams of the second alloy (50% copper) to make the final alloy with 45% copper.
The equation for the total weight is: x + y = 1000 grams. (Equation 1)
The equation for the total amount of copper is: 0.25x + 0.50y = 0.45(1000). (Equation 2)
To solve this system of equations, we can use the substitution or elimination method. Let's use the elimination method in this case:
Multiply Equation 1 by 0.25 to align the coefficients of x:
0.25(x + y) = 0.25(1000)
0.25x + 0.25y = 250 (Equation 3)
Now subtract Equation 3 from Equation 2 to eliminate x:
(0.25x + 0.50y) - (0.25x + 0.25y) = 112.5 - 250
0.25y - 0.25y = -137.5
0 = -137.5
This tells us that 0 = -137.5, which is not possible. Therefore, there is no solution for this system of equations.
Hence, it is not possible to use these two alloys in the given proportions to make an alloy that is 45% copper.
To determine how much of each alloy should be used to make an alloy with a desired copper content, we can use a weighted average formula. Let's denote:
x = grams of the 25% copper alloy
y = grams of the 50% copper alloy
We need to find values for x and y that satisfy the following conditions:
1) The total weight of the alloys is 1000 grams: x + y = 1000
2) The desired copper content of the resulting alloy is 45%: (0.25x + 0.50y) / 1000 = 0.45
Now, let's solve the system of equations using substitution or elimination:
From the first equation, we can express x in terms of y: x = 1000 - y
Substituting x in the second equation:
(0.25(1000 - y) + 0.50y) / 1000 = 0.45
Simplifying:
(250 - 0.25y + 0.50y) / 1000 = 0.45
(250 + 0.25y) / 1000 = 0.45
250 + 0.25y = 0.45 * 1000
250 + 0.25y = 450
0.25y = 450 - 250
0.25y = 200
y = 200 / 0.25
y = 800
Substituting y back into the first equation to find x:
x = 1000 - y
x = 1000 - 800
x = 200
Therefore, to make 1000 grams of an alloy that is 45% copper, we need to use 200 grams of the 25% copper alloy and 800 grams of the 50% copper alloy.