Properties of Proportions

The properties of proportions are fundamental and help students understand and work with ratios. These properties derive from basic principles of mathematics. There are several properties of proportions, and some of them are mentioned below:

• Property 1: Cross-Multiplication If \(\frac{a}{b} = \frac{c}{d}\), then \(a \times d = b \times c\).
   
• Property 2: Reciprocal Property If \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{b}{a} = \frac{d}{c}\).
   
• Property 3: Means-Extremes In a proportion, the product of the means equals the product of the extremes.
   
• Property 4: Scaling Multiplying or dividing both terms of a ratio by the same non-zero number does not change the proportion.
   
• Property 5: Addition and Subtraction If \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{a + b}{b} = \frac{c + d}{d}\) and \(\frac{a - b}{b} = \frac{c - d}{d}\).

Students may confuse or make mistakes while learning the properties of proportions. To avoid such confusion, we can follow these tips and tricks:

• Cross-Multiplication: Students should remember that if two ratios are equal, the cross-multiplication of their terms is equal. This is a quick way to verify proportionality.
   
• Reciprocal Property: Understanding that a proportion remains valid if both ratios are inverted can help when rearranging equations.
   
• Scaling: If you multiply or divide both terms of a ratio by the same number, the proportion remains unchanged. This can simplify calculations.

Students may get confused when understanding the properties of proportions, leading to mistakes when solving problems. Here are some common mistakes and solutions.