Problem 25 - Elementary Algebra

Question

\(\frac{90-n}{n}=10+\frac{2}{n}\)

Step-by-Step Solution

Answer

\(n = 8\).

Step 1: Eliminate Fractions

To simplify the equation, eliminate fractions by multiplying every term by the denominator, which is \(n\). This gives: \[ n \left( \frac{90-n}{n} \right) = n(10) + n \left( \frac{2}{n} \right) \] which simplifies to: \[ 90-n = 10n + 2 \].

Step 2: Rearrange Terms

Next, rearrange the terms to bring all terms involving \(n\) on one side of the equation. This results in: \[ 90 - n = 10n + 2 \] Add \(n\) to both sides and subtract 2 from both sides:\[ 90 - 2 = 10n + n \].

Step 3: Simplify

Now, simplify both sides:\[ 88 = 11n \].

Step 4: Solve for \(n\)

Divide both sides by 11 to solve for \(n\):\[ n = \frac{88}{11} \].

Step 5: Simplification

Perform the division to find the value of \(n\):\[ n = 8 \].

Key Concepts

Eliminating Fractions in Equations

Fractions in equations can often make calculations more complex and confusing. Fortunately, there’s a straightforward method to eliminate them and simplify the problem. In algebra, you can remove fractions by finding a common denominator and multiplying all terms by it. This not only helps in simplifying the equation but also makes solving much easier.
In our example, we started with the equation \( \frac{90-n}{n} = 10 + \frac{2}{n} \). The common denominator here is \( n \). By multiplying each term by \( n \), we turn

• \( \frac{90-n}{n} \) into just \( 90-n \)
• \( 10 \cdot n \) remains as \( 10n \)
• \( \frac{2}{n} \cdot n \) becomes \( 2 \)

This results in: \( 90-n = 10n + 2 \). This step effectively eliminates all the fractions, making the equation much easier to handle. The key is always multiplying every term by the common denominator.

Solving Linear Equations

Once fractions are out of the equation, solving a linear equation becomes a lot more manageable. Solving involves finding the value of the variable that makes the equation true. In this equation, \( 90 - n = 10n + 2 \), our goal is to determine the value of \( n \).
The general strategy:

• Isolate the variable on one side.
• Move all other terms to the opposite side.

Start by rearranging terms so we can focus on \( n \). Add \( n \) to both sides of the equation. This gives us \( 90 = 10n + n + 2 \).
Then subtract 2 from both sides to simplify further, resulting in \( 88 = 11n \). With the variable isolated, our next step will be to solve for it.

Rearranging Algebraic Equations

Rearranging equations is about re-positioning terms to isolate the variable in question. This is a fundamental skill needed in algebra.
The principle is to move terms around to group like terms or perform operations that make solving easier. In our example, after rearranging and simplifying, we had \( 88 = 11n \).
To find the value of \( n \), divide both sides of the equation by the coefficient of \( n \), which was 11. This step is crucial as it isolates \( n \):

• \( \frac{88}{11} = n \)
• Simplify to find \( n = 8 \)

Rearranging helps in logically structuring the sequence of operations. Once rearranged, solving the equation becomes straightforward, bringing us to the solution that \( n = 8 \).