To solve this problem, we need to model the situation using a system of equations.
Denote:
- [tex]\( d \)[/tex] as the number of dimes
- [tex]\( n \)[/tex] as the number of nickels
We are provided with two key pieces of information:
1. A total of 22 coins (dimes + nickels)
2. A total value of [tex]$1.90 from these coins
Let's break this down step by step:
1. Total Number of Coins:
- We are told that there are 22 coins in total. This can be expressed mathematically as:
\[
d + n = 22
\]
2. Total Value of Coins:
- Each dime is worth $[/tex]0.10.
- Each nickel is worth [tex]$0.05.
- The total value of the 22 coins is $[/tex]1.90.
- Therefore, the sum of the value of the dimes and the value of the nickels should equal $1.90. This can be expressed mathematically as:
[tex]\[
0.10d + 0.05n = 1.90
\][/tex]
Putting these two equations together, the system of equations that models the given situation is:
[tex]\[
\begin{cases}
d + n = 22 \\
0.10d + 0.05n = 1.90
\end{cases}
\][/tex]
Now, let's identify the correct answer from the given options:
A.
[tex]\(
\begin{cases}
d + n = 1.9 \\
0.1d + 0.05n = 22
\end{cases}
\)[/tex]
- This is incorrect.
B.
[tex]\(
\begin{cases}
d + n = 22 \\
0.1d + 0.05n = 1.9
\end{cases}
\)[/tex]
- This is correct.
C.
[tex]\(
\begin{cases}
d + n = 1.9 \\
0.1d + 0.5n = 22
\end{cases}
\)[/tex]
- This is incorrect.
D.
[tex]\(
\begin{cases}
d + n = 22 \\
0.1d + 0.5n = 19
\end{cases}
\)[/tex]
- This is incorrect.
Therefore, the correct answer is B.
