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Anatoliy has a combination of 104 nickels and quarters totaling [tex]$\$[/tex] 22$. Which system of linear equations can be used to find the number of nickels, [tex]n[/tex], and the number of quarters, [tex]q[/tex], Anatoliy has?

A.
[tex]
\begin{array}{l}
n+q=22 \\
0.05n+0.25q=104
\end{array}
[/tex]

B.
[tex]
\begin{array}{l}
n+q=104 \\
5n+25q=22
\end{array}
[/tex]

C.
[tex]
\begin{array}{l}
n+q=104 \\
0.05n+0.25q=22
\end{array}
[/tex]

D.
[tex]
\begin{array}{l}
n+q=22 \\
5n+25q=104
\end{array}
[/tex]



Answer :

GinnyAnswer
Let's break down the problem step-by-step:

1. Anatoliy has a combination of nickels and quarters totaling [tex]$104$[/tex] coins:
- This means the sum of the number of nickels [tex]\( n \)[/tex] and the number of quarters [tex]\( q \)[/tex] is 104.
- We can express this relationship as the equation [tex]\( n + q = 104 \)[/tex].

2. Anatoliy's nickels and quarters together amount to $22, which is equivalent to 2200 cents:
- The value of a nickel is [tex]\( 0.05 \)[/tex] dollars (or 5 cents), and the value of a quarter is [tex]\( 0.25 \)[/tex] dollars (or 25 cents).
- The total value of the nickels and quarters can be expressed as [tex]\( 0.05n + 0.25q = 22 \)[/tex] dollars.

Combining these two pieces of information, we form the following system of linear equations to describe the problem:

[tex]\[ \begin{cases} n + q = 104 \\ 0.05n + 0.25q = 22 \end{cases} \][/tex]

Upon examination of the given choices:
1. [tex]\( \begin{cases} n + q = 22 \\ 0.05n + 0.25q = 104 \end{cases} \)[/tex]
2. [tex]\( \begin{cases} n + q = 104 \\ 5n + 25q = 22 \end{cases} \)[/tex]
3. [tex]\( \begin{cases} n + q = 104 \\ 0.05n + 0.25q = 22 \end{cases} \)[/tex]
4. [tex]\( \begin{cases} n + q = 22 \\ 5n + 25q = 104 \end{cases} \)[/tex]

The correct system of linear equations that represent the problem is:

[tex]\[ \begin{cases} n + q = 104 \\ 0.05n + 0.25q = 22 \end{cases} \][/tex]

Thus, the correct answer is the third option:

[tex]\[ \begin{cases} n + q = 104 \\ 0.05n + 0.25q = 22 \end{cases} \][/tex]