- Let \(n\) be the number of nickels, so \(175 - n\) is the number of dimes.
- Assign the values to each type of coin: nickels are \(\$0.05\) and dimes are \(\$0.10\).
- Create the equation based on total value: \(0.05n + 0.10(175 - n) = 13.30\).
- Identify this matches the fourth option: \(0.1(175 - n) + 0.05n = 13.30\).
- Final answer: \(\boxed{0.1(175 - n) + 0.05n = 13.30}\)
Explanation
- Analyzing the Problem Data
Bodhi has a total of 175 coins, which are either dimes or nickels. We are asked to define an equation to determine the number of nickels, which we represent by \(n\). Since the total number of coins is 175, if there are \(n\) nickels, the remaining coins must be dimes. Thus, the number of dimes is \((175 - n)\).
- Constructing the Equation
The value of a nickel is \(\$0.05\), and the value of a dime is \(\$0.10\). The total value of the collection is \(\$13.30\). We can construct the equation by summing the total value contributed by the nickels and the total value contributed by the dimes.
The value of the nickels is \(\$0.05 \times n\).
The value of the dimes is \(\$0.10 \times (175 - n)\).
Summing these gives the total value:
\[0.05n + 0.10(175 - n) = 13.30\]
- Comparing with Given Options
We now compare our derived equation, \(0.05n + 0.10(175 - n) = 13.30\), with the given options. Our equation is equivalent to \(0.1(175 - n) + 0.05n = 13.30\), which corresponds exactly to the fourth option.
Examples
Understanding how to model relationships between quantities like coins or other resources using algebraic equations is a fundamental skill in mathematics and daily finance. For example, if you are organizing a school fundraiser and have a specific number of items that cost different amounts, setting up an equation like this helps you accurately calculate the count of each item needed to hit a specific revenue target, ensuring your budget or inventory is perfectly balanced.