Understanding Proportions in Fractions and Ratios

Proportions are an essential concept in mathematics, often used to compare quantities or solve real-world problems. When dealing with fractions or ratios, it is crucial to understand when two ratios are proportional. Proportional ratios are equal when expressed in simplest form. This guide will explore how to determine if given ratios are proportional and provide examples to clarify the process.

How to Determine Proportions in Ratios

To determine if two ratios or fractions are proportional, the following steps can be followed:

Express each ratio in its simplest form. Check if the cross-multiplication of the ratios results in equal products.

Examples of Proportional and Non-Proportional Ratios

Example 1: Analyzing Option A

Option A: 4/5, 8/10

Express each ratio in its simplest form:

4/5 is already in simplest form. 8/10 simplifies to 4/5 by dividing both numerator and denominator by 2.

Since 4/5 4/5, these ratios are proportional.

Example 2: Analyzing Option B

Option B: 16/10, 4/2

Express each ratio in its simplest form:

16/10 simplifies to 8/5 by dividing both numerator and denominator by 2. 4/2 simplifies to 2/1 by dividing both numerator and denominator by 2.

Since 8/5 ≠ 2/1, these ratios are not proportional.

Example 3: Analyzing Option C

Option C: 6/2, 12/4

Express each ratio in its simplest form:

6/2 simplifies to 3 by dividing both numerator and denominator by 2. 12/4 simplifies to 3 by dividing both numerator and denominator by 4.

Since 3 3, these ratios are proportional.

Example 4: Analyzing Option D

Option D: 20/30, 4–6

Express 20/30 in its simplest form:

20/30 simplifies to 2/3 by dividing both numerator and denominator by 10. 4–6 is not a valid ratio. If the intended operation is 4/6, it simplifies to 2/3 by dividing both numerator and denominator by 2.

Since 2/3 2/3, these ratios are proportional if 4–6 is meant to be 4/6.

Conclusion

From the analysis above, Option B (16/10, 4/2) is the pair of ratios that does not show proportion. Understanding proportions in fractions and ratios is a fundamental skill in mathematics, crucial for advanced topics and problem-solving.

Keywords: fractions, proportions, ratios