Equivalent Expressions For 2 3/4 × 8/9 A Step-by-Step Guide
In this article, we will delve into the concept of equivalent expressions, focusing on the multiplication of a mixed number and a fraction. Specifically, we aim to identify expressions that are equivalent to 2 3/4 × 8/9. This exploration will involve converting mixed numbers to improper fractions, performing multiplication, and simplifying the resulting fractions. We will also examine how different representations of the same value can be obtained through various mathematical operations. Our goal is to provide a comprehensive understanding of equivalent expressions and how to determine them accurately. This topic is crucial for mastering fraction arithmetic and algebraic manipulations, forming a solid foundation for more advanced mathematical concepts. By the end of this discussion, you will be able to confidently identify equivalent expressions and apply these skills to solve a variety of mathematical problems.
Converting Mixed Numbers to Improper Fractions
The first step in determining equivalent expressions for 2 3/4 × 8/9 is to convert the mixed number, 2 3/4, into an improper fraction. A mixed number is a combination of a whole number and a fraction, while an improper fraction has a numerator that is greater than or equal to its denominator. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator of the fractional part and add the numerator. This result becomes the new numerator, and the denominator remains the same. For 2 3/4, we multiply 2 by 4, which equals 8, and then add the numerator 3, resulting in 11. Therefore, the improper fraction equivalent to 2 3/4 is 11/4. This conversion is a fundamental step in simplifying and performing operations with mixed numbers. Understanding this process allows us to work more easily with fractions in multiplication, division, and other arithmetic operations. It is essential to practice this conversion to ensure accuracy and speed in mathematical calculations. Furthermore, this skill is not only useful in basic arithmetic but also in more complex algebraic problems where fractions are involved. Mastering the conversion of mixed numbers to improper fractions is a key building block in mathematical proficiency.
Multiplying Fractions: A Step-by-Step Guide
After converting the mixed number to an improper fraction, we can proceed with the multiplication. The original expression 2 3/4 × 8/9 now becomes 11/4 × 8/9. To multiply fractions, we multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator. In this case, we multiply 11 by 8, which equals 88, and multiply 4 by 9, which equals 36. Thus, the result of the multiplication is 88/36. This process of multiplying fractions is straightforward but requires careful attention to detail to avoid errors. It is a fundamental operation in arithmetic and is used extensively in various mathematical contexts. Understanding how to multiply fractions correctly is essential for solving problems involving ratios, proportions, and algebraic equations. Furthermore, this skill is crucial in practical applications such as cooking, measuring, and financial calculations. By mastering fraction multiplication, you can confidently tackle more complex mathematical challenges and real-world scenarios that require these skills. The ability to multiply fractions accurately and efficiently is a cornerstone of mathematical competence.
Simplifying Fractions: Reducing to Lowest Terms
The result of multiplying 11/4 × 8/9 is 88/36, but this fraction can be simplified to its lowest terms. Simplifying a fraction means reducing it to an equivalent fraction where the numerator and the denominator have no common factors other than 1. To simplify 88/36, we need to find the greatest common divisor (GCD) of 88 and 36. The GCD is the largest number that divides both numbers without leaving a remainder. The factors of 88 are 1, 2, 4, 8, 11, 22, 44, and 88, while the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The greatest common divisor of 88 and 36 is 4. To simplify the fraction, we divide both the numerator and the denominator by the GCD. Dividing 88 by 4 gives us 22, and dividing 36 by 4 gives us 9. Therefore, the simplified fraction is 22/9. Simplifying fractions is an important skill in mathematics as it allows us to express fractions in their simplest form, making them easier to understand and work with. This process is crucial in various mathematical operations and problem-solving scenarios. By simplifying fractions, we can also more easily compare different fractions and determine their relative sizes. This skill is fundamental in algebra, calculus, and other advanced mathematical fields. Mastering the simplification of fractions is essential for achieving mathematical proficiency and confidence.
Evaluating the Given Options
Now that we have simplified the expression 2 3/4 × 8/9 to 22/9, we can evaluate the given options to identify which ones are equivalent. We will examine each option individually to determine if it simplifies to 22/9 or another equivalent form. This process involves converting mixed numbers to improper fractions, performing multiplication, and simplifying the results. By comparing each option to the simplified form of the original expression, we can accurately identify the equivalent expressions. This step is crucial in reinforcing our understanding of fraction arithmetic and equivalent forms. It also highlights the importance of accuracy and attention to detail in mathematical calculations. By systematically evaluating each option, we can confidently determine the correct answers and gain a deeper understanding of the relationships between different mathematical expressions. This skill is not only valuable in academic settings but also in practical situations where mathematical comparisons are necessary.
Option 1: 9/4 × 8/9
Let's evaluate the first option: 9/4 × 8/9. To multiply these fractions, we multiply the numerators together and the denominators together. This gives us (9 × 8) / (4 × 9) = 72/36. Now, we need to simplify this fraction. The greatest common divisor (GCD) of 72 and 36 is 36. Dividing both the numerator and the denominator by 36, we get 72/36 = 2/1 = 2. Comparing this result to the simplified form of the original expression, 22/9, we can see that they are not equivalent. Therefore, this option is not a correct answer. This process of evaluating each option step-by-step is crucial in ensuring accuracy and avoiding errors. By carefully performing the calculations and simplifying the results, we can confidently determine whether each option is equivalent to the original expression. This approach also reinforces our understanding of fraction arithmetic and simplification techniques. In this case, we have clearly shown that 9/4 × 8/9 is not equivalent to 22/9, demonstrating the importance of thorough evaluation.
Option 2: 11/4 × 8/9
The second option is 11/4 × 8/9. This expression represents the multiplication of the improper form of 2 3/4 (which is 11/4) by 8/9. As we calculated earlier, multiplying these fractions involves multiplying the numerators (11 × 8 = 88) and the denominators (4 × 9 = 36), resulting in 88/36. Simplifying this fraction requires finding the greatest common divisor (GCD) of 88 and 36, which is 4. Dividing both the numerator and the denominator by 4, we get 88/4 = 22 and 36/4 = 9. Thus, the simplified fraction is 22/9. This result matches the simplified form of our original expression, which we found to be 22/9. Therefore, this option is equivalent to the original expression. This confirms that correctly converting the mixed number to an improper fraction and performing the multiplication yields the expected result. The step-by-step evaluation process is crucial in ensuring accuracy and understanding the underlying mathematical principles. This example demonstrates the importance of careful calculation and simplification in determining equivalent expressions.
Option 3: 72/36
The third option to consider is 72/36. To determine if this fraction is equivalent to the original expression, we need to simplify it. The greatest common divisor (GCD) of 72 and 36 is 36. Dividing both the numerator and the denominator by 36, we get 72/36 = 2/1 = 2. Comparing this simplified form to the simplified form of the original expression, which is 22/9, we can see that they are not equal. Therefore, 72/36 is not equivalent to 2 3/4 × 8/9. This evaluation reinforces the importance of simplifying fractions to their lowest terms to accurately compare them. It also highlights the potential for confusion if fractions are not simplified, as 72/36 might initially seem related to the multiplication performed in the original expression but ultimately represents a different value. This step-by-step analysis ensures a clear understanding of the relationships between different fractional expressions.
Option 4: 88/36
Our fourth option is 88/36. As we calculated earlier when evaluating Option 2 (11/4 × 8/9), multiplying the fractions resulted in 88/36 before simplification. This fraction represents the direct result of multiplying the numerators and denominators of 11/4 and 8/9. To determine if it is equivalent to the original expression, we need to simplify it. The greatest common divisor (GCD) of 88 and 36 is 4. Dividing both the numerator and the denominator by 4, we get 88/4 = 22 and 36/4 = 9. Thus, the simplified fraction is 22/9. This matches the simplified form of our original expression, confirming that 88/36 is indeed equivalent to 2 3/4 × 8/9. This evaluation reinforces the concept that different forms of a fraction can represent the same value, and simplification is key to identifying these equivalencies. The process of verifying this option highlights the importance of understanding the steps involved in fraction multiplication and simplification.
Option 5: 2 4/9
The final option is the mixed number 2 4/9. To determine if this is equivalent to the original expression, we need to convert it to an improper fraction and compare it to our simplified result of 22/9. To convert 2 4/9 to an improper fraction, we multiply the whole number (2) by the denominator (9) and add the numerator (4). This gives us (2 × 9) + 4 = 18 + 4 = 22. The denominator remains the same, so the improper fraction is 22/9. Comparing this to the simplified form of the original expression, 22/9, we can see that they are the same. Therefore, the mixed number 2 4/9 is equivalent to 2 3/4 × 8/9. This evaluation demonstrates the relationship between mixed numbers and improper fractions and reinforces the importance of being able to convert between these forms to compare and simplify expressions. This final step completes our analysis of all the given options and provides a comprehensive understanding of the equivalent expressions.
Conclusion
In conclusion, after converting the mixed number to an improper fraction, performing the multiplication, and simplifying the results, we identified the expressions equivalent to 2 3/4 × 8/9. The equivalent expressions are:
- 11/4 × 8/9
- 88/36
- 2 4/9
These expressions all simplify to 22/9, which is the simplified form of the original expression. This exercise highlights the importance of understanding fraction arithmetic, including converting mixed numbers to improper fractions, multiplying fractions, and simplifying fractions to their lowest terms. By mastering these skills, you can confidently identify equivalent expressions and solve a variety of mathematical problems involving fractions. The ability to work with fractions accurately and efficiently is crucial for success in mathematics and in many real-world applications.
