Problem Description:
The task involves simplifying algebraic fractions. The provided example in the image is:
$$
\frac{15x^2}{30x}
$$
We are tasked with simplifying this fraction step by step.
---
Solution:
#### Step 1: Understand the Problem
We need to simplify the algebraic fraction:
$$
\frac{15x^2}{30x}
$$
This involves identifying and canceling out common factors between the numerator and the denominator.
---
#### Step 2: Factorize Numerator and Denominator
-
Numerator: \( 15x^2 \)
- Factorize \( 15 \): \( 15 = 3 \times 5 \)
- Factorize \( x^2 \): \( x^2 = x \times x \)
- So, \( 15x^2 = 3 \times 5 \times x \times x \)
-
Denominator: \( 30x \)
- Factorize \( 30 \): \( 30 = 2 \times 3 \times 5 \)
- Factorize \( x \): \( x = x \)
- So, \( 30x = 2 \times 3 \times 5 \times x \)
Thus, the fraction becomes:
$$
\frac{15x^2}{30x} = \frac{3 \times 5 \times x \times x}{2 \times 3 \times 5 \times x}
$$
---
#### Step 3: Identify Common Factors
Now, identify the common factors in the numerator and the denominator:
- Both have a factor of \( 3 \).
- Both have a factor of \( 5 \).
- Both have a factor of \( x \).
So, the common factors are \( 3 \), \( 5 \), and \( x \).
---
#### Step 4: Cancel Out Common Factors
Cancel out the common factors from both the numerator and the denominator:
$$
\frac{3 \times 5 \times x \times x}{2 \times 3 \times 5 \times x} = \frac{\cancel{3} \times \cancel{5} \times \cancel{x} \times x}{2 \times \cancel{3} \times \cancel{5} \times \cancel{x}}
$$
After canceling, we are left with:
$$
\frac{x}{2}
$$
---
#### Step 5: Write the Final Answer
The simplified form of the fraction is:
$$
\boxed{\frac{x}{2}}
$$
---
Explanation:
1.
Factorization: Breaking down the terms into their prime factors helps identify what can be canceled.
2.
Common Factors: Identifying and canceling out the common factors ensures the fraction is in its simplest form.
3.
Simplification: After canceling, only the remaining terms are left in the numerator and denominator.
This method ensures that the fraction is fully simplified while maintaining mathematical accuracy.
Thus, the final answer is:
$$
\boxed{\frac{x}{2}}
$$
Parent Tip: Review the logic above to help your child master the concept of algebraic fractions worksheet.
