A metallurgist has one alloy containing 43% aluminum and another containing 63% aluminum. How many pounds of each alloy must he use to make 43 pounds of a third alloy containing 49% aluminum?

Question

A metallurgist has one alloy containing 43% aluminum and another containing 63% aluminum. How many pounds of each alloy must he use to make 43 pounds of a third alloy containing 49% aluminum? 

Answer 1

A metallurgist has one alloy containing 43% aluminum and another containing 63% aluminum. How many pounds of each alloy must he use to make 43 pounds of a third alloy containing 49% aluminum? 

This is a mixture problem. A chart might be helpful, in order to assist you with coming up with the equations to solve this problem.

Items	Amount of Each Item	% for each item (as a decimal)	Total Value of Item
43% aluminum	x	.43	.43x
63% aluminum	43 - x	.63	.63(43 – x)
Mixture (49% aluminum	43	.49	43*.49 = .43x + .63(43 – x)

With mixture problems such as this one that have to do with percentages, you can multiply across and add down in order to obtain your equations. Therefore, the Amount of Each Item * % for each item (as a decimal) = Total Value of Item.

Again, the chart is provided only to assist one with coming up with the two equations. Since the total mixture (third allow) will have 43 pounds, then you can call one of the amounts x, since you do not know what that is, and the other amount will be the total amount in the mixture (the third allow) minus x. Therefore, if you call the Amount of the 43% aluminum used x, then the amount of the 63% aluminum used must be 43 – x.

Next, you can multiple across using the formula. Therefore, the Total Value of the 43% aluminum would be: x * .43 = .43x. This is because 43% as a decimal is .43 because to change from a percent to a decimal, you can divide the number given by 100, or you can move the decimal point back two decimal places. By the same formula, the Total Value of the 63% aluminum would be: (43 – x) * .63 = .63(43 – x) = 27.09 - .63x because 63% as a decimal is .63. Also by the same formula, the Total Value of the Mixture (the third alloy) would be 43 * .49 = 21.07, because 49% as a decimal is .49.

Now you can add down, to make an equation from the last column, because the Total Value from the 43% aluminum plus the total value from the 63% aluminum will equal the total value from the mixture (the third alloy that is 49% aluminum). Therefore, .43x + 27.09 - .63x = 21.07. We can now use this equation to solve for x. Since .43x + 27.09 - .63x = 21.07, we can collect our like terms on the left hand side of the equation. Like terms have the same variable and the same exponent. Therefore, .43x - .63x = -.20x. Now, we have a new equation of -.20x + 27.09 = 21.07. Now, we need to get the number on the other side of the equation from the variable term. The opposite of +27.09 – 27.09. Therefore, we can subtract 27.09 from both sides of the equation, and now -.20x + 27.09 – 27.09 = 21.07 – 27.09. Since 27.09 – 27.09 = 0 and 21.07 – 27.09 = -6.02, we now have a new equation of -.20x = -6.02. Once you are down to one term on each side of the equation, then you can divide. The opposite of multiplying x by -.20 is dividing x by -.20. Thus, you can now divide both sides of the equation by -.20. Therefore, -.20x / -.20 = -6.02 / -.20, which means x = -6.02 / - .20. This simplifies to x = 30.1.

Therefore, since x was the amount of the 43% aluminum, the amount of the 43% aluminum in the mixture is 30.1 pounds. To obtain the amount of the 63% aluminum, simply subtract 43 – x, which means 43 – 30.1 = 12.9. Therefore, the amount of the 63% aluminum in the mixture (the third alloy which is 49% aluminum) is 12.9 pounds.

To check your answer, you can multiply .43 * 30.1, which is 12.943. You can then multiply .63 * 12.9, which is 8.127. Then, check to see if when you add the two amounts, you obtain the 21.07 for the total value of the mixture (third alloy with 49% aluminum). 12.943 + 8.127 = 21.07. Therefore, your answers are correct and you have solved the problem.