Ratios are fundamental tools in mathematics used to compare quantities. Understanding How To Compare Ratios is crucial in various real-life situations, from cooking and mixing ingredients to understanding proportions in business and science. When faced with two or more ratios, you might need to determine which ratio is larger, smaller, or if they are equivalent. Fortunately, there are straightforward methods to achieve this. This guide will walk you through two primary methods: the LCM (Least Common Multiple) Method and the Cross Multiplication Method, providing you with clear steps and examples to master ratio comparison.
Comparing Ratios Using the LCM Method
The LCM Method is a reliable way to compare ratios by finding a common denominator, making direct comparison easy. Here’s how to do it:
Step 1: Simplify the Ratios
Ensure that the ratios you are comparing are in their simplest form. For instance, if you have the ratios 6:8 and 3:5, simplify 6:8 to 3:4 by dividing both terms by their greatest common divisor, which is 2. So now we are comparing 3:4 and 3:5.
Step 2: Find the LCM of the Denominators
Consider the ratios as fractions. The second term in each ratio acts as the denominator. In our example, the denominators are 4 and 5. Find the Least Common Multiple (LCM) of these denominators. The LCM of 4 and 5 is 20.
Step 3: Adjust the Ratios to Have a Common Denominator
Convert each ratio into an equivalent fraction with the LCM as the new denominator.
For the ratio 3:4 (or (frac{3}{4})), to get the denominator 20, multiply both the numerator and the denominator by 5 (since (frac{20}{4} = 5)):
(frac{3 times 5}{4 times 5} = frac{15}{20})
For the ratio 3:5 (or (frac{3}{5})), to get the denominator 20, multiply both the numerator and the denominator by 4 (since (frac{20}{5} = 4)):
(frac{3 times 4}{5 times 4} = frac{12}{20})
Step 4: Compare the Numerators
Now that both ratios have the same denominator (20), you can directly compare their numerators. We have (frac{15}{20}) and (frac{12}{20}). Comparing the numerators, 15 and 12, we see that 15 is greater than 12.
Step 5: Determine the Larger Ratio
The ratio with the larger numerator is the greater ratio. Since 15 > 12, (frac{15}{20} > frac{12}{20}). Therefore, the original ratio 3:4 is greater than 3:5.
Comparing Ratios Using the Cross Multiplication Method
The Cross Multiplication Method offers a quicker approach to comparing ratios, especially when dealing with just two ratios. Here’s how it works:
Step 1: Simplify the Ratios
As with the LCM method, start by simplifying the ratios to their simplest form. Let’s take two different ratios for this method, say 2:3 and 4:7. Both are already in their simplest forms.
Step 2: Cross Multiply
To compare two ratios (frac{a}{b}) and (frac{c}{d}), cross multiply as follows:
Multiply the numerator of the first ratio (a) by the denominator of the second ratio (d). This gives you ‘ad’.
Multiply the denominator of the first ratio (b) by the numerator of the second ratio (c). This gives you ‘bc’.
For our ratios (frac{2}{3}) and (frac{4}{7}):
Calculate ad: 2 x 7 = 14
Calculate bc: 3 x 4 = 12
Step 3: Compare the Products
Compare the results of the cross multiplication (ad and bc):
- If ad > bc, then (frac{a}{b} > frac{c}{d})
- If ad < bc, then (frac{a}{b} < frac{c}{d})
- If ad = bc, then (frac{a}{b} = frac{c}{d})
In our example, ad = 14 and bc = 12. Since 14 > 12, we conclude that (frac{2}{3} > frac{4}{7}). Therefore, the ratio 2:3 is greater than the ratio 4:7.
Conclusion
Both the LCM Method and the Cross Multiplication Method are effective for comparing ratios. The LCM method is particularly useful when you are comparing more than two ratios as it sets a common base for all of them. Cross multiplication is generally faster for comparing just two ratios. Choose the method that best suits the problem at hand and remember that understanding how to compare ratios is a valuable skill in mathematics and beyond.
