Andy has 22 coins made up of quarters and half dollars, and their total value is \( \$ 9.75 \). How many quarters does he have? A. 19 quarters B. 5 quarters C. 10 quarters D. 17 quarters

To solve this problem, we can set up a system of equations. Let \( q \) represent the number of quarters and \( h \) represent the number of half dollars. We know: 1. The total number of coins is \( q + h = 22 \). 2. The total value of the coins is \( 0.25q + 0.50h = 9.75 \). Now, we can simplify the second equation by multiplying by 100 to eliminate the decimals: \[ 25q + 50h = 975. \] Now, we can rewrite the first equation to express \( h \) in terms of \( q \): \[ h = 22 - q. \] Substituting \( h \) into the value equation gives us: \[ 25q + 50(22 - q) = 975. \] Expanding and simplifying: \[ 25q + 1100 - 50q = 975, \] \[ -25q + 1100 = 975, \] \[ -25q = 975 - 1100, \] \[ -25q = -125, \] \[ q = \frac{125}{25} = 5. \] Therefore, Andy has 5 quarters. The answer is B. 5 quarters. For further fun and insights: Back in the late 1700s, coins were the primary way of conducting transactions, and each coin had its unique design reflecting the nation's heritage. Today, every coin — like Andy's quarters and half dollars — continues to represent a piece of history, showcasing figures, events, and symbols that tell tales of the nation's past! In real-world applications, this kind of problem-solving is vital! You might encounter similar scenarios while budgeting your savings or figuring out change. Always keep your equations straightforward and double-check your substitutions to avoid common pitfalls — like mixing up product values or miscounting your total coins!