Problem 60 - College Algebra with Corequisite Support

Question

In a bag, a child has 325 coins worth $$\$ 19.50$$. There were three types of coins: pennies, nickels, and dimes. If the bag contained the same number of nickels as dimes, how many of each type of coin was in the bag?

Step-by-Step Solution

Answer

The child has 75 pennies, 125 nickels, and 125 dimes.

Step 1: Define Variables

Let's define the variables for the number of coins of each type in the bag. Let \( p \) represent the number of pennies, \( n \) represent the number of nickels, and \( d \) represent the number of dimes. According to the problem, there are the same number of nickels and dimes, so we can write \( n = d \).

Step 2: Set Up Equations Based on Total Coin Count

The child has a total of 325 coins. Thus, the sum of pennies, nickels, and dimes is equal to 325. We can write this equation as:\[p + n + d = 325\]Using the relationship \( n = d \), we can substitute \( d \) in our coin equation: \[p + n + n = 325 \ p + 2n = 325\]

Step 3: Set Up Equations Based on Total Value

The total value of coins is \$19.50. This value comes from pennies (\(0.01 \times p\)), nickels (\(0.05 \times n\)), and dimes (\(0.10 \times d\)). We can write this relation as:\[0.01p + 0.05n + 0.10d = 19.50\]Again, using \( n = d \), substitute into the value equation: \[0.01p + 0.05n + 0.10n = 19.50 \0.01p + 0.15n = 19.50\]

Step 4: Solve the System of Equations

Now we have the following system of equations:1. \( p + 2n = 325 \)2. \( 0.01p + 0.15n = 19.50 \)From equation (1), solve for \( p \):\[p = 325 - 2n\]Substituting \( p \) into equation (2):\[0.01(325 - 2n) + 0.15n = 19.50 \3.25 - 0.02n + 0.15n = 19.50 \0.13n = 19.50 - 3.25 \0.13n = 16.25\]Solve for \( n \):\[n = \frac{16.25}{0.13} \approx 125\]

Step 5: Find Number of Pennies and Dimes

Using \( n = 125 \), the number of dimes is also \( d = 125 \) (since \( n = d \)). Now, calculate \( p \) using equation:\[p = 325 - 2 \times 125 \p = 325 - 250 \p = 75\]

Step 6: Verify the Solution

Let's confirm the total monetary value using these numbers. Calculate the value:\(75\) pennies: \(0.01 \times 75 = 0.75\)\(125\) nickels: \(0.05 \times 125 = 6.25\)\(125\) dimes: \(0.10 \times 125 = 12.50\)Sum: \(0.75 + 6.25 + 12.50 = 19.50\), which matches the total value given in the problem.

Key Concepts

Coin Problems

Coin problems in mathematics often involve determining the number or value of different coins based on specific conditions. In our problem, we are dealing with three types of coins: pennies, nickels, and dimes. Each coin has a distinct value, such as

• Pennies are worth $0.01.
• Nickels are worth $0.05.
• Dimes are worth $0.10.

The challenge is to determine how many of each type of coin exists given certain conditions. For example, we know the total number of coins is 325 and their combined value is $19.50. Additionally, we are told that the number of nickels is the same as the number of dimes, which simplifies the calculations. Solving coin problems usually involves setting up equations based on these conditions.

Linear Equations

Linear equations are fundamental tools for solving problems involving relationships between variables. In the context of our coin problem, we used linear equations to translate the information given into mathematical terms. We have two main equations:

• The equation that represents the total number of coins: \( p + n + d = 325 \).
• The equation that represents the total value of the coins: \( 0.01p + 0.05n + 0.10d = 19.50 \).

By using the relationship between nickels and dimes, \( n = d \), we simplified these into a system of linear equations:

• \( p + 2n = 325 \)
• \( 0.01p + 0.15n = 19.50 \)

These types of equations are linear because their graphs are straight lines, and they involve variables raised to the power of one. Solving them involves methods like substitution or elimination to find the value of each variable.

Value Problems

Value problems focus on finding the worth or price of items based on given conditions and relationships. In our exercise, the value of the coins was a key aspect. We were asked to find out how the total monetary value of different coins adds up to a specific amount, which was $19.50 in this case.

• Firstly, dissect the given relationships and establish equations representing these values.
• Then, use these quantitative relationships to form equations that encapsulate all conditions provided.

Understanding value problems requires a solid grasp of how to represent different quantities mathematically and manipulate these representations to find solutions. These problems can range from simple to complex, but they generally involve foundational math skills to explore the relationships between numbers and values.