Problem 6 - Elementary Algebra

Question

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question. One-fourth of the candies in a bag of M\&M's are red. If there are 23 red candies, how many candies are in the bag?

Step-by-Step Solution

Answer

There are 92 candies in the bag.

Step 1 - Understand the problem

Read the problem carefully. We know that one-fourth of the candies are red and there are 23 red candies. We need to find the total number of candies in the bag.

Step 2 - Define the variable

Let the total number of candies in the bag be represented by the variable Let \(x\) be the total number of candies in the bag.

Step 3 - Set up the equation

According to the problem, one-fourth of the candies are red. So, we can set up the equation as follows: \[ \frac{1}{4}x = 23 \]

Step 4 - Solve the equation

To find the value of \(x\), multiply both sides of the equation by 4: \[ x = 23 \times 4 \] So, \( x = 92 \).

Step 5 - Verify the solution

One-fourth of 92 is: \[ \frac{1}{4} \times 92 = 23 \] This matches the number of red candies given in the problem.

Step 6 - Write the answer in a complete sentence

Therefore, there are 92 candies in the bag.

Key Concepts

Elementary Algebra

Elementary algebra is about solving equations that involve unknowns, often called variables. In this problem, we need to find the total number of candies in a bag when one-fourth of them are red and we are given that there are 23 red candies. We start by reading the problem carefully and identifying what we know and what we need to find. This is a foundation of algebra: understanding given information and defining what we need to solve for.

Solving Equations

In solving equations, we transform our word problem into a mathematical equation. Here, we let the total number of candies be represented by the variable. So \( x \) is used to denote the total number of candies.

We know that one-fourth of the total candies is equal to 23 red candies. This gives us the equation: \[ \frac{1}{4} x = 23 \]

To isolate \( x \), we multiply both sides by 4, resulting in: \[ x = 23 \times 4 = 92 \]

This solution method is logical and straightforward. By this process, we find that there are 92 candies in total.

Problem-Solving Strategies

Solving word problems involves several strategies.

• First, read the problem carefully to understand what is being asked. This helps identify the knowns and unknowns.
• Next, define variables to represent unknown quantities. In this case, we defined \( x \) as the total number of candies.
• Then, set up equations based on the relationships described in the problem. We used the fact that one-fourth of the candies equates to 23 red candies.
• Finally, solve the equation and check your work. Verifying that one-fourth of 92 is indeed 23 confirms our solution.

Step-by-step problem-solving ensures that each part of the question is addressed methodically.

Variables

Variables are symbols used to represent unknown quantities in algebra. They are a core part of forming equations and solving them. In our problem, the variable \( x \) represented the total number of candies.

By defining \( x \) clearly and constructing an equation around it, we make the problem solvable. This is a crucial step in turning words into a solvable mathematical statement. Using variables introduces flexibility and power in expressing and solving a wide array of problems.