Solving Word Problems Using Algebra: A Practical Example

Algebra is a powerful tool in translating everyday situations into mathematical equations. This article will walk you through solving a specific problem related to coins, demonstrating how to use algebraic equations to find the solution. We will tackle a problem involving quarters, dimes, and nickels.

The Problem

Paul has $4.75 in coins. He has some quarters, one more dime than quarters, and three less nickels than quarters. How many dimes does Paul have?

Understanding the Problem

Given the problem statement, we need to set up equations that can help us find out how many dimes Paul has. Let's start by defining the number of quarters as Q.

Setting Up Equations

Let Q represent the number of quarters:

The value of the quarters is 0.25Q. The number of dimes is one more than the number of quarters, so it is Q 1. The value of the dimes is 0.10(Q 1). The number of nickels is three less than the number of quarters, so it is Q-3. The value of the nickels is 0.05(Q-3).

We are given that the total value is $4.75. So, we can write the equation:

0.25Q 0.10(Q 1) 0.05(Q-3) 4.75

Solving the Equation

Let's simplify and solve the equation step by step:

Distribute the values: Combine like terms: Isolate the term with Q: Solve for Q: Find the number of dimes:

Step 1: Distribute the values:

0.25Q 0.10Q 0.10 0.05Q - 0.15 4.75

Step 2: Combine like terms:

(0.25Q 0.10Q 0.05Q) (0.10 - 0.15) 4.75

0.40Q - 0.05 4.75

Step 3: Add 0.05 to both sides:

0.40Q 4.80

Step 4: Divide by 0.40:

Q 12

Step 5: Find the number of dimes:

Q 1 12 1 13

Hence, Paul has 13 dimes.

Additional Tips and Strategies

To make solving such problems easier, follow these steps and tips:

Define your variables clearly. Write down the relationships given in the problem using variables. Create equations based on the relationships. simplify the equations and solve step by step. Double-check your solutions by plugging back into the original problem.

Using algebra to solve real life problems like this one is not only useful in homework but also in many practical applications. Remember, practice makes perfect!