Problem 90 - Prealgebra

Question

Solve for \(n\). $$25=0.40 n$$

Step-by-Step Solution

Answer

The solution is \(n = 62.5\).

Step 1: Understanding the Equation

We are given the equation \(25 = 0.40n\). Our goal is to solve for \(n\), which means we need to isolate \(n\) on one side of the equation.

Step 2: Isolating n

To isolate \(n\), we need to get rid of the coefficient \(0.40\) that is multiplied by \(n\). We can do this by dividing both sides of the equation by \(0.40\).

Step 3: Performing the Division

Divide both sides of the equation by \(0.40\):\[\frac{25}{0.40} = n\]

Step 4: Calculating the Division

Calculate \(\frac{25}{0.40}\) by performing the division. \(25\) divided by \(0.40\) equals \(62.5\).

Step 5: Solution Verification

Let's verify our solution by substituting \(n = 62.5\) back into the original equation. Check that \(0.40 \times 62.5 = 25\). This shows our solution is correct.

Key Concepts

Isolation Method

Solving linear equations often involves the isolation method, where we aim to get the variable by itself on one side of the equation. In our exercise, the equation is given as \( 25 = 0.40n \). To isolate \( n \), we need to "free" it from any other numbers attached. This requires us to "undo" any operation affecting \( n \); in this case, it’s multiplication by 0.40.

Think of this like peeling layers off an onion to reveal the core. The ultimate goal is to have just \( n \) on one side and the rest on the opposite. By methodically eliminating other numbers, we simplify the equation. Here, dividing both sides by 0.40 does the trick. We're essentially reversing the multiplication to strip down the equation to its simplest form.

Division in Equations

Division plays a crucial role when solving equations, especially when dealing with coefficients. After identifying the need to isolate \( n \) by removing the 0.40, we divide the entire equation by this coefficient. In this example, we divide both sides by 0.40 to isolate \( n \).

The operation is as follows:

• Divide \( 25 \) by 0.40.
• Perform the division to find the value for \( n \).

Doing this correctly provides \( n = 62.5 \). Always remember: what you do to one side, you must do to the other. It ensures the equation remains balanced and the solution valid.

Equation Verification

Once you've arrived at a potential solution, it's crucial to verify it. This is done by checking if the solution satisfies the original equation. Substitute back the value you found into the initial equation to see if it holds true.

For our particular problem, we substitute \( n = 62.5 \) back into \( 25 = 0.40n \) and calculate the product:

• Calculate \( 0.40 \times 62.5 \) and see if it equals \( 25 \).
• If both sides match, this confirms the solution is correct.

Verification acts as a safeguard against mistakes, ensuring reliability and accuracy in your solution process.