Question

1. a collection of nickels and quarters has a combined value of three dollars. it contains a total of 32 coins. which system of linear equations models this scenario? let q = the number of quarters and let n = the number of nickels. option 1: \\(\

$$\begin{cases} n + q = 32 \\\\ 0.05n + 0.25q = 3 \\end{cases}$$

\\) option 2: \\(\

$$\begin{cases} n + q = 32 \\\\ 0.5n + 0.25q = 3 \\end{cases}$$

\\) option 3: \\(\

$$\begin{cases} n + q = 32 \\\\ 5n + 25q = 3 \\end{cases}$$

\\) option 4: \\(\

$$\begin{cases} n + q = 3 \\\\ 5n + 25q = 32 \\end{cases}$$

\\)

Explanation:

Step1: Analyze total number of coins

The total number of nickels (\(N\)) and quarters (\(Q\)) is 32, so the first equation is \(N + Q = 32\).

Step2: Analyze total value in dollars

A nickel is worth $0.05 and a quarter is worth $0.25. The total value is $3. So the second equation is LXI0. We can also think in cents: a nickel is 5 cents, a quarter is 25 cents, and $3 is 300 cents. Then the equation would be \(5N + 25Q = 300\), but dividing both sides by 100 gives \(0.05N + 0.25Q = 3\), which matches the first way. Now check the options:

• First option: \(N + Q = 32\) and \(0.05N + 0.25Q = 3\) – matches.
• Second option: \(0.5N\) is incorrect (nickel is $0.05, not $0.5).
• Third option: \(5N + 25Q = 3\) is wrong (should be 300 cents, not 3 cents).
• Fourth option: \(N + Q = 3\) is wrong (total coins are 32, not 3).

Answer:

$$\begin{cases} N + Q = 32 \\ 0.05N + 0.25Q = 3 \end{cases}$$

(the first option among the given choices)