Answer :
To identify which pairs [tex]\((d, n)\)[/tex] are solutions to the equation [tex]\(0.10d + 0.05n = 22.25\)[/tex], we will substitute each provided pair into the equation and check whether the left-hand side equals 22.25.
1. Checking pair [tex]\((0, 445)\)[/tex]:
- Substitute [tex]\(d = 0\)[/tex] and [tex]\(n = 445\)[/tex] into the equation:
[tex]\[ 0.10(0) + 0.05(445) = 0 + 22.25 = 22.25 \][/tex]
- The left-hand side is 22.25, which equals the right-hand side. Thus, [tex]\((0, 445)\)[/tex] is a solution.
2. Checking pair [tex]\((0.50, 435)\)[/tex]:
- Substitute [tex]\(d = 0.50\)[/tex] and [tex]\(n = 435\)[/tex] into the equation:
[tex]\[ 0.10(0.50) + 0.05(435) = 0.05 + 21.75 = 21.80 \][/tex]
- The left-hand side is 21.80, which is not equal to the right-hand side. Thus, [tex]\((0.50, 435)\)[/tex] is not a solution.
3. Checking pair [tex]\((233, 21)\)[/tex]:
- Substitute [tex]\(d = 233\)[/tex] and [tex]\(n = 21\)[/tex] into the equation:
[tex]\[ 0.10(233) + 0.05(21) = 23.30 + 1.05 = 24.35 \][/tex]
- The left-hand side is 24.35, which is not equal to the right-hand side. Thus, [tex]\((233, 21)\)[/tex] is not a solution.
4. Checking pair [tex]\((118, 209)\)[/tex]:
- Substitute [tex]\(d = 118\)[/tex] and [tex]\(n = 209\)[/tex] into the equation:
[tex]\[ 0.10(118) + 0.05(209) = 11.80 + 10.45 = 22.25 \][/tex]
- The left-hand side is 22.25, which equals the right-hand side. Thus, [tex]\((118, 209)\)[/tex] is a solution.
5. Checking pair [tex]\((172, 101)\)[/tex]:
- Substitute [tex]\(d = 172\)[/tex] and [tex]\(n = 101\)[/tex] into the equation:
[tex]\[ 0.10(172) + 0.05(101) = 17.20 + 5.05 = 22.25 \][/tex]
- The left-hand side is 22.25, which equals the right-hand side. Thus, [tex]\((172, 101)\)[/tex] is a solution.
Therefore, the pairs [tex]\((d, n)\)[/tex] that are solutions to the given equation are:
- [tex]\((0, 445)\)[/tex]
- [tex]\((118, 209)\)[/tex]
- [tex]\((172, 101)\)[/tex]
So, the correct selections are:
- A. [tex]\((0, 445)\)[/tex]
- D. [tex]\((118, 209)\)[/tex]
- E. [tex]\((172, 101)\)[/tex]
1. Checking pair [tex]\((0, 445)\)[/tex]:
- Substitute [tex]\(d = 0\)[/tex] and [tex]\(n = 445\)[/tex] into the equation:
[tex]\[ 0.10(0) + 0.05(445) = 0 + 22.25 = 22.25 \][/tex]
- The left-hand side is 22.25, which equals the right-hand side. Thus, [tex]\((0, 445)\)[/tex] is a solution.
2. Checking pair [tex]\((0.50, 435)\)[/tex]:
- Substitute [tex]\(d = 0.50\)[/tex] and [tex]\(n = 435\)[/tex] into the equation:
[tex]\[ 0.10(0.50) + 0.05(435) = 0.05 + 21.75 = 21.80 \][/tex]
- The left-hand side is 21.80, which is not equal to the right-hand side. Thus, [tex]\((0.50, 435)\)[/tex] is not a solution.
3. Checking pair [tex]\((233, 21)\)[/tex]:
- Substitute [tex]\(d = 233\)[/tex] and [tex]\(n = 21\)[/tex] into the equation:
[tex]\[ 0.10(233) + 0.05(21) = 23.30 + 1.05 = 24.35 \][/tex]
- The left-hand side is 24.35, which is not equal to the right-hand side. Thus, [tex]\((233, 21)\)[/tex] is not a solution.
4. Checking pair [tex]\((118, 209)\)[/tex]:
- Substitute [tex]\(d = 118\)[/tex] and [tex]\(n = 209\)[/tex] into the equation:
[tex]\[ 0.10(118) + 0.05(209) = 11.80 + 10.45 = 22.25 \][/tex]
- The left-hand side is 22.25, which equals the right-hand side. Thus, [tex]\((118, 209)\)[/tex] is a solution.
5. Checking pair [tex]\((172, 101)\)[/tex]:
- Substitute [tex]\(d = 172\)[/tex] and [tex]\(n = 101\)[/tex] into the equation:
[tex]\[ 0.10(172) + 0.05(101) = 17.20 + 5.05 = 22.25 \][/tex]
- The left-hand side is 22.25, which equals the right-hand side. Thus, [tex]\((172, 101)\)[/tex] is a solution.
Therefore, the pairs [tex]\((d, n)\)[/tex] that are solutions to the given equation are:
- [tex]\((0, 445)\)[/tex]
- [tex]\((118, 209)\)[/tex]
- [tex]\((172, 101)\)[/tex]
So, the correct selections are:
- A. [tex]\((0, 445)\)[/tex]
- D. [tex]\((118, 209)\)[/tex]
- E. [tex]\((172, 101)\)[/tex]
