Problem 75 - Elementary Algebra

Question

In the following exercises, translate and solve. 28 is 25 % of what number?

Step-by-Step Solution

Answer

112

Step 1: Understand the problem

The problem states that 28 is 25% of some unknown number. This can be translated into a mathematical equation.

Step 2: Set up the equation

Let the unknown number be denoted by the variable \(x\). According to the problem, 28 is 25% of \(x\). This translates to the equation: \[ 28 = 0.25x \]

Step 3: Solve the equation

To solve for \(x\), divide both sides of the equation by 0.25: \[ x = \frac{28}{0.25} \] Calculating this gives: \[ x = 112 \]

Step 4: Verify the solution

To ensure the solution is correct, check if 25% of 112 equals 28: \[ 0.25 \times 112 = 28 \] Since this is true, the solution is verified.

Key Concepts

Translating Word Problems

When facing a word problem in mathematics, the first step is to understand what the problem is asking. This involves identifying the key figures and their relationships. In our problem, we need to find an unknown number where 28 is 25% of it. Let's break this down:

• Identify the percentage: 25%
• Identify the part: 28
• Find the whole: unknown number

These elements can be translated into a mathematical expression. The phrase '28 is 25% of what number' can be represented as the equation: \[28 = 0.25x\]. Here, we use the decimal form of 25%, which is 0.25. This step is critical because it transforms the word problem into a solvable equation.

Solving Equations

Once the word problem has been translated into an equation, the next step is to solve it. The equation based on our problem is: \[28 = 0.25x\]. To find the value of the unknown number \(x\), we need to isolate \(x\) on one side of the equation. This can be done by dividing both sides by 0.25:

• Original equation: \[28 = 0.25x\]
• Divide both sides by 0.25: \[x = \frac{28}{0.25}\]

Perform the division to get: \[x = 112\]. Here, we see that the unknown number, which is 25% equates to 28, is actually 112.

Verification of Solution

After solving the equation, verifying the solution ensures everything is correct. For our problem, the verification step involves confirming that 25% of 112 equals 28. Here's how we do it:

• Start with the solution: \(112\)
• Multiply it by 0.25: \[0.25 \times 112\]

When you compute \[0.25 \times 112\], it equals 28. Since the calculation confirms our earlier steps, we know our solution is correct.
Verification is a crucial step because it helps catch mistakes and reinforces the understanding of the concept. Always recheck by substituting the solution back into the original context of the problem.