Problem 62 - Prealgebra

Question

Solve each equation. Check your solution. $$125 n=15$$

Step-by-Step Solution

Answer

The solution to the equation is \(n = \frac{3}{25}\).

Step 1: Isolate the Variable

The goal is to solve for \(n\). To do this, divide both sides of the equation by 125 to isolate \(n\):\[125n = 15\]This becomes:\[n = \frac{15}{125}\]

Step 2: Simplify the Fraction

Now, simplify the fraction \(\frac{15}{125}\). Both the numerator and the denominator can be divided by their greatest common divisor, which is 5:\[n = \frac{15 \div 5}{125 \div 5} = \frac{3}{25}\]

Step 3: Check the Solution

Substitute \(n = \frac{3}{25}\) back into the original equation to verify the solution:\[125 \cdot \frac{3}{25} = 15\]Simplify the left-hand side:\[125 \cdot \frac{3}{25} = 125 \div 25 \cdot 3 = 5 \cdot 3 = 15\]Since both sides of the equation are equal, the solution is verified.

Key Concepts

Understanding Solving Equations

In prealgebra, mastering how to solve equations is crucial. The fundamental idea is to find the value of the variable that makes the equation true. In our exercise, we start with the equation \(125n = 15\). Here, the goal is to isolate the variable \(n\) on one side of the equation.
To accomplish this, we employ a balance strategy where we perform the same operation on both sides of the equation to maintain equality. In this problem, the coefficient of \(n\) is 125. Thus, we divide both sides by this coefficient.
This operation gives us \(n = \frac{15}{125}\). Now \(n\) is isolated, meaning we've successfully solved for \(n\), albeit in fractional form, which requires further simplification. Understanding this step involves knowing the mathematical operations that maintain the balance of the equation, like addition, subtraction, multiplication, and division.

The Art of Fraction Simplification

Once you have isolated the variable and end up with a fraction, the next step is simplification. Simplifying fractions is about reducing the fraction to its lowest terms, making it simpler and easier to interpret.
For example, after isolating \(n\), we obtain \(n = \frac{15}{125}\). Both the numerator (15) and the denominator (125) share a greatest common divisor (GCD), which is 5.
By dividing both the numerator and denominator by their GCD, we simplify the fraction:

• Divide 15 by 5 to get 3
• Divide 125 by 5 to get 25

So, \(n = \frac{3}{25}\). Simplifying fractions helps in achieving the most concise representation of a solution and is a valuable skill in all areas of math.

Why Checking Solutions is Important

Checking solutions is a vital step in the problem-solving process. It ensures accuracy and confirms that the solution truly satisfies the original equation.
In this exercise, we check our solution by plugging \(n = \frac{3}{25}\) back into the original equation \(125n = 15\). This validation step involves recalculating to see if both sides equal each other:

• Multiply 125 by \(\frac{3}{25}\)
• This simplifies to \(5 \times 3 = 15\)

Since the left side equals the right side, \(15 = 15\), we have confirmed that our solution \(n = \frac{3}{25}\) is correct. This process is essential for catching errors and truly understanding that the value found is the one that solves the equation.