In mathematics, ratios are used to compare quantities. Understanding how to Compare The Ratios is a fundamental skill. Ratios are often expressed in their simplest form, and to effectively compare them, especially when dealing with more than two ratios, specific methods are employed. This article will explore two primary methods for comparing ratios: the Least Common Multiple (LCM) Method and the Cross Multiplication Method. Both methods provide a systematic approach to determine which of the given ratios is larger or if they are equal. Let’s delve into each method to understand how they work.
Comparing Ratios Using the LCM Method
The LCM method is a robust approach to compare the ratios by making their denominators the same. This allows for a direct comparison of the numerators to determine the relationship between the ratios. Here’s a step-by-step guide to using the LCM method:
Step 1: Simplify the Ratios
Begin by ensuring that the given ratios are in their simplest form. This means that the terms of each ratio should not have any common factors other than 1. For instance, if you need to compare 4:5 and 1:7, they are already in their simplest forms.
Step 2: Find the LCM of the Denominators
Identify the denominators of the ratios when expressed as fractions. For ratios 4:5 and 1:7, the denominators are 5 and 7 respectively. Calculate the LCM of these denominators. The LCM of 5 and 7 is 35 because 5 and 7 are prime numbers, and their LCM is simply their product (5 × 7 = 35).
Step 3: Determine the Multiplication Factor for Each Ratio
Divide the LCM by the denominator of each ratio.
For the ratio 4:5 (or (frac{4}{5})), divide the LCM (35) by the denominator (5): (frac{35}{5} = 7).
For the ratio 1:7 (or (frac{1}{7})), divide the LCM (35) by the denominator (7): (frac{35}{7} = 5).
Step 4: Convert Ratios to Equivalent Fractions with the LCM as the Common Denominator
Express each ratio as a fraction and multiply both the numerator and the denominator by the factor calculated in Step 3.
For 4:5, multiply both parts by 7: (frac{4 times 7}{5 times 7} = frac{28}{35}).
For 1:7, multiply both parts by 5: (frac{1 times 5}{7 times 5} = frac{5}{35}).
Step 5: Compare the Numerators
Now that both fractions have the same denominator (35), you can compare the ratios by comparing their numerators. The ratio with the larger numerator is the larger ratio.
Step 6: Conclude the Comparison
Compare the numerators 28 and 5. Since 28 > 5, it follows that (frac{28}{35} > frac{5}{35}). Therefore, we can conclude that (frac{4}{5} > frac{1}{7}), meaning the ratio 4:5 is greater than the ratio 1:7.
Comparing Ratios Using the Cross Multiplication Method
The cross multiplication method offers a quicker way to compare the ratios, especially when dealing with just two ratios. It avoids the need to find the LCM and directly compares products derived from the ratios. Here’s how to use the cross multiplication method:
Step 1: Simplify the Ratios
Ensure that the given ratios are in their simplest form, just like in the LCM method.
Step 2: Cross Multiply
For two ratios expressed as fractions (frac{a}{b}) and (frac{c}{d}), perform cross multiplication. This involves multiplying the numerator of the first ratio by the denominator of the second ratio (a × d), and the denominator of the first ratio by the numerator of the second ratio (b × c).
Step 3: Compare the Products
Compare the two products obtained from cross multiplication:
- If (a × d) > (b × c), then (frac{a}{b} > frac{c}{d}).
- If (a × d) < (b × c), then (frac{a}{b} < frac{c}{d}).
- If (a × d) = (b × c), then (frac{a}{b} = frac{c}{d}).
For example, let’s compare the ratios 3:4 and 2:3 using cross multiplication.
Here, (frac{a}{b} = frac{3}{4}) and (frac{c}{d} = frac{2}{3}).
Cross multiply:
a × d = 3 × 3 = 9
b × c = 4 × 2 = 8
Comparing the products: 9 > 8.
Therefore, according to the rule, since 9 > 8, then (frac{3}{4} > frac{2}{3}). This indicates that the ratio 3:4 is greater than the ratio 2:3.
Both the LCM method and the cross multiplication method are effective tools to compare the ratios. The LCM method is particularly useful when comparing more than two ratios as it provides a common denominator for all, making comparison straightforward. Cross multiplication is quicker for comparing just two ratios and is a handy shortcut. Understanding and applying both methods will enhance your ability to work with and compare the ratios in various mathematical contexts.
