Problem 58 - Prealgebra

Question

Solve each equation. Check your solution. $$20 m=4$$

Step-by-Step Solution

Answer

The solution is \( m = \frac{1}{5} \).

Step 1: Isolate the variable

Given equation is \( 20m = 4 \). To solve for \( m \), we need to isolate \( m \) by dividing both sides of the equation by 20. This gives us \( m = \frac{4}{20} \).

Step 2: Simplify the fraction

Now simplify the fraction \( \frac{4}{20} \). Both 4 and 20 are divisible by 4, so dividing the numerator and the denominator by 4, we get \( \frac{4}{20} = \frac{1}{5} \). Thus, \( m = \frac{1}{5} \).

Step 3: Verification

Substitute \( m = \frac{1}{5} \) back into the original equation to verify our solution. Calculating \( 20m \) gives \( 20 \times \frac{1}{5} = 4 \), which matches the right side of the original equation. Therefore, the solution is confirmed.

Key Concepts

Isolating the Variable

In any equation, isolating the variable is a crucial first step to finding the solution. The goal here is to get the variable, in this case, \( m \), by itself on one side of the equation. This involves performing operations that will cancel out other numbers around it until the variable stands alone.

• Start with the equation: \( 20m = 4 \).
• To isolate \( m \), divide both sides by 20.
• This operation is based on the property that dividing both sides of an equation by the same non-zero number keeps the equation balanced.

Executing this step, we have: \( m = \frac{4}{20} \). Remember, the essence of isolation is to focus on simplifying the equation step by step, getting closer to a simple form that presents the variable clearly. It's like playing a game of gently peeling away the layers around your desired goal—the variable.

Simplifying Fractions

Once you have the variable isolated in the form of a fraction, it's essential to simplify the fraction to its lowest terms. Simplification makes the equation more understandable and easier to interpret.
Let's take a closer look at our fraction \( \frac{4}{20} \). Both terms in this fraction are divisible by the same number, and finding the greatest common factor is key to simplifying quickly.

• Identify the greatest common divisor (GCD) of 4 and 20, which is 4.
• Divide both the numerator and the denominator by this number.

Performing the division gives us: \( \frac{4 \div 4}{20 \div 4} = \frac{1}{5} \). Hence, the fraction reduces to \( \frac{1}{5} \). This simplified expression represents the same value, just in an easier form, making calculations and verification smoother.

Verification of Solution

Verification of your solution is a crucial step in solving any equation as it ensures the accuracy of your work. Through verification, you confirm that your solution not only mathematically satisfies the equation but also aligns with all logical steps taken prior.
To verify our solution for \( m = \frac{1}{5} \), substitute this value back into the original equation:

1. Replace \( m \) in the equation \( 20m = 4 \) with \( \frac{1}{5} \).
2. Compute \( 20 \times \frac{1}{5} \).
3. If the result equals the original number on the other side of the equation, your solution is correct.

Calculating gives: \( 20 \times \frac{1}{5} = 4 \), which matches the original equation perfectly. This step confirms that \( m = \frac{1}{5} \) is indeed the correct solution. By verifying, you're demonstrating rigor and precision in your mathematical endeavors, ensuring the confidence of rightness in potential real-world applications.