Solving Coin Problems: A Practical Example with Nickels, Dimes, and Pennies

In this article, we'll work through a real-world problem involving coins, demonstrating how to solve it using algebra. We'll walk you through the process of determining the number of nickels, dimes, and pennies a girl has if she has a total of $4.62, with a specific relationship between the quantities of each type of coin.

The Problem: A Girl's Coin Collection

Let's start by defining the variables and the relationships between the coins. The girl has:

4.62 dollars 3 times as many dimes as nickels 6 more pennies than dimes

To solve this problem, we'll follow a step-by-step approach, defining the number of each type of coin, then setting up and solving the equation.

Step 1: Define the Variables

We start by defining the number of nickels as:

The number of dimes is three times the number of nickels, so it can be expressed as:

The number of pennies is six more than the number of dimes, which can be expressed as:

3n 6

Step 2: Calculate the Total Value of the Coins

The total value of the coins in dollars is given by the sum of the values of the nickels, dimes, and pennies. The value of each type of coin is as follows:

Nickels: 0.05n Dimes: 0.10(3n) 0.30n Pennies: 0.01(3n 6) 0.03n 0.06

So, the total value equation becomes:

0.05n 0.30n 0.03n 0.06 4.62

Step 3: Solve the Equation

Combining the terms on the left side gives us:

0.05n 0.30n 0.03n 0.06 4.62

This simplifies to:

0.38n 0.06 4.62

Subtract 0.06 from both sides:

0.38n 4.56

Divide both sides by 0.38 to find n:

n 4.56 / 0.38 ≈ 12

Step 4: Find the Number of Each Type of Coin

Now that we have the number of nickels (n 12), we can find the number of dimes and pennies:

Number of dimes: 3n 3 * 12 36 Number of pennies: 3n 6 3 * 12 6 42

Summary

The girl has:

12 nickels 36 dimes 42 pennies

The total value of these coins is indeed 4.62 dollars, confirming our solution.

Alternative Method: Using a Table

An alternative method to solve the problem is by creating a table to organize the numbers and verify the solution:

N D P Total 12 36 42 0.60 3.60 0.42 4.62

This table confirms that the girl has 12 nickels, 36 dimes, and 42 pennies, totaling 4.62 dollars.

Conclusion

Solving problems involving coins can be straightforward with the right approach. By defining the variables, setting up the equation, and solving step-by-step, we can determine the exact number of each type of coin. Whether using algebraic methods or tables, the process is systematic and easily understandable.