Comparing ratios is a fundamental mathematical skill used in various applications, from cooking to financial analysis. This article explains two common methods for comparing two ratios: the Least Common Multiple (LCM) method and the Cross-Multiplication method. Understanding these techniques will equip you to confidently determine which ratio represents a larger or smaller proportion.
Comparing Ratios Using the LCM Method
The LCM method involves finding the least common multiple of the denominators to make comparison straightforward. Here’s a step-by-step guide:
Step 1: Simplify the Ratios: Ensure both ratios are in their simplest form. For instance, if you have 6:8, simplify it to 3:4. Let’s use the example of comparing 4:5 and 1:7.
Step 2: Find the LCM of the Denominators: Determine the LCM of the denominators of both ratios. In our example, the denominators are 5 and 7, and their LCM is 35.
Step 3: Divide LCM by Each Denominator: Divide the LCM by each ratio’s denominator. For 4:5, 35/5 = 7. For 1:7, 35/7 = 5.
Step 4: Create Equivalent Fractions: Multiply both the numerator and denominator of each ratio by the result obtained in the previous step. This creates equivalent fractions with a common denominator.
- For 4:5, multiply by 7: (4×7)/(5×7) = 28/35
- For 1:7, multiply by 5: (1×5)/(7×5) = 5/35
Step 5: Compare the Numerators: Now that both fractions have the same denominator (35), compare the numerators. The larger numerator indicates the larger ratio.
Step 6: Determine the Larger Ratio: In our example, 28 > 5, therefore 28/35 > 5/35, which means 4:5 > 1:7.
Comparing Ratios Using the Cross-Multiplication Method
Cross-multiplication offers a quicker way to compare ratios:
Step 1: Simplify (if necessary): Ensure the ratios are in their simplest form.
Step 2: Cross-Multiply: Multiply the numerator of the first ratio by the denominator of the second, and vice versa. Let’s compare a/b and c/d:
- Multiply a x d
- Multiply b x c
Step 3: Compare the Products: Three scenarios can arise:
- If ad > bc: then a/b > c/d
- If ad < bc: then a/b < c/d
- If ad = bc: then a/b = c/d
Using our previous example (4/5 and 1/7):
- 4 x 7 = 28
- 5 x 1 = 5
Since 28 > 5, we conclude that 4/5 > 1/7.
Conclusion
Both the LCM and Cross-Multiplication methods provide effective ways to compare two ratios. While the LCM method offers a step-by-step approach to understanding the comparison process, cross-multiplication provides a faster route to the solution. Choose the method that best suits your understanding and the complexity of the ratios you’re comparing. Understanding these methods will enhance your ability to analyze and interpret proportional relationships in diverse contexts.
