To solve the problem of combining a metal containing [tex]\(45\%\)[/tex] nickel with [tex]\(6\)[/tex] mg of pure nickel to form an alloy containing [tex]\(78\%\)[/tex] nickel, let's go through the steps required to derive a suitable equation.
### Step-by-Step Solution
1. Define Variables
- Let [tex]\( a \)[/tex] be the amount of the metal containing [tex]\(45\%\)[/tex] nickel.
- Since we know the amount of pure nickel added is 6 mg, we use that directly in our equation.
2. Calculate Nickel Content
- In [tex]\( a \)[/tex] mg of metal containing [tex]\(45\%\)[/tex] nickel, the amount of nickel is [tex]\(0.45a\)[/tex] mg.
- In 6 mg of pure nickel, the amount of nickel is [tex]\(6\)[/tex] mg.
3. Formulate Equation for Total Nickel Content
- The total weight of the alloy will be [tex]\(a + 6\)[/tex] mg.
- The total amount of nickel in the alloy will be [tex]\(0.45a + 6\)[/tex] mg.
- We want the final alloy to contain [tex]\(78\%\)[/tex] nickel, so the content of nickel in the alloy should be [tex]\(0.78\)[/tex] times the total weight of the alloy.
4. Set Up Equality
- Set up the equation relating the amount of nickel in the alloy to the percentage of nickel desired:
[tex]\[
\frac{0.45a + 6}{a + 6} = 0.78
\][/tex]
5. Identify the Correct Equation
- Compare this equation with the given answer choices:
- [tex]\(\frac{a(0.45)+6(1.00)}{c^{a+6}}=\frac{0.78}{100}\)[/tex]
- [tex]\(\frac{a(0.45)+6(100)}{a+6}=\frac{78}{100}\)[/tex]
- [tex]\(\frac{a(0.45)+6(1.00)}{1.45}=\frac{78}{100}\)[/tex]
- [tex]\(\frac{a(0.45)+6(1.00)}{a+6}=\frac{78}{100}\)[/tex]
- Clearly, the correct equation matching our derived equation is:
[tex]\[
\frac{a(0.45)+6(1.00)}{a+6}=\frac{78}{100}
\][/tex]
This is the equation that can be used to solve for [tex]\(a\)[/tex].
### Solution for [tex]\(a\)[/tex]
When solving this equation, we find that:
[tex]\[
a = 4 \text{ mg}
\][/tex]
Therefore, 4 mg of the metal containing [tex]\(45\%\)[/tex] nickel must be combined with 6 mg of pure nickel to form an alloy containing [tex]\(78\%\)[/tex] nickel.
