Solving Word Problems: A Guide with Examples and Explanation

Word problems are a crucial aspect of algebra that help students understand how to apply mathematical concepts to real-world situations. In this article, we will provide a detailed guide on how to solve a specific type of word problem, along with various examples to illustrate the process. Let's delve into the details.

Understanding the Problem

Let's consider the following problem: A number is reduced by 7 and then multiplied by 5, resulting in 100. What is the original number? To solve this problem, we first need to break down the problem into a series of mathematical steps. Let's denote the unknown number as (x). The problem can be broken down as follows: - The number is reduced by 7: (x - 7).- The result is then multiplied by 5: (5(x - 7)).- The final result is 100: (5(x - 7) 100).

Setting Up the Equation

We can now set up the equation and solve for (x). Here are the steps: 1. **Write the equation**: (5(x - 7) 100). 2. **Distribute the 5**: (5x - 35 100). 3. **Add 35 to both sides**: (5x - 35 35 100 35) or (5x 135). 4. **Divide both sides by 5**: (x frac{135}{5}) or (x 27). Let's go through each step in detail: 1. **Step 1: Write the equation** We start with the original equation: (5(x - 7) 100). 2. **Step 2: Distribute the 5** Distributing the 5 on the left side gives us: (5x - 35 100). 3. **Step 3: Add 35 to both sides** Adding 35 to both sides to isolate the term with (x): (5x - 35 35 100 35) or (5x 135). 4. **Step 4: Divide both sides by 5** Dividing both sides by 5 to solve for (x): (x frac{135}{5}) or (x 27). Thus, the original number is 27.

Additional Examples

Let's look at some additional examples to solidify our understanding.

Example 1

Given the equation (m - 7 20) and then (5m - 35 100), let's solve for (m): 1. **Start with (m - 7 20)** 2. **Add 7 to both sides**: (m 20 7) or (m 27). 3. **Substitute (m 27) into (5m - 35 100)** 4. **Verify**: (5(27) - 35 135 - 35 100). Therefore, (m 27).

Example 2

Let's solve the equation (5x - 7 100): 1. **Add 7 to both sides**: (5x 107). 2. **Divide by 5**: (x frac{107}{5} 21.4). Therefore, (x 21.4).

Example 3

Let's solve the equation (x - 7 times 5 100): 1. **Simplify the left side**: (5x - 35 100). 2. **Add 35 to both sides**: (5x 135). 3. **Divide by 5**: (x 27). Therefore, (x 27).

By following these detailed steps and solving similar problems, you can build a strong foundation in algebra and problem-solving techniques. If you have any questions or need further clarification, feel free to ask!