Problem 60 - College Algebra

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

75 pennies, 125 nickels, and 125 dimes.

Step 1: Understanding the Variables

Let's denote the number of pennies as \( p \), nickels as \( n \), and dimes as \( d \). We are given three types of coins: pennies (1 cent), nickels (5 cents), and dimes (10 cents). We are also told that the number of nickels is equal to the number of dimes, i.e., \( n = d \).

Step 2: Formulating Equations

We have two key pieces of information: 1. The total number of coins is 325, therefore, \( p + n + d = 325 \). 2. The total value of coins is $19.50, therefore, \( 0.01p + 0.05n + 0.10d = 19.50 \). Alternatively, we can multiply throughout by 100 to get:\( p + 5n + 10d = 1950 \).

Step 3: Substituting the Known Relationship

Since \( n = d \), we can replace \( n \) and \( d \) in both equations:1. \( p + 2d = 325 \) (since \( n + d = 2d \)).2. \( p + 5d + 10d = 1950 \), which simplifies to \( p + 15d = 1950 \).

Step 4: Solving the System of Equations

Now, solve the system of equations:- From \( p + 2d = 325 \), we get \( p = 325 - 2d \).- Substitute into \( p + 15d = 1950 \), giving \( 325 - 2d + 15d = 1950 \).- Simplify to get \( 13d = 1625 \).- Solve for \( d \), giving \( d = \/ 125 \).

Step 5: Finding the Number of Nickels and Pennies

Using \( d = 125 \):- From \( n = d \), thus, \( n = 125 \).- Use \( p = 325 - 2d \) from the first equation to find \( p \).- Substitute \( d = 125 \), resulting in \( p = 325 - 2 \times 125 = 75 \).

Step 6: Verifying the Solution

Check the total value: - Value from pennies: \( 75 \times 0.01 = 0.75 \).- Value from nickels: \( 125 \times 0.05 = 6.25 \).- Value from dimes: \( 125 \times 0.10 = 12.50 \).- Total value: \( 0.75 + 6.25 + 12.50 = 19.50 \) which matches the given condition.- The solution satisfies all the conditions: correct total number, equal nickels and dimes, and correct total value.

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning they involve only addition, subtraction, and/or multiplication of variables raised to the power of one.
In a system of linear equations, we often have two or more equations working together to describe a relationship between variables.
In our coin problem, we devised linear equations to represent the number of coins and their total value.Here’s how it works:

• Each equation we create must adhere to the conditions or constraints given in the problem.
• In this case, the total number of coins was represented by the equation: \( p + n + d = 325 \), where \( p \), \( n \), and \( d \) stand for pennies, nickels, and dimes respectively.
• We also had a value equation: \( 0.01p + 0.05n + 0.10d = 19.50 \), which describes how the total monetary value of these coins sum up to $19.50.

By solving these linear equations, we can find the specific values for each type of coin that satisfies all given conditions. Knowing how to build and solve these equations is key to solving many real-life problems, like our coin problem.

Coin Problem

The coin problem is a classic example used in math to practice solving systems of equations.
It revolves around determining the quantity of different coins based on their combined value and number. Here's a simplified breakdown:

• To successfully solve a coin problem, begin by identifying what is given and what is unknown. In our problem, the total value and total number of coins are known, but the number of each type of coin is not.


• We also knew that the number of nickels is equal to the number of dimes, simplifying the relationships between the coins.
• The challenge is to translate these conditions into mathematical equations that are solvable, as we did by using linear equations.

The coin problem not only improves algebraic skills but also teaches logical reasoning and careful reading of problem statements to find hidden clues necessary for solution.

Substitution Method

The substitution method is one effective way to solve a system of equations, including the kind found in the coin problem.
It involves expressing one variable in terms of another and then substituting this into the other equations.Follow these steps:

• First, find an equation you can solve easily for one of the variables—in our case, the equation \( p + n + d = 325 \) was simplified to \( p = 325 - 2d \).


• Then, replace the variable in another equation with the expression you found—so \( 325 - 2d \) was substituted into \( p + 15d = 1950 \), simplifying to \( 13d = 1625 \).
• Once you solve for one variable, substitute back to find the others. We found \( d = 125 \) and used it to calculate the number of nickels and pennies as well.

This method is powerful because it reduces the number of variables in an equation, making it simpler to solve. By practicing substitution, you become adept at tackling a wide range of algebraic problems.