Problem 10 - Algebra 2 and Trigonometry

Question

Write each ratio in simplest form. \(\frac{10 x}{35 x}, x \neq 0\)

Step-by-Step Solution

Answer

The simplest form of \( \frac{10x}{35x} \) is \( \frac{2}{7} \).

Step 1: Identify the Common Factor

To simplify the ratio \( \frac{10x}{35x} \), first identify the greatest common factor of the numerator and the denominator. Here, the common factor is \(5x\).

Step 2: Divide by the Common Factor

Divide both the numerator and the denominator by the common factor \(5x\): \[ \frac{10x \div 5x}{35x \div 5x} = \frac{2}{7} \]

Step 3: Simplify and Verify

After dividing, we get \( \frac{2}{7} \). This cannot be simplified further, confirming that the ratio is in its simplest form.

Key Concepts

Greatest Common Factor

When simplifying ratios or fractions, finding the Greatest Common Factor (GCF) is a crucial step. The GCF is the largest factor that divides both the numerator and denominator without leaving any remainder. For instance, consider the ratio \(\frac{10x}{35x}\). Here, both terms, \(10x\) and \(35x\), have common factors. The factors of \(10x\) are \(1, 2, 5, 10, x\), and the factors of \(35x\) are \(1, 5, 7, 35, x\). The largest factor they share is \(5x\). By identifying the GCF, you find the key to simplifying the expression effectively and correctly. Finding the GCF ensures that you reduce the ratio to its simplest form accurately.

Numerator and Denominator

In any fraction or ratio \(\frac{a}{b}\), \(a\) is called the numerator, and \(b\) is called the denominator. The numerator represents how many parts you have, while the denominator indicates the total possible parts. Consider the ratio \(\frac{10x}{35x}\) once again. In this example:

• The numerator is \(10x\).
• The denominator is \(35x\).

Recognizing and understanding these parts is vital for performing operations such as simplification. The aim is to find a common factor between both the numerator and denominator, as this allows for easy simplification. By focusing on these components, you ensure that any simplification maintains the proper mathematical relationship between them without changing the overall value of the fraction or ratio.

Simplified Form

Simplifying a ratio or fraction to its simplest form means reducing it to the lowest possible terms while maintaining its original value. This process involves dividing both the numerator and denominator by their Greatest Common Factor (GCF). For the ratio \(\frac{10x}{35x}\), we determined the GCF to be \(5x\). Thus, by dividing both terms by \(5x\), we obtain:

• \(10x \div 5x = 2\)
• \(35x \div 5x = 7\)

So, the simplified form of \(\frac{10x}{35x}\) is \(\frac{2}{7}\). After this operation, verify by checking if the new numerator and denominator have any common factors further. With \(\frac{2}{7}\), no further simplification is possible, confirming that the ratio is as simple as it gets. Simplification clarifies relationships and comparisons, allowing for more straightforward analysis and understanding in mathematics.