A metalworker has a metal alloy that is ​30% copper and another alloy that is 70​% copper. How many kilograms of each alloy should the metalworker combine to create 120kg of a 62​% copper​ alloy?

Question

Please someone help me

Answer 1

Samuel, I'm going to help you, but I want you to read carefully everything I write so that you will be able to do these problems on your own in the future. They aren't easy, but you can do it!!

First, read the word problem and use your powers of reading comprehension and your understanding of math to name the terms. I am going to call my terms, my unknowns, F and G.

F: amount in kilograms of the first alloy, which is 30% copper and 70% of the second alloy

G: amount in kilograms of the second alloy, which is 70% copper

Now, you need to set up a couple of equations. The fact that you have two unknowns is a hint that you need a minimum of two equations.

So we know that the metalwork wants to create 120 kilograms of a third alloy, using the first and second alloys. This leads to the equation: F + G = 120

We have this specification: That the 120 kilograms of the newly created alloy needs to be 62% copper. Let's calculate what 62% (= 0.62) of 120 kilograms is. Remember "of," in this case, translates as multiplication. That problem is 0.62 times 120, which equals 74.4 kilograms.

Remember: 30% (=0.30 = 0.3) of F is copper, and 70% (=0.70 = 0.7)of G is copper. This leads to our second equation: 0.3F + 0.7G = 74.4

Now we have two equations and two unknowns (F and G). You solve first for one unknown (F or G) and then solve for the other. Then, you state the answer in a sentence, something like, "The metalworker would need ?? kilograms of the alloy that is 30% copper and ??? kilograms of the alloy that is 70% copper to get 120 kilograms of a third alloy that is 62% copper."

I do not like to do my students' homework for them. I am assuming that you can solve these two equations that I will rewrite below. To solve, you can use graphing if you have a graphing calculator and your teacher allows it. You can solve by substitution, or you can solve by elimination. Here are the equations:

F + G = 120

0.3F + 0.7G = 74.4

HINT: G > F

Answer 2

Hi Samuel! I'd love to help you with this.

We have two different types of information in this problem, mass (kilograms) and composition (% copper), so we need two different equations to solve this.

We can start with mass. We need a total of 120 kilograms and we have two different alloys. Since the amount of each alloy is unknown (and what we're solving for), we use variables to express this. So if we say that the amount of alloy 1 (the 30% copper alloy) is x kilograms and that the amount of alloy 2 (the 70% copper alloy) is y kilograms, then we can write our first equation: x + y = 120

Notice that all parts of this equation have the same units (kilograms).

For our second equation, we need to deal with the percent copper in each alloy. We need our final alloy to be 62% copper and knowing that we can write percentages as decimals, we can express this as an equation like so: 0.3x + 0.7y = 0.62*120.

Notice again that all parts of this equation have the same units (percent multiplied by kilograms).

Now we have a system of equations; two equations to solve two different variables.

x + y = 120

0.3x + 0.7y = 0.62*120

To solve this set of equations, we can use substitution, elimination, or graphing. However, since we have an equation that has variable with no coefficients, substitution will be the easiest.

Let's solve for x (you can also solve for y) in the first equation: x = 120-y

Now we can plug this value for x into our second equation: 0.3(120-y) + 0.7y = 0.62*120

Expand: 36 - 0.3y + 0.7y = 74.4

Combine like terms: 0.4y = 74.4 - 36 ---> 0.4y = 38.4

Divide: 0.4y/0.4 = 38.4/0.4 ---> y = 96

Now that we know one variable, we can find x by plugging y into either of the two original equations. For simplicity, lets plug it into the first equation (x + y = 120)

x + (96) = 120

so, x = 120 - 96 = 24

Answer: The metalworker should use 24 kilograms of the 30% copper alloy and 96 kilograms of the 70% copper alloy.

Answer 3

This is a systems of equations problem.

Let's assume X= "The Metal Alloy with 30% Copper" and Y=" The Metal Alloy with 70% Copper".

We have two variables so we will need two equations.

First Equation: X+Y=120 (The total amount of kilograms of both metal alloys)

Second Equation: 0.30X+0.70Y=0.62(120) (The amount of copper present in the metal alloys X and Y to be 62% of 120kg)

Manipulate the First Equation to obtain Y-120-X, then substitute into the second equation

.30X+.70(120-X)=74.4

.30X+84-.70X =74.4

84-0.4X =74.4

...So X=24 kg

24+Y=120

...So Y=96 kg