- Identify the variables: \(d\) for dimes and \(n\) for nickels.
- Establish the system of equations based on the count and total value: \(d + n = 175\) and \(0.10d + 0.05n = 13.30\).
- Use substitution by expressing \(d\) as \(175 - n\).
- Substitute into the value equation to obtain the final equation in terms of \(n\): \(\boxed{0.1(175 - n) + 0.05n = 13.30}\).
Explanation
- Analysis of the Problem Data
We are given that there are 175 coins in total, consisting of dimes and nickels. Let \(d\) be the number of dimes and \(n\) be the number of nickels. We are also given the total value of these coins, which is \(\$13.30\). Since a dime is worth \(\$0.10\) and a nickel is worth \(\$0.05\), we can set up a system of linear equations to represent these relationships:
- The total number of coins: \(d + n = 175\)
- The total value of the coins: \(0.10d + 0.05n = 13.30\)
- Expressing Dimes in Terms of Nickels
To find the number of nickels \(n\), we want to express the total value equation in terms of only the variable \(n\). From the first equation, we can isolate \(d\):
\[d = 175 - n\]
This equation shows that the number of dimes is the total number of coins minus the number of nickels.
- Formulating the Equation for n
Now, substitute the expression \(d = 175 - n\) into the value equation \(0.10d + 0.05n = 13.30\). Replacing \(d\) with \(175 - n\), we get:
\[0.10(175 - n) + 0.05n = 13.30\]
This equation now uses only the variable \(n\), allowing us to solve for the number of nickels in the collection. By comparing this to the provided options, we see that it matches the fourth option.
Examples
Imagine you are organizing a small fundraiser and collecting donations in a jar. You know you have exactly 175 coins in the jar, all of which are nickels and dimes, and the total value is \(\$13.30\). By setting up a system of equations like the one we solved, you can easily calculate how many of each coin you have without needing to count them individually. This approach is fundamental in algebra for solving real-world problems where you have multiple constraints on a set of items, such as budgeting for supplies, managing inventories, or balancing accounts.