Problem 28 - Intermediate Algebra

Question

A box contains $$\$ 2$$ in nickels, dimes, and quarters. There are 19 coins in all, and there are twice as many nickels as dimes. How many coins of each kind are there?

Step-by-Step Solution

Answer

There are 10 nickels, 5 dimes, and 4 quarters.

Step 1: Define the Variables

Let \( n \) be the number of nickels, \( d \) be the number of dimes, and \( q \) be the number of quarters. We need to determine the values of \( n \), \( d \), and \( q \).

Step 2: Set Up the Equations

From the problem, we know that:1. The total number of coins is 19: \( n + d + q = 19 \).2. The total value of the coins is \$2, which is 200 cents: \( 5n + 10d + 25q = 200 \). 3. There are twice as many nickels as dimes: \( n = 2d \).

Step 3: Substitute Using Known Relations

Use the relation \( n = 2d \) in the first equation:\[ 2d + d + q = 19 \]This simplifies to:\[ 3d + q = 19 \]Now substitute \( n = 2d \) in the value equation:\[ 5(2d) + 10d + 25q = 200 \]Which simplifies to:\[ 10d + 10d + 25q = 200 \]Or:\[ 20d + 25q = 200 \]

Step 4: Solve the Simplified System of Equations

We now have two equations:1. \( 3d + q = 19 \)2. \( 20d + 25q = 200 \)From the first equation, solve for \( q \):\[ q = 19 - 3d \]Substitute this into the second equation:\[ 20d + 25(19 - 3d) = 200 \]\[ 20d + 475 - 75d = 200 \]Simplify:\[ -55d + 475 = 200 \]\[ -55d = 200 - 475 \]\[ -55d = -275 \]\( d = \frac{-275}{-55} = 5 \)

Step 5: Find the Number of Nickels and Quarters

Now that we have \( d = 5 \), we use \( n = 2d \) to find:\[ n = 2(5) = 10 \]And substitute \( d = 5 \) into the equation for \( q \):\[ q = 19 - 3(5) = 4 \]

Step 6: Verify the Solution

Check the total number of coins: \( 10 + 5 + 4 = 19 \).Check the total value: \( 5(10) + 10(5) + 25(4) = 50 + 50 + 100 = 200 \) cents.Both conditions of the problem are satisfied.

Key Concepts

Systems of Equations

When solving problems involving multiple variables and conditions, systems of equations become essential. They enable us to represent and solve real-world problems simultaneously. In this context, we used three equations to describe the scenario:

• The total number of coins was expressed by: \( n + d + q = 19 \).
• The total value of the coins in cents was: \( 5n + 10d + 25q = 200 \).
• The relation between nickels and dimes was: \( n = 2d \).

Each equation represents a different aspect of the problem. By substituting and rearranging these equations, you can effectively reduce the number of variables you're solving for, ultimately leading to a solution. This approach is key to solving problems involving systems of equations.

Coin Problems

Coin problems are a type of algebraic problem where the number and type of coins in a collection (like nickels, dimes, and quarters) need to be determined. They often involve setting up equations based on both the quantity and value of the coins. Understand the following key concepts for solving coin problems:

• Variable Definition: Clearly define what each variable represents, for instance, \( n \) for nickels, \( d \) for dimes, and \( q \) for quarters.
• Setting Up Equations: Establish equations based on given conditions. Here, we utilized total coin count and monetary value.
• Relations: Identify any additional relationships between the variables, such as twice as many nickels as dimes.

Coin problems help strengthen algebraic reasoning and the ability to set up and manipulate equations.

Algebraic Reasoning

Algebraic reasoning allows us to identify relationships and connections between variables or situations, making it a central part of solving equations and problems like the one given. It involves:

• Logical Thinking: Analyzing the problem to determine what is being asked and how the different pieces of information relate to one another.
• Substitution: Replacing one part of an equation with another, simpler expression. This technique was used to eliminate and solve for variables like using \( n = 2d \) in the example equation.
• Simplification: Breaking down complex equations into simpler ones. This is achieved by combining like terms and making careful substitutions.

This kind of reasoning doesn't just apply to math problems—it's a skill that enhances problem-solving capabilities in everyday scenarios.