To determine the system of equations that correctly models the situation where there is a total of 22 dimes and nickels that have a combined value of \[tex]$1.90, we can follow these steps:
1. Identify the numbers of coins:
- Let \( d \) be the number of dimes.
- Let \( n \) be the number of nickels.
Since there is a total of 22 coins, we can write the first equation as:
\[
d + n = 22
\]
2. Identify the total value of the coins:
- The value of each dime is \$[/tex]0.10.
- The value of each nickel is \[tex]$0.05.
Since the total value is \$[/tex]1.90, we can write the second equation as:
[tex]\[
0.1d + 0.05n = 1.90
\][/tex]
Now let's compare the options:
Option A:
[tex]\[
d + n = 1.9 \\
0.1d + 0.05n = 22
\][/tex]
This option does not make sense because the number of coins cannot equal 1.9 when the total number of coins is 22.
Option B:
[tex]\[
d + n = 22 \\
0.1d + 0.05n = 0.1 .9
\][/tex]
The second equation has an error because 0.1 1.9 (where probably the intention was to write 0.1 19) does not match the correct total value of 1.90 dollars. Hence this is incorrect for our scenario.
Option C:
[tex]\[
d + n = 19 \\
0.1d + 0.5n = 22
\][/tex]
The first equation here incorrectly represents the number of coins as 19 instead of 22, so this option is incorrect.
Option D:
[tex]\[
d + n = 22 \\
0.1d + 0.5n = 1.9
\][/tex]
Here, the first equation correctly states that the combined number of coins is 22. However, there is an error in the second equation as the coefficient for nickels should be [tex]\( 0.05 \)[/tex] and not [tex]\( 0.5 \)[/tex]:
[tex]\[
0.1d + 0.05n = 1.9
\][/tex]
Since none of the given options perfectly match the correct set of equations, we understand that the correct modeling equations should be close to:
[tex]\[
d + n = 22
\][/tex]
[tex]\[
0.1d + 0.05n = 1.9
\][/tex]
Based on the descriptions given in the question, the closest and most plausible matching pair to describe our initial modeling would be:
[tex]\[
D: \quad d + n = 22 \quad \text{and} \quad 0.1d + 0.5n = 1.9
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\text{D}}
\][/tex]
