Sofia has 25 coins in nickels and dimes in her pocket for a total of [tex]\$1.65[/tex]. How many of each type of coin does she have?

Complete the equations where [tex]N[/tex] stands for nickels and [tex]D[/tex] stands for dimes.

\[
\begin{array}{c}
1.65 = 0.05N + 0.10D \\
N + D = 25
\end{array}
\]

Reminder: Nickels are worth [tex]\$0.05[/tex] and dimes are worth [tex]\$0.10[/tex].



Answer :

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]