Question

Anne and Nancy use a metal alloy that is 19.8% copper to make jewelry. How many ounces of an alloy that is 10% copper must be mixed with an alloy that is 24% copper to form 70 ounces of the desired alloy?

Answer 1

Anne and Nancy must mix 21 ounces of a 10% copper alloy with 49 ounces of a 24% copper alloy to create 70 ounces of the desired 19.8% copper alloy.

Step-by-step explanation:

To solve the problem, we will set up a system of equations based on the percentage of copper in the final alloy and the total mass of the alloy. Anne and Nancy want to make 70 ounces of alloy that is 19.8% copper. Let's denote x as the number of ounces of the 10% copper alloy and y as the number of ounces of the 24% copper alloy.

The total mass of the alloy must be 70 ounces:

• x + y = 70 (equation 1)
  

The total percentage of copper in the final alloy is 19.8%, which is given by:

• 
  0.10x + 0.24y = 0.198 × 70 (equation 2)
  

Solving these equations simultaneously, first multiply equation 2 by 100 to clear the decimals:

• 
  10x + 24y = 19.8 × 70
  

Now multiply equation 1 by 10 and subtract it from the modified equation 2 to eliminate x:

• 
  14y = 9.8 × 70
  
• 
  y = (9.8 × 70) / 14
  
• 
  y = 49 ounces (approx.)

Substitute the value of y back into equation 1 to find x:

• 
  x = 70 – 49
  
• 
  x = 21 ounces

Therefore, Anne and Nancy must mix 21 ounces of the 10% copper alloy with 49 ounces of the 24% copper alloy to get 70 ounces of the desired 19.8% copper alloy.

Answer 2

To create 70 ounces of a 19.8% copper alloy, one must find the correct proportions of a 10% copper alloy and a 24% copper alloy. This involves solving a system of equations that represents the total amount of the mixture (70 ounces) and the total copper content (19.8% of 70 ounces).

Step-by-step explanation:

To solve the problem of mixing different percentages of copper alloys to form a desired alloy, we can set up a system of equations based on the amount of copper in each alloy and the total desired amount of the final alloy. We want to mix some amount of 10% copper alloy with some amount of 24% copper alloy to get 70 ounces of a 19.8% copper alloy.

Let's denote x as the amount of the 10% copper alloy and y as the amount of the 24% copper alloy to be mixed.

Our system of equations will be based on the following two conditions:
1. The total amount of the mixture should be 70 ounces, so: x + y = 70
2. The total amount of copper in the mixture should be 19.8% of 70 ounces:

• (0.10x) + (0.24y) = 0.198 × 70

We can solve the system of equations to find the values of x and y.