Sure, let's go through the solution step-by-step.
Sofia has 25 coins in total, consisting of nickels (N) and dimes (D), and the combined value of these coins is \[tex]$1.65. We will use these pieces of information to set up two equations.
Step 1: Setting up the equations
1. The first equation relates to the value of the coins. Since a nickel is worth \$[/tex]0.05 and a dime is worth \$0.10, the total value of Sofia's coins can be expressed as:
[tex]\[
0.05N + 0.10D = 1.65
\][/tex]
2. The second equation relates to the total number of coins. Since Sofia has 25 coins in total, we have:
[tex]\[
N + D = 25
\][/tex]
So, we have the following system of equations:
[tex]\[
\begin{cases}
0.05N + 0.10D = 1.65 \\
N + D = 25
\end{cases}
\][/tex]
Step 2: Solving the equations
We can solve this system of equations by substitution or elimination. Let's use the substitution method.
1. From the second equation, we can express [tex]\( N \)[/tex] in terms of [tex]\( D \)[/tex]:
[tex]\[
N = 25 - D
\][/tex]
2. Substitute [tex]\( N = 25 - D \)[/tex] into the first equation:
[tex]\[
0.05(25 - D) + 0.10D = 1.65
\][/tex]
3. Simplify the equation:
[tex]\[
0.05 \times 25 - 0.05D + 0.10D = 1.65
\][/tex]
[tex]\[
1.25 - 0.05D + 0.10D = 1.65
\][/tex]
[tex]\[
1.25 + 0.05D = 1.65
\][/tex]
4. Solve for [tex]\( D \)[/tex]:
[tex]\[
0.05D = 1.65 - 1.25
\][/tex]
[tex]\[
0.05D = 0.40
\][/tex]
[tex]\[
D = \frac{0.40}{0.05}
\][/tex]
[tex]\[
D = 8
\][/tex]
5. Now, use [tex]\( D = 8 \)[/tex] to find [tex]\( N \)[/tex]:
[tex]\[
N = 25 - D
\][/tex]
[tex]\[
N = 25 - 8
\][/tex]
[tex]\[
N = 17
\][/tex]
Conclusion
Sofia has 17 nickels and 8 dimes.
Equations completed:
[tex]\[
\begin{array}{c}
1.65 = 0.05 N + 0.10 D \\
N + D = 25
\end{array}
\][/tex]
