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Bodhi has a collection of 175 dimes and nickels. The collection is worth $\$ 13.30$. Which equation can be used to find $n$, the number of nickels in the collection?


A$0.1 n+0.05(n-175)=13.30$
B$0.1 n+0.05(175-n)=13.30$
C$0.1(n-175)+0.05=13.30$
D$0.1(175-n)+0.05 n=13.30$

Verified Answer :

  • Define the variables for the coins: let \(n\) be the number of nickels and \(D = 175 - n\) be the number of dimes.
  • Multiply the number of each type of coin by its respective monetary value: \(\$0.10\) for dimes and \(\$0.05\) for nickels.
  • Formulate the total value equation: \(0.10(175 - n) + 0.05n = 13.30\).
  • The correct equation is \(\boxed{0.1(175 - n) + 0.05n = 13.30}\).

Explanation

  1. Analyzing the Problem Data

We are given a collection of \(175\) coins, which are a mix of dimes and nickels. Let \(n\) be the number of nickels and \(D\) be the number of dimes. We know that the total number of coins is \(175\), so the relationship between the number of coins is \(D + n = 175\). This allows us to express the number of dimes in terms of the number of nickels: \(D = 175 - n\). Each dime has a value of \(\$0.10\) and each nickel has a value of \(\$0.05\). The total value of the collection is \(\$13.30\).

  1. Formulating the Equation

We construct the equation for the total value of the coins. The total value is the sum of the value of the dimes and the value of the nickels:
\[(\text{Value of a dime} \times \text{Number of dimes}) + (\text{Value of a nickel} \times \text{Number of nickels}) = \text{Total Value}\]
Substituting the known values and the expression for \(D\) derived in the previous step, we get:
\[0.10 \times (175 - n) + 0.05 \times n = 13.30\]
This equation accounts for the value of each coin type, with the number of dimes represented by \((175 - n)\) and the number of nickels represented by \(n\).

  1. Conclusion

Comparing our derived equation to the provided options, we see that it matches the fourth choice:
\[0.1(175 - n) + 0.05n = 13.30\]
Thus, this equation correctly models the total value of the coin collection based on the number of nickels \(n\).

Examples

Understanding how to model coin collections is a fundamental skill in algebra that applies to many real-world situations, such as managing cash registers, balancing budgets, or calculating the total value of collected items. For instance, if you were organizing a fundraiser and needed to count various donations in different coin denominations, you could use this type of equation to quickly determine the number of each coin type without having to count every single one individually. This approach not only saves time but also ensures accuracy in accounting and resource management.

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