Problem 58 - Algebra 2

Question

Simplify each expression. $$ \frac{1}{5 x}+\frac{1}{10 x} $$

Step-by-Step Solution

Answer

After simplifying the expression, we get \( \frac{3}{10x} \)

Step 1: Identify common denominator

In order to add these fractions, we need to first find a common denominator. We can see that both denominators are multiples of \(x\). Therefore, our common denominator will be the number that both 5x and 10x can go into, which is 10x.

Step 2: Rewrite fractions with common denominator

The first fraction already has \(x\) in its denominator and we want to have \(10x\). So, we multiple the first fraction (both numerator and denominator) by 2. Our expression then becomes: \( \frac{2}{10x} + \frac{1}{10x} \)

Step 3: Combine fractions

Now that both fractions have the same denominator, we can add the fractions together: \( \frac{2 + 1}{10x} \)

Step 4: Simplify

Doing the addition in the numerator, we find the simplified expression to be: \( \frac{3}{10x} \)

Key Concepts

Finding a Common Denominator

When adding fractions, having a common denominator is crucial. This makes it possible to combine the fractions into a single expression. A common denominator is simply a shared multiple of the individual denominators. In this exercise, the denominators are \(5x\) and \(10x\). To find the smallest common denominator:

• Identify the least common multiple (LCM) of the numerical parts (i.e., 5 and 10), which is 10.
• Retain the common variable part \(x\), making the common denominator \(10x\).

This step ensures both fractions stand on equal footing, easing the process of adding them together.

Simplifying Expressions

Simplifying expressions involves expressing equations in their simplest form for easier understanding and calculation. In terms of fractions, simplifying means:

• Finding a common denominator, as explained earlier.
• Rewriting fractions so they match this denominator without changing their value.

In our example, the first fraction \(\frac{1}{5x}\) is adjusted to have \(10x\) as its denominator. We achieve this by multiplying both the numerator and denominator by 2, becoming \(\frac{2}{10x}\). This step simplifies the process of adding the fractions, as they now share a common denominator.

Adding Fractions

Once fractions have a common denominator, they can be easily summed. The process of adding fractions involves:

• Adding the numerators together while keeping the denominator the same.
• Simplifying the result if needed.

In this case:

• The expression becomes \(\frac{2}{10x} + \frac{1}{10x}\).
• We add the numerators: \(2 + 1 = 3\).
• Therefore, the resulting fraction is \(\frac{3}{10x}\).

This simplified result is the final answer, neatly presented and easy to understand.