Rename Each Pair Of Fractions By Using Common Denominators. Use $\ \textgreater \ $, $=$, Or $\ \textless \ $ To Compare The Fractions.1. $\frac{4}{6} \longrightarrow \frac{3}{4}$2. $\frac{1}{2}
Introduction
In mathematics, fractions are a way to represent a part of a whole. When comparing or adding fractions, it's essential to have a common denominator to ensure accurate calculations. In this article, we will explore how to rename each pair of fractions by using common denominators and compare them using symbols such as $\textgreater$, $=$, or $\textless$.
What are Common Denominators?
A common denominator is the least common multiple (LCM) of the denominators of two or more fractions. It's the smallest number that both denominators can divide into evenly. For example, if we have two fractions with denominators 4 and 6, the least common multiple of 4 and 6 is 12. Therefore, 12 is the common denominator for these fractions.
Renaming Fractions with Common Denominators
To rename a fraction with a common denominator, we need to multiply the numerator and denominator by the same number. This number is the ratio of the common denominator to the original denominator. Let's consider an example:
Example 1: Renaming $\frac{4}{6}$ with a Common Denominator
Suppose we want to rename $\frac{4}{6}$ with a common denominator of 12. To do this, we need to multiply the numerator and denominator by 2, since $\frac{12}{6} = 2$.
Therefore, $\frac{4}{6}$ is equivalent to $\frac{8}{12}$.
Example 2: Renaming $\frac{3}{4}$ with a Common Denominator
Now, let's consider renaming $\frac{3}{4}$ with a common denominator of 12. To do this, we need to multiply the numerator and denominator by 3, since $\frac{12}{4} = 3$.
Therefore, $\frac{3}{4}$ is equivalent to $\frac{9}{12}$.
Comparing Fractions with Common Denominators
Now that we have renamed the fractions with common denominators, we can compare them using symbols such as $\textgreater$, $=$, or $\textless$. Let's consider an example:
Example 3: Comparing $\frac{8}{12}$ and $\frac{9}{12}$
Since both fractions have the same denominator (12), we can compare the numerators directly. The numerator of $\frac{8}{12}$ is 8, and the numerator of $\frac{9}{12}$ is 9. Since 9 is greater than 8, we can conclude that $\frac{9}{12}$ is greater than $\frac{8}{12}$.
Conclusion
In conclusion, renaming fractions with common denominators is an essential skill in mathematics. By using common denominators, we can compare fractions accurately and perform calculations with confidence. In this article, we have explored how to rename fractions with common denominators and compare them using symbols such as $\textgreater$, $=$, or $\textless$. We have also provided examples to illustrate the concept.
Common Denominators: A Summary
- A common denominator is the least common multiple (LCM) of the denominators of two or more fractions.
- To rename a fraction with a common denominator, we need to multiply the numerator and denominator by the same number.
- The ratio of the common denominator to the original denominator is used to multiply the numerator and denominator.
- Comparing fractions with common denominators involves comparing the numerators directly.
Frequently Asked Questions
Q: What is a common denominator?
A: A common denominator is the least common multiple (LCM) of the denominators of two or more fractions.
Q: How do I rename a fraction with a common denominator?
A: To rename a fraction with a common denominator, we need to multiply the numerator and denominator by the same number, which is the ratio of the common denominator to the original denominator.
Q: How do I compare fractions with common denominators?
A: To compare fractions with common denominators, we need to compare the numerators directly.
References
- [1] Khan Academy. (n.d.). Least Common Multiple (LCM). Retrieved from https://www.khanacademy.org/math/pre-algebra/pre-algebra-lcm
- [2] Math Open Reference. (n.d.). Fractions. Retrieved from https://www.mathopenref.com/fractions.html
Glossary
- Common Denominator: The least common multiple (LCM) of the denominators of two or more fractions.
- Least Common Multiple (LCM): The smallest number that two or more numbers can divide into evenly.
- Numerator: The number above the line in a fraction.
- Denominator: The number below the line in a fraction.
Frequently Asked Questions: Renaming Fractions with Common Denominators ====================================================================
Q: What is a common denominator?
A: A common denominator is the least common multiple (LCM) of the denominators of two or more fractions. It's the smallest number that both denominators can divide into evenly.
Q: Why do we need to use common denominators?
A: We need to use common denominators to compare fractions accurately. When fractions have different denominators, it's difficult to compare them directly. By using a common denominator, we can compare the numerators directly and determine which fraction is greater or smaller.
Q: How do I find the common denominator of two fractions?
A: To find the common denominator of two fractions, you need to find the least common multiple (LCM) of the two denominators. You can use a calculator or a formula to find the LCM.
Q: What is the least common multiple (LCM)?
A: The least common multiple (LCM) is the smallest number that two or more numbers can divide into evenly. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that both 4 and 6 can divide into evenly.
Q: How do I rename a fraction with a common denominator?
A: To rename a fraction with a common denominator, you need to multiply the numerator and denominator by the same number. This number is the ratio of the common denominator to the original denominator.
Q: What is the ratio of the common denominator to the original denominator?
A: The ratio of the common denominator to the original denominator is the number that you need to multiply the numerator and denominator by to get the new fraction. For example, if the common denominator is 12 and the original denominator is 6, the ratio is 2, because 12 divided by 6 is 2.
Q: How do I compare fractions with common denominators?
A: To compare fractions with common denominators, you need to compare the numerators directly. The fraction with the larger numerator is greater, and the fraction with the smaller numerator is smaller.
Q: What if the numerators are equal?
A: If the numerators are equal, the fractions are equal. For example, if the numerators are both 3, the fractions are equal, even if the denominators are different.
Q: Can I use a calculator to find the common denominator?
A: Yes, you can use a calculator to find the common denominator. Most calculators have a function to find the least common multiple (LCM) of two numbers.
Q: Can I use a formula to find the common denominator?
A: Yes, you can use a formula to find the common denominator. The formula is:
LCM(a, b) = (a × b) / GCD(a, b)
where LCM is the least common multiple, a and b are the two numbers, and GCD is the greatest common divisor.
Q: What is the greatest common divisor (GCD)?
A: The greatest common divisor (GCD) is the largest number that two or more numbers can divide into evenly. For example, the GCD of 4 and 6 is 2, because 2 is the largest number that both 4 and 6 can divide into evenly.
Q: Can I use a calculator to find the GCD?
A: Yes, you can use a calculator to find the GCD. Most calculators have a function to find the greatest common divisor (GCD) of two numbers.
Q: Can I use a formula to find the GCD?
A: Yes, you can use a formula to find the GCD. The formula is:
GCD(a, b) = (a × b) / LCM(a, b)
where GCD is the greatest common divisor, a and b are the two numbers, and LCM is the least common multiple.
Conclusion
In conclusion, renaming fractions with common denominators is an essential skill in mathematics. By using common denominators, we can compare fractions accurately and perform calculations with confidence. We have also provided answers to frequently asked questions to help you understand the concept better.
Common Denominators: A Summary
- A common denominator is the least common multiple (LCM) of the denominators of two or more fractions.
- To rename a fraction with a common denominator, we need to multiply the numerator and denominator by the same number.
- The ratio of the common denominator to the original denominator is used to multiply the numerator and denominator.
- Comparing fractions with common denominators involves comparing the numerators directly.
- The least common multiple (LCM) is the smallest number that two or more numbers can divide into evenly.
- The greatest common divisor (GCD) is the largest number that two or more numbers can divide into evenly.