Let's analyze the given options to find which pairs [tex]\((d, n)\)[/tex], where [tex]\(d\)[/tex] represents the number of dimes and [tex]\(n\)[/tex] represents the number of nickels, satisfy the equation [tex]\(0.10d + 0.05n = 22.25\)[/tex].
### Checking Option A: [tex]\((0, 445)\)[/tex]
For [tex]\(d = 0\)[/tex] and [tex]\(n = 445\)[/tex]:
[tex]\[
0.10(0) + 0.05(445) = 0 + 22.25 = 22.25
\][/tex]
Since the left side equals the right side of the equation, [tex]\((0, 445)\)[/tex] is a solution.
### Checking Option B: [tex]\((0.5, 435)\)[/tex]
For [tex]\(d = 0.5\)[/tex] and [tex]\(n = 435\)[/tex]:
[tex]\[
0.10(0.5) + 0.05(435) = 0.05 + 21.75 = 21.80
\][/tex]
Since the left side does not equal the right side of the equation, [tex]\((0.5, 435)\)[/tex] is not a solution.
### Checking Option C: [tex]\((233, 21)\)[/tex]
For [tex]\(d = 233\)[/tex] and [tex]\(n = 21\)[/tex]:
[tex]\[
0.10(233) + 0.05(21) = 23.30 + 1.05 = 24.35
\][/tex]
Since the left side does not equal the right side of the equation, [tex]\((233, 21)\)[/tex] is not a solution.
### Checking Option D: [tex]\((118, 209)\)[/tex]
For [tex]\(d = 118\)[/tex] and [tex]\(n = 209\)[/tex]:
[tex]\[
0.10(118) + 0.05(209) = 11.80 + 10.45 = 22.25
\][/tex]
Since the left side equals the right side of the equation, [tex]\((118, 209)\)[/tex] is a solution.
### Checking Option E: [tex]\((172, 101)\)[/tex]
For [tex]\(d = 172\)[/tex] and [tex]\(n = 101\)[/tex]:
[tex]\[
0.10(172) + 0.05(101) = 17.20 + 5.05 = 22.25
\][/tex]
Since the left side equals the right side of the equation, [tex]\((172, 101)\)[/tex] is a solution.
Thus, the pairs [tex]\((d, n)\)[/tex] that are solutions to the equation [tex]\(0.10d + 0.05n = 22.25\)[/tex] are:
[tex]\[
\boxed{(0, 445), (118, 209), (172, 101)}
\][/tex]
