Problem 58 - Precalculus

Question

A copy machine accepts nickels, dimes, and quarters. After 1 hour, there are 30 coins total and their value is \(\$ 4.60 .\) If there are four more quarters than nickels, how many nickels, quarters, and dimes are in the machine?

Step-by-Step Solution

Answer

There are 10 nickels, 6 dimes, and 14 quarters.

Step 1: Define Variables

Let's define variables to represent the number of each type of coin. Let \( n \) be the number of nickels, \( d \) be the number of dimes, and \( q \) be the number of quarters.

Step 2: Set Up Equations Based on Coin Count and Value

We know that the total number of coins is 30, so our first equation is: \[ n + d + q = 30 \]The total value from the coins is \( \$4.60 \), which is 460 cents, giving our second equation: \[ 5n + 10d + 25q = 460 \]Additionally, there are four more quarters than nickels, so: \[ q = n + 4 \]

Step 3: Substitute and Solve the System of Equations

First, substitute \( q = n + 4 \) into the first equation:\[ n + d + (n + 4) = 30 \]Simplifying gives:\[ 2n + d + 4 = 30 \]\[ 2n + d = 26 \] Now substitute \( q = n + 4 \) into the second equation:\[ 5n + 10d + 25(n + 4) = 460 \]Expanding and simplifying:\[ 5n + 10d + 25n + 100 = 460 \]\[ 30n + 10d = 360 \]\[ 3n + d = 36 \]

Step 4: Solve for One Variable

Subtract the equation \( 2n + d = 26 \) from \( 3n + d = 36 \):\[ (3n + d) - (2n + d) = 36 - 26 \]\[ n = 10 \]

Step 5: Solve for Remaining Variables

Substitute \( n = 10 \) into \( 2n + d = 26 \):\[ 2(10) + d = 26 \]\[ 20 + d = 26 \]\[ d = 6 \]Then substitute \( n = 10 \) into \( q = n + 4 \):\[ q = 10 + 4 = 14 \]

Step 6: Verify the Solution

Verify that all conditions are met. The total coins: \[ 10 + 6 + 14 = 30 \]. The total value:\[ 5(10) + 10(6) + 25(14) = 50 + 60 + 350 = 460 \] cents All conditions are satisfied, confirming the solution is correct.

Key Concepts

Coin Problems

Coin problems are a classic type of algebra problem, often requiring the solver to find the number of coins of different denominations that add up to a certain total value. In this exercise, we deal with nickels, dimes, and quarters.
The goal is to understand how many of each type of coin are there, based on certain conditions.
With coin problems, you usually have:

• A total count of coins
• A total monetary value
• Relationships between the different types of coins

Breaking down these components helps formulate equations that represent the problem. Then, by solving these equations, we can find out how many coins of each type there are.

Variable Substitution

Variable substitution is a powerful technique in solving systems of equations. It involves replacing one variable with another expression to simplify equations.
In our problem, we have the relationship between quarters (\( q \)) and nickels (\( n \)) as \( q = n + 4 \). By substituting \( q \) in the main equations, we reduce the number of variables, making them easier to solve.
Using substitution aids in reducing complexity by eliminating a variable from the equation. This step-by-step reduction ensures that we approach the solution logically and systematically, making it less prone to errors.

Equation Setup

Setting up the equations is a critical step in solving any problem involving systems of equations. Here, it involves translating the conditions and relationships of the problem into mathematical expressions.
When setting up equations:

• Define clear variables to represent unknowns (e.g., \( n \) for nickels, \( d \) for dimes, \( q \) for quarters).
• Use given conditions to form equations, such as total coins and total value.
• Incorporate any additional relationships, like the number of quarters being four more than nickels.

By carefully setting up equations, we create a clear framework to work through the problem methodically. This approach not only helps in obtaining the right solution but also strengthens problem-solving skills.