Let's solve the given problem step-by-step:
1. Understand the Equations:
We have two types of coins: nickels (n) and quarters (q). We are given:
- The total number of coins is 108:
[tex]\[ n + q = 108 \][/tex]
- The total value of the coins is $21:
[tex]\[ 0.05n + 0.25q = 21 \][/tex]
2. Express [tex]\( q \)[/tex] in Terms of [tex]\( n \)[/tex] Using the First Equation:
From the equation [tex]\( n + q = 108 \)[/tex], we can solve for [tex]\( q \)[/tex]:
[tex]\[ q = 108 - n \][/tex]
3. Substitute [tex]\( q \)[/tex] into the Second Equation:
We'll substitute [tex]\( q \)[/tex] with [tex]\( 108 - n \)[/tex] in the second equation [tex]\( 0.05n + 0.25q = 21 \)[/tex]:
[tex]\[ 0.05n + 0.25(108 - n) = 21 \][/tex]
4. Simplify and Solve the Equation:
[tex]\[ 0.05n + 0.25 \times 108 - 0.25n = 21 \][/tex]
[tex]\[ 0.05n + 27 - 0.25n = 21 \][/tex]
Combine like terms:
[tex]\[ 27 - 0.20n = 21 \][/tex]
Isolate the term with [tex]\( n \)[/tex]:
[tex]\[ 27 - 21 = 0.20n \][/tex]
[tex]\[ 6 = 0.20n \][/tex]
Solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{6}{0.20} \][/tex]
[tex]\[ n = 30 \][/tex]
5. Find [tex]\( q \)[/tex]:
Since [tex]\( n = 30 \)[/tex] and [tex]\( q = 108 - n \)[/tex]:
[tex]\[ q = 108 - 30 \][/tex]
[tex]\[ q = 78 \][/tex]
6. Check the Possible Values to Replace [tex]\( q \)[/tex] in the Chart:
The possible values to replace [tex]\( q \)[/tex] in the chart are given as:
- 21
- 108
- [tex]\( 21 - n \)[/tex]
- [tex]\( 108 - n \)[/tex]
From the substitution [tex]\( q = 108 - n \)[/tex], this directly tells us that the correct replacement for [tex]\( q \)[/tex] is [tex]\( 108 - n \)[/tex].
Therefore, the value that can replace [tex]\( q \)[/tex] in the chart is:
[tex]\[ \boxed{108 - n} \][/tex]
