Chapter 7.3, Problem 90E

Solution

To state: The weight of each alloy, in grams, that is required to produce 80g of a new alloy that is 34% silver if the amount of 30% alloy is twice the amount of 22% alloy used. Stewart’s Metals has three silver alloys, first is 22% silver, another is 30% silver, and the third is 42% silver.

The weight of each alloy is 14.55g of the first alloy, 29.10g of the second alloy, and 36.35 of the third alloy.

Given information:

Stewart’s Metals ha three silver alloys, first is 22% silver, another is 30% silver, and the third is 42% silver.

Explanation:

Let’s take x as the number of grams of the 22% silver alloy, y as the number of grams of the 30% silver alloy, and z as the number of grams of the 42% silver alloy.

Now make equations using the given information.

Total amount of the new alloy is 80g which consists of all the given three alloys:

  $x+y+z=80$

New alloy contains 34% silver:

  $\begin{array}{c}\left(0.22\right)x+\left(0.30\right)y+\left(0.42\right)z=\left(0.34\right)80\\ 0.22x+0.30y+0.42z=27.2\end{array}$

In the new alloy, the amount of 30% alloy is twice the amount of 22% alloy used: $y=2x$

The three equations are:

  $\begin{array}{c}x+y+z=80\cdots \left(1\right)\\ 0.22x+0.30y+0.42z=27.2\cdots \left(2\right)\\ y=2x\cdots \left(3\right)\end{array}$

Multiply the first equation by -0.42,

  $\begin{array}{l}\left(x+y+z=80\right)\times \left(-0.42\right)\\ -0.42x-0.42y-0.42z=-33.6\cdots \left(4\right)\end{array}$

Now add equations (2) and (4),

  $\begin{array}{l}-0.42x-0.42y-0.42z=-33.6\\ \underset{¯}{0.22x+0.30y+0.42z=27.2}\\ -0.20x-0.12y=-0.64\end{array}$

Substitute the value of $y=2x\text{in}-0.20x-0.12y=-0.64$ ,

  $\begin{array}{c}-0.20x-0.12\left(2x\right)=-0.64\\ -0.20x-0.24x=-0.64\\ -0.44x=-0.64\\ x=14.55\end{array}$

Now substitute the value of $x=14.55\text{in}y=2x$ ,

  $\begin{array}{l}y=2x\\ y=2\left(14.55\right)\\ y=29.10\end{array}$

Now substitute the values of x and y in equation (1),

  $\begin{array}{c}x+y+z=80\\ 14.55+29.10+z=80\\ z=36.35\end{array}$

Therefore, the weight of each alloy, in grams, that is required to produce 80g of a new alloy that is 34% silver is 14.55g of the first alloy, 29.10g of the second alloy, and 36.35g of the third alloy.