Certainly! Let's determine how many nickels (N) and dimes (D) Janiel has with the following information:
1. The total number of coins is 23.
2. The total value of the coins is [tex]$2.20.
Let's set up the equations based on this information:
1. The first equation relates to the total value of the coins. Since each nickel is worth $[/tex]0.05 and each dime is worth $0.10, the equation for the total value is:
[tex]\[
0.05N + 0.10D = 2.20
\][/tex]
2. The second equation relates to the total number of coins. Janiel has 23 coins in total, so the equation is:
[tex]\[
N + D = 23
\][/tex]
Now, we need to solve this system of equations.
First equation:
[tex]\[
0.05N + 0.10D = 2.20
\][/tex]
Second equation:
[tex]\[
N + D = 23
\][/tex]
To solve these equations, follow these steps:
1. From the second equation, we can express [tex]\( N \)[/tex] in terms of [tex]\( D \)[/tex]:
[tex]\[
N = 23 - D
\][/tex]
2. Substitute [tex]\( N = 23 - D \)[/tex] into the first equation:
[tex]\[
0.05(23 - D) + 0.10D = 2.20
\][/tex]
3. Distribute the 0.05:
[tex]\[
1.15 - 0.05D + 0.10D = 2.20
\][/tex]
4. Combine like terms:
[tex]\[
1.15 + 0.05D = 2.20
\][/tex]
5. Subtract 1.15 from both sides:
[tex]\[
0.05D = 1.05
\][/tex]
6. Divide both sides by 0.05 to find [tex]\( D \)[/tex]:
[tex]\[
D = 21
\][/tex]
7. Substitute [tex]\( D = 21 \)[/tex] back into [tex]\( N = 23 - D \)[/tex]:
[tex]\[
N = 23 - 21
\][/tex]
[tex]\[
N = 2
\][/tex]
Therefore, Janiel has 2 nickels and 21 dimes.
In the given equations, the values are:
[tex]\[
\begin{array}{c}
0.05N + 0.10D = 2.20 \\
N + D = 23
\end{array}
\][/tex]
where [tex]\( N = 2 \)[/tex] and [tex]\( D = 21 \)[/tex].
