Cross multiplying definitely works as a shortcut to compare fractions, but Does Cross Multiplying Always Work To Compare Fractions? Yes, cross multiplication always works to compare fractions, provided that the denominators of both fractions are positive. At COMPARE.EDU.VN, we clarify how cross multiplication simplifies fraction comparison and variable solving, but caution against misuse with negative denominators or inequalities. Understanding its reliable application ensures confident problem-solving in math and beyond.
1. Understanding Cross Multiplication: A Comprehensive Guide
Cross multiplication is a valuable tool for comparing fractions, solving proportions, and even identifying equivalent ratios. It’s a technique widely taught in schools, but a deeper understanding of its principles and limitations is crucial for its correct and effective application. Let’s dissect this technique step by step.
1.1. What is Cross Multiplication?
Cross multiplication is a mathematical procedure primarily used to compare two fractions or solve proportions. Given two fractions, (frac{a}{b}) and (frac{c}{d}), the process involves multiplying the numerator of the first fraction by the denominator of the second, and vice versa.
This yields two products:
- (a times d)
- (b times c)
By comparing these products, one can determine the relationship between the original fractions, whether they are equal, or which one is larger.
1.2. The Mechanics of Cross Multiplication
The procedure is straightforward. If we have (frac{a}{b}) and (frac{c}{d}), then cross multiplication involves the following steps:
- Multiply (a) by (d) to get (ad).
- Multiply (b) by (c) to get (bc).
- Compare the results:
- 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}).
1.3. Why Does Cross Multiplication Work?
To understand why cross multiplication works, we need to delve into the underlying mathematical principles. Cross multiplication is essentially a shortcut for clearing denominators to compare fractions more easily. Here’s the breakdown:
Suppose we want to compare two fractions (frac{a}{b}) and (frac{c}{d}). To do this rigorously, we can find a common denominator and compare the numerators. The common denominator would be (bd). Therefore, we convert both fractions to have this denominator:
- (frac{a}{b} = frac{a times d}{b times d} = frac{ad}{bd})
- (frac{c}{d} = frac{c times b}{d times b} = frac{bc}{bd})
Now, we can easily compare the fractions by comparing their numerators (ad) and (bc). If (ad > bc), then (frac{a}{b} > frac{c}{d}). This is exactly what cross multiplication tells us, making it a quick way to achieve the same result.
1.4. Assumptions and Limitations
While cross multiplication is a powerful tool, it comes with certain assumptions and limitations that must be considered:
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Positive Denominators: The most critical assumption is that the denominators (b) and (d) are positive. If either denominator is negative, the direction of the inequality must be reversed when multiplying, which can complicate the process.
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Comparing Only Two Fractions: Cross multiplication is designed to compare two fractions at a time. When dealing with three or more fractions, other methods like finding a common denominator are more appropriate.
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Not Applicable to All Operations: Cross multiplication is specifically for comparing fractions or solving proportions. It cannot be applied to addition, subtraction, multiplication, or division of fractions.
Understanding these nuances helps avoid common pitfalls and ensures the correct application of cross multiplication in various mathematical contexts. At COMPARE.EDU.VN, we emphasize these details to help users make informed decisions based on reliable methods.
2. Practical Examples of Cross Multiplication
To fully grasp the utility and mechanics of cross multiplication, let’s explore several practical examples. These examples will cover various scenarios, from simple fraction comparisons to solving for unknown variables in proportions.
2.1. Comparing Two Fractions
Example 1: Determining Which Fraction is Larger
Consider the fractions (frac{3}{4}) and (frac{5}{7}). We want to determine which fraction is greater without converting them to decimals or finding a common denominator.
Using cross multiplication:
- Multiply (3) by (7): (3 times 7 = 21)
- Multiply (4) by (5): (4 times 5 = 20)
- Compare the results: Since (21 > 20), then (frac{3}{4} > frac{5}{7}).
This simple comparison quickly tells us that (frac{3}{4}) is the larger fraction.
Example 2: Verifying Equality of Fractions
Let’s check if (frac{6}{8}) and (frac{9}{12}) are equal.
- Multiply (6) by (12): (6 times 12 = 72)
- Multiply (8) by (9): (8 times 9 = 72)
- Compare the results: Since (72 = 72), then (frac{6}{8} = frac{9}{12}).
Thus, we confirm that the two fractions are indeed equal.
2.2. Solving Proportions
Cross multiplication is particularly useful in solving proportions where one of the values is unknown.
Example 3: Solving for an Unknown Variable
Suppose we have the proportion (frac{4}{x} = frac{8}{16}). We need to find the value of (x).
Using cross multiplication:
- Multiply (4) by (16): (4 times 16 = 64)
- Multiply (x) by (8): (8 times x = 8x)
- Set the results equal: (64 = 8x)
- Solve for (x): (x = frac{64}{8} = 8)
Therefore, the value of (x) that makes the proportion true is (8).
Example 4: Solving a More Complex Proportion
Consider the proportion (frac{2}{5} = frac{x}{35}).
- Multiply (2) by (35): (2 times 35 = 70)
- Multiply (5) by (x): (5 times x = 5x)
- Set the results equal: (70 = 5x)
- Solve for (x): (x = frac{70}{5} = 14)
Hence, (x = 14) is the solution to the proportion.
2.3. Real-World Applications
Cross multiplication isn’t just a theoretical concept; it has numerous practical applications in everyday life.
Example 5: Scaling Recipes
A recipe calls for (frac{2}{3}) cup of flour for every (frac{1}{2}) cup of sugar. If you want to use 2 cups of sugar, how much flour do you need?
Set up the proportion: (frac{text{flour}}{text{sugar}} = frac{frac{2}{3}}{frac{1}{2}} = frac{x}{2})
Using cross multiplication:
- Multiply (frac{2}{3}) by (2): (frac{2}{3} times 2 = frac{4}{3})
- Multiply (frac{1}{2}) by (x): (frac{1}{2} times x = frac{1}{2}x)
- Set the results equal: (frac{4}{3} = frac{1}{2}x)
- Solve for (x): (x = frac{4}{3} div frac{1}{2} = frac{4}{3} times 2 = frac{8}{3})
So, you need (frac{8}{3}) cups of flour, or 2 (frac{2}{3}) cups, for 2 cups of sugar.
Example 6: Calculating Distances on a Map
On a map, 1 inch represents 50 miles. If two cities are (3.5) inches apart on the map, what is the actual distance between them?
Set up the proportion: (frac{text{inches}}{text{miles}} = frac{1}{50} = frac{3.5}{x})
Using cross multiplication:
- Multiply (1) by (x): (1 times x = x)
- Multiply (50) by (3.5): (50 times 3.5 = 175)
- Set the results equal: (x = 175)
Therefore, the actual distance between the two cities is 175 miles.
These examples illustrate the versatility of cross multiplication in solving various problems, from academic exercises to real-world scenarios. At COMPARE.EDU.VN, we provide these detailed examples to ensure our users can confidently apply these techniques.
3. Addressing Common Misconceptions
Despite its simplicity and utility, cross multiplication is often misunderstood and misused. Clearing up these misconceptions is crucial for its effective application. Let’s address some of the most common errors and misunderstandings.
3.1. Misconception 1: Cross Multiplication Works With Negative Denominators
One of the most pervasive misconceptions is that cross multiplication can be applied without considering the sign of the denominators. This is incorrect. Cross multiplication is valid only when both denominators are positive.
Why This is Wrong: When a denominator is negative, multiplying across without accounting for the negative sign can lead to incorrect comparisons. For instance, consider (frac{1}{-2}) and (frac{1}{3}).
If we ignore the negative sign and cross multiply directly, we get:
- (1 times 3 = 3)
- (-2 times 1 = -2)
Comparing these results might lead us to believe that (frac{1}{-2} > frac{1}{3}), which is false. The correct comparison requires us to recognize that (frac{1}{-2} = -frac{1}{2}), and (-frac{1}{2} < frac{1}{3}).
Correct Approach: Always ensure that the denominators are positive. If a fraction has a negative denominator, multiply both the numerator and denominator by -1 to make the denominator positive.
For example, (frac{1}{-2}) should be converted to (frac{-1}{2}) before applying cross multiplication.
3.2. Misconception 2: Cross Multiplication Works With Inequalities Directly
Another common mistake is applying cross multiplication directly to inequalities without considering the signs of the terms involved.
Why This is Wrong: When dealing with inequalities, the direction of the inequality sign must be reversed if you multiply or divide by a negative number. Cross multiplying without considering this rule can lead to incorrect conclusions.
Consider the inequality (frac{a}{b} < frac{c}{d}). If either (b) or (d) is negative, the resulting inequality after cross multiplication must be adjusted accordingly.
Correct Approach: Before cross multiplying an inequality, ensure that both denominators are positive. If one or both are negative, multiply the entire fraction by -1 to make the denominator positive, and remember to flip the inequality sign if necessary.
For example, if you have (frac{x}{-3} < frac{2}{5}), first rewrite it as (frac{-x}{3} < frac{2}{5}). Now you can cross multiply safely:
- (-x times 5 < 2 times 3)
- (-5x < 6)
- (x > -frac{6}{5})
3.3. Misconception 3: Cross Multiplication Can Be Used for Addition and Subtraction
Many students mistakenly try to apply cross multiplication to add or subtract fractions.
Why This is Wrong: Cross multiplication is designed for comparing or solving proportions, not for performing arithmetic operations like addition or subtraction. These operations require a common denominator.
For example, attempting to add (frac{1}{2} + frac{1}{3}) using cross multiplication would lead to:
- (1 times 3 + 1 times 2 = 3 + 2 = 5)
This result doesn’t give us the correct sum. The correct approach involves finding a common denominator:
- (frac{1}{2} + frac{1}{3} = frac{3}{6} + frac{2}{6} = frac{5}{6})
Correct Approach: To add or subtract fractions, always find a common denominator first.
3.4. Misconception 4: Cross Multiplication is Always the Fastest Method
While cross multiplication is often quicker than finding a common denominator, it isn’t always the most efficient method, especially when dealing with multiple fractions or complex expressions.
Why This is Wrong: For simple comparisons, cross multiplication is efficient. However, when you have several fractions to compare or when the fractions involve complex algebraic expressions, other methods may be more appropriate.
Correct Approach: Evaluate the complexity of the problem. For multiple fractions, finding a common denominator and comparing numerators may be more straightforward. For complex algebraic fractions, simplification or other algebraic techniques might be more efficient.
Understanding and avoiding these common misconceptions is crucial for the correct and effective use of cross multiplication. At COMPARE.EDU.VN, we aim to provide clear explanations and examples to help users avoid these pitfalls and use mathematical tools with confidence.
4. Advanced Applications and Considerations
Beyond the basic use cases, cross multiplication has several advanced applications and considerations that can enhance its utility and applicability.
4.1. Cross Multiplication with Algebraic Expressions
Cross multiplication is not limited to simple numerical fractions. It can also be applied to algebraic expressions, making it a versatile tool in algebra.
Example 1: Solving for a Variable in an Algebraic Proportion
Consider the equation (frac{x + 1}{4} = frac{x – 2}{3}). To solve for (x), we can use cross multiplication:
- Multiply ((x + 1)) by (3): (3(x + 1) = 3x + 3)
- Multiply (4) by ((x – 2)): (4(x – 2) = 4x – 8)
- Set the results equal: (3x + 3 = 4x – 8)
- Solve for (x):
- Subtract (3x) from both sides: (3 = x – 8)
- Add (8) to both sides: (x = 11)
Thus, the solution to the equation is (x = 11).
Example 2: Solving for a Variable with More Complex Expressions
Consider the equation (frac{2x}{x + 3} = frac{4}{5}).
- Multiply (2x) by (5): (2x times 5 = 10x)
- Multiply ((x + 3)) by (4): (4(x + 3) = 4x + 12)
- Set the results equal: (10x = 4x + 12)
- Solve for (x):
- Subtract (4x) from both sides: (6x = 12)
- Divide by (6): (x = 2)
Therefore, the value of (x) is (2).
4.2. Using Cross Multiplication to Identify Equivalent Ratios
Cross multiplication can efficiently determine if two ratios are equivalent, which is useful in various fields, including statistics and finance.
Example 3: Determining Equivalent Ratios
Are the ratios (3:4) and (6:8) equivalent?
- Express the ratios as fractions: (frac{3}{4}) and (frac{6}{8})
- Cross multiply:
- (3 times 8 = 24)
- (4 times 6 = 24)
- Compare the results: Since (24 = 24), the ratios are equivalent.
Example 4: Determining Non-Equivalent Ratios
Are the ratios (2:5) and (4:9) equivalent?
- Express the ratios as fractions: (frac{2}{5}) and (frac{4}{9})
- Cross multiply:
- (2 times 9 = 18)
- (5 times 4 = 20)
- Compare the results: Since (18 neq 20), the ratios are not equivalent.
4.3. Advanced Considerations: Domain Restrictions
When working with algebraic fractions, it’s important to consider domain restrictions. These are values of the variable that would make the denominator equal to zero, rendering the fraction undefined.
Example 5: Identifying Domain Restrictions
Consider the equation (frac{x}{x – 2} = frac{3}{4}).
- Identify the domain restriction: The denominator (x – 2) cannot be zero. Therefore, (x neq 2).
- Cross multiply:
- (x times 4 = 4x)
- (3 times (x – 2) = 3x – 6)
- Set the results equal: (4x = 3x – 6)
- Solve for (x):
- Subtract (3x) from both sides: (x = -6)
Since (x = -6) does not violate the domain restriction (x neq 2), it is a valid solution.
Example 6: Handling Domain Restrictions in More Complex Equations
Consider the equation (frac{1}{x + 1} = frac{x}{x^2 – 1}).
- Identify the domain restrictions:
- (x + 1 neq 0), so (x neq -1)
- (x^2 – 1 neq 0), which factors to ((x + 1)(x – 1) neq 0), so (x neq 1) and (x neq -1)
- Cross multiply:
- (1 times (x^2 – 1) = x^2 – 1)
- (x times (x + 1) = x^2 + x)
- Set the results equal: (x^2 – 1 = x^2 + x)
- Solve for (x):
- Subtract (x^2) from both sides: (-1 = x)
However, (x = -1) violates the domain restriction (x neq -1). Therefore, this equation has no solution.
These advanced applications and considerations highlight the depth and versatility of cross multiplication. At COMPARE.EDU.VN, we strive to provide comprehensive insights that enable users to tackle complex problems with confidence.
5. Alternatives to Cross Multiplication
While cross multiplication is a handy tool, it’s not always the best or only option. Knowing alternative methods can provide flexibility and a deeper understanding of fraction comparisons.
5.1. Finding a Common Denominator
One of the most fundamental methods for comparing fractions is to find a common denominator. This involves converting the fractions to have the same denominator and then comparing their numerators.
How It Works:
- Identify the Least Common Multiple (LCM): Find the LCM of the denominators of the fractions you want to compare.
- Convert the Fractions: Multiply the numerator and denominator of each fraction by the factor needed to make the denominator equal to the LCM.
- Compare the Numerators: Once the fractions have the same denominator, compare their numerators. The fraction with the larger numerator is the larger fraction.
Example:
Compare (frac{3}{4}) and (frac{5}{6}).
- The LCM of 4 and 6 is 12.
- Convert the fractions:
- (frac{3}{4} = frac{3 times 3}{4 times 3} = frac{9}{12})
- (frac{5}{6} = frac{5 times 2}{6 times 2} = frac{10}{12})
- Compare the numerators: Since (10 > 9), (frac{10}{12} > frac{9}{12}), so (frac{5}{6} > frac{3}{4}).
Advantages:
- Works for any number of fractions.
- Straightforward and easy to understand.
Disadvantages:
- Can be more time-consuming than cross multiplication for comparing just two fractions.
- Requires finding the LCM, which can be challenging for large denominators.
5.2. Converting to Decimals
Another way to compare fractions is to convert them to decimal form. This method is particularly useful when you have a calculator available or when the fractions are easily convertible to decimals.
How It Works:
- Divide: Divide the numerator by the denominator for each fraction.
- Compare the Decimals: Compare the resulting decimal values. The fraction with the larger decimal value is the larger fraction.
Example:
Compare (frac{3}{8}) and (frac{2}{5}).
- Convert to decimals:
- (frac{3}{8} = 0.375)
- (frac{2}{5} = 0.4)
- Compare the decimals: Since (0.4 > 0.375), (frac{2}{5} > frac{3}{8}).
Advantages:
- Simple and straightforward, especially with a calculator.
- Easy to compare multiple fractions.
Disadvantages:
- Some fractions result in repeating decimals, which can be difficult to compare accurately.
- May not be suitable for exact comparisons without a calculator.
5.3. Benchmarking
Benchmarking involves comparing fractions to a common reference point, such as (frac{1}{2}) or 1. This method is useful for quick, approximate comparisons.
How It Works:
- Choose a Benchmark: Select a common fraction like (frac{1}{2}) or 1.
- Compare to the Benchmark: Determine whether each fraction is greater than, less than, or equal to the benchmark.
- Compare the Results: Use the comparisons to the benchmark to infer the relationship between the original fractions.
Example:
Compare (frac{4}{7}) and (frac{5}{9}).
- Benchmark: (frac{1}{2})
- Compare to the benchmark:
- (frac{4}{7} > frac{1}{2}) because (4 > frac{7}{2} = 3.5)
- (frac{5}{9} > frac{1}{2}) because (5 > frac{9}{2} = 4.5)
- Since both fractions are greater than (frac{1}{2}), we need to refine the comparison.
Let’s try comparing how much each fraction is away from (frac{1}{2}):
- (frac{4}{7} – frac{1}{2} = frac{8 – 7}{14} = frac{1}{14})
- (frac{5}{9} – frac{1}{2} = frac{10 – 9}{18} = frac{1}{18})
Since (frac{1}{14} > frac{1}{18}), (frac{4}{7}) is slightly further away from (frac{1}{2}) than (frac{5}{9}), meaning (frac{4}{7}) is larger but close. We can confirm using another method if needed.
Advantages:
- Quick and easy for approximate comparisons.
- Useful for developing number sense.
Disadvantages:
- May not provide precise comparisons.
- Requires some intuition and familiarity with fraction values.
5.4. Visual Models
Visual models, such as area models or number lines, can be used to compare fractions, especially for students who are visual learners.
How It Works:
- Create a Model: Draw a visual representation of each fraction, such as dividing a rectangle into equal parts and shading the appropriate number of parts.
- Compare the Models: Visually compare the shaded areas or positions on the number line to determine which fraction is larger.
Example:
Compare (frac{2}{3}) and (frac{3}{4}).
- Create models: Draw two rectangles of the same size. Divide one into 3 equal parts and shade 2 parts to represent (frac{2}{3}). Divide the other into 4 equal parts and shade 3 parts to represent (frac{3}{4}).
- Compare the models: Visually, it’s clear that (frac{3}{4}) has a larger shaded area than (frac{2}{3}), so (frac{3}{4} > frac{2}{3}).
Advantages:
- Intuitive and easy to understand, especially for visual learners.
- Helpful for building conceptual understanding of fractions.
Disadvantages:
- Not practical for comparing fractions with large denominators.
- May not be precise enough for exact comparisons.
These alternative methods provide a range of options for comparing fractions, each with its own strengths and weaknesses. At COMPARE.EDU.VN, we encourage users to explore these methods to find the most suitable approach for their needs.
6. Real-World Applications Beyond Mathematics
While cross multiplication is a mathematical technique, its underlying principles extend beyond the classroom and into various real-world scenarios. Understanding these applications can highlight the practical relevance of this tool.
6.1. Business and Finance
In business and finance, proportions and ratios are frequently used to analyze financial statements, compare performance metrics, and make strategic decisions. Cross multiplication can be a valuable tool in these contexts.
Example 1: Analyzing Profit Margins
Suppose a company has a profit margin of (15%) on sales of $500,000. If the company wants to increase its sales to $750,000 while maintaining the same profit margin, what should the new profit be?
Set up the proportion: (frac{text{profit}}{text{sales}} = frac{15}{100} = frac{x}{750,000})
Using cross multiplication:
- Multiply (15) by (750,000): (15 times 750,000 = 11,250,000)
- Multiply (100) by (x): (100 times x = 100x)
- Set the results equal: (11,250,000 = 100x)
- Solve for (x): (x = frac{11,250,000}{100} = 112,500)
So, the new profit should be $112,500 to maintain the (15%) profit margin.
Example 2: Currency Exchange Rates
If the exchange rate between the US dollar and the Euro is (1.10) Euros per dollar, how many Euros would you get for $500?
Set up the proportion: (frac{text{Euros}}{text{Dollars}} = frac{1.10}{1} = frac{x}{500})
Using cross multiplication:
- Multiply (1.10) by (500): (1.10 times 500 = 550)
- Multiply (1) by (x): (1 times x = x)
- Set the results equal: (x = 550)
Therefore, you would get 550 Euros for $500.
6.2. Cooking and Baking
In cooking and baking, maintaining the correct ratios of ingredients is essential for achieving the desired results. Cross multiplication can help adjust recipes for different serving sizes.
Example 3: Scaling a Recipe
A recipe for cookies calls for (frac{2}{3}) cup of flour and (frac{1}{4}) cup of sugar. If you want to double the recipe, how much flour and sugar do you need?
For flour: (frac{text{flour}}{text{original}} = frac{frac{2}{3}}{1} = frac{x}{2})
- Multiply (frac{2}{3}) by (2): (frac{2}{3} times 2 = frac{4}{3})
- Multiply (1) by (x): (1 times x = x)
- So, (x = frac{4}{3}) cups of flour.
For sugar: (frac{text{sugar}}{text{original}} = frac{frac{1}{4}}{1} = frac{x}{2})
- Multiply (frac{1}{4}) by (2): (frac{1}{4} times 2 = frac{2}{4} = frac{1}{2})
- Multiply (1) by (x): (1 times x = x)
- So, (x = frac{1}{2}) cup of sugar.
Therefore, you need (frac{4}{3}) cups of flour and (frac{1}{2}) cup of sugar to double the recipe.
6.3. Healthcare and Medicine
In healthcare, ratios and proportions are used to calculate dosages, interpret test results, and monitor patient health.
Example 4: Calculating Medication Dosage
A doctor prescribes a medication at a dosage of 5 mg per kg of body weight. If a patient weighs 75 kg, what is the required dosage?
Set up the proportion: (frac{text{dosage}}{text{weight}} = frac{5}{1} = frac{x}{75})
Using cross multiplication:
- Multiply (5) by (75): (5 times 75 = 375)
- Multiply (1) by (x): (1 times x = x)
- So, (x = 375) mg.
Therefore, the required dosage is 375 mg.
6.4. Construction and Engineering
In construction and engineering, proportions are used to scale blueprints, calculate material quantities, and ensure structural integrity.
Example 5: Scaling Blueprints
On a blueprint, 1 inch represents 10 feet. If a wall is 2.5 inches long on the blueprint, what is its actual length?
Set up the proportion: (frac{text{inches}}{text{feet}} = frac{1}{10} = frac{2.5}{x})
Using cross multiplication:
- Multiply (1) by (x): (1 times x = x)
- Multiply (10) by (2.5): (10 times 2.5 = 25)
- So, (x = 25) feet.
Therefore, the actual length of the wall is 25 feet.
These examples demonstrate the broad applicability of cross multiplication in various fields beyond mathematics. At compare.edu.vn, we aim to highlight these practical uses to underscore the value of understanding fundamental mathematical concepts.
7. Tips and Tricks for Mastering Cross Multiplication
To become proficient in cross multiplication, it’s essential to practice regularly and understand some key tips and tricks. These strategies can help you avoid common errors and solve problems more efficiently.
7.1. Practice Regularly
Like any mathematical skill, proficiency in cross multiplication comes with consistent practice. Work through a variety of problems, from simple fraction comparisons to complex algebraic equations.
Strategies:
- Start with Basics: Begin with simple numerical fractions and gradually increase the complexity.
- Use Worksheets: Utilize online resources and textbooks to find practice worksheets.
- Real-World Problems: Apply cross multiplication to real-world scenarios to reinforce your understanding.
7.2. Always Check for Positive Denominators
One of the most critical steps in cross multiplication is to ensure that the denominators are positive. If a denominator is negative, multiply both the numerator and denominator by -1 before proceeding.
Example:
Compare (frac{3}{-4}) and (frac{5}{6}).
- Rewrite (frac{3}{-4}) as (frac{-3}{4}).
- Now, cross multiply: (-3 times 6 = -18) and (4 times 5 = 20).
- Since (-18 < 20), (frac{-3}{4} < frac{5}{6}).
7.3. Simplify Before Cross Multiplying
Simplifying fractions before cross multiplying can make the calculations easier and reduce the chances of error.
Example:
Compare (frac{6}{8}) and (frac{9}{12}).
- Simplify (frac{6}{8}) to (frac{3}{4}) and (frac{9}{12}) to (frac
