What Is The Least Common Denominator Of $\frac{7}{11}, \frac{1}{8},$ And $\frac{1}{3}$?

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Introduction

In mathematics, the least common denominator (LCD) is the smallest multiple that is common to the denominators of a set of fractions. It is an essential concept in arithmetic and algebra, particularly when working with fractions and mixed numbers. In this article, we will explore the concept of the least common denominator and provide a step-by-step guide on how to find the LCD of three fractions: $\frac{7}{11}, \frac{1}{8},$ and $\frac{1}{3}$.

What is the Least Common Denominator?

The least common denominator is the smallest multiple that is common to the denominators of a set of fractions. It is denoted by the symbol LCD. The LCD is an essential concept in mathematics, particularly when working with fractions and mixed numbers. It is used to add, subtract, multiply, and divide fractions with different denominators.

Finding the Least Common Denominator

To find the least common denominator, we need to find the prime factorization of each denominator. The prime factorization of a number is the expression of that number as the product of its prime factors.

Prime Factorization of 11

The prime factorization of 11 is 11, since 11 is a prime number.

Prime Factorization of 8

The prime factorization of 8 is $2^3$, since 8 can be expressed as the product of three 2's.

Prime Factorization of 3

The prime factorization of 3 is 3, since 3 is a prime number.

Finding the Least Common Denominator of $\frac{7}{11}, \frac{1}{8},$ and $\frac{1}{3}$

To find the least common denominator of $\frac{7}{11}, \frac{1}{8},$ and $\frac{1}{3}$, we need to find the product of the highest power of each prime factor that appears in the prime factorization of the denominators.

Step 1: Identify the Prime Factors

The prime factors of 11 are 11, the prime factors of 8 are $2^3$, and the prime factors of 3 are 3.

Step 2: Find the Highest Power of Each Prime Factor

The highest power of 2 that appears in the prime factorization of the denominators is $2^3$. The highest power of 3 that appears in the prime factorization of the denominators is 3. The highest power of 11 that appears in the prime factorization of the denominators is 11.

Step 3: Find the Product of the Highest Power of Each Prime Factor

The product of the highest power of each prime factor is $2^3 \times 3 \times 11 = 264$.

Conclusion

The least common denominator of $\frac{7}{11}, \frac{1}{8},$ and $\frac{1}{3}$ is 264. This is the smallest multiple that is common to the denominators of the three fractions. The least common denominator is an essential concept in mathematics, particularly when working with fractions and mixed numbers.

Example Problems

Problem 1

Find the least common denominator of $\frac{3}{5}, \frac{1}{6},$ and $\frac{1}{4}$.

Solution

The prime factorization of 5 is 5, the prime factorization of 6 is $2 \times 3$, and the prime factorization of 4 is $2^2$. The product of the highest power of each prime factor is $2^2 \times 3 \times 5 = 60$. Therefore, the least common denominator of $\frac{3}{5}, \frac{1}{6},$ and $\frac{1}{4}$ is 60.

Problem 2

Find the least common denominator of $\frac{2}{3}, \frac{1}{9},$ and $\frac{1}{6}$.

Solution

The prime factorization of 3 is 3, the prime factorization of 9 is $3^2$, and the prime factorization of 6 is $2 \times 3$. The product of the highest power of each prime factor is $2 \times 3^2 = 18$. Therefore, the least common denominator of $\frac{2}{3}, \frac{1}{9},$ and $\frac{1}{6}$ is 18.

Final Thoughts

The least common denominator is an essential concept in mathematics, particularly when working with fractions and mixed numbers. It is used to add, subtract, multiply, and divide fractions with different denominators. In this article, we provided a step-by-step guide on how to find the least common denominator of three fractions: $\frac{7}{11}, \frac{1}{8},$ and $\frac{1}{3}$. We also provided example problems to help illustrate the concept.

Introduction

In our previous article, we discussed the concept of the least common denominator (LCD) and provided a step-by-step guide on how to find the LCD of three fractions. In this article, we will answer some frequently asked questions (FAQs) about the LCD.

Q&A

Q1: What is the least common denominator?

A1: The least common denominator (LCD) is the smallest multiple that is common to the denominators of a set of fractions.

Q2: Why is the least common denominator important?

A2: The least common denominator is important because it allows us to add, subtract, multiply, and divide fractions with different denominators.

Q3: How do I find the least common denominator?

A3: To find the least common denominator, you need to find the prime factorization of each denominator and then find the product of the highest power of each prime factor.

Q4: What is the prime factorization of a number?

A4: The prime factorization of a number is the expression of that number as the product of its prime factors.

Q5: How do I find the prime factorization of a number?

A5: To find the prime factorization of a number, you need to divide the number by prime numbers starting from 2 until you reach 1.

Q6: What is the difference between the least common denominator and the greatest common divisor?

A6: The least common denominator (LCD) is the smallest multiple that is common to the denominators of a set of fractions, while the greatest common divisor (GCD) is the largest number that divides each of the numbers in a set of numbers.

Q7: Can I use a calculator to find the least common denominator?

A7: Yes, you can use a calculator to find the least common denominator. However, it is also important to understand the concept and be able to do it manually.

Q8: How do I use the least common denominator to add fractions?

A8: To add fractions, you need to find the least common denominator and then convert each fraction to have the same denominator.

Q9: Can I use the least common denominator to subtract fractions?

A9: Yes, you can use the least common denominator to subtract fractions. The process is similar to adding fractions.

Q10: Can I use the least common denominator to multiply fractions?

A10: Yes, you can use the least common denominator to multiply fractions. The process is similar to adding and subtracting fractions.

Example Problems

Problem 1

Find the least common denominator of $\frac{3}{5}, \frac{1}{6},$ and $\frac{1}{4}$.

Solution

The prime factorization of 5 is 5, the prime factorization of 6 is $2 \times 3$, and the prime factorization of 4 is $2^2$. The product of the highest power of each prime factor is $2^2 \times 3 \times 5 = 60$. Therefore, the least common denominator of $\frac{3}{5}, \frac{1}{6},$ and $\frac{1}{4}$ is 60.

Problem 2

Find the least common denominator of $\frac{2}{3}, \frac{1}{9},$ and $\frac{1}{6}$.

Solution

The prime factorization of 3 is 3, the prime factorization of 9 is $3^2$, and the prime factorization of 6 is $2 \times 3$. The product of the highest power of each prime factor is $2 \times 3^2 = 18$. Therefore, the least common denominator of $\frac{2}{3}, \frac{1}{9},$ and $\frac{1}{6}$ is 18.

Final Thoughts

The least common denominator is an essential concept in mathematics, particularly when working with fractions and mixed numbers. It is used to add, subtract, multiply, and divide fractions with different denominators. In this article, we answered some frequently asked questions (FAQs) about the LCD and provided example problems to help illustrate the concept.

Additional Resources

• Mathway: A online math problem solver that can help you find the least common denominator.
• Khan Academy: A online learning platform that provides video lessons and practice exercises on mathematics, including the least common denominator.
• Wolfram Alpha: A online calculator that can help you find the least common denominator and other mathematical concepts.

Conclusion

The least common denominator is an essential concept in mathematics, particularly when working with fractions and mixed numbers. It is used to add, subtract, multiply, and divide fractions with different denominators. In this article, we answered some frequently asked questions (FAQs) about the LCD and provided example problems to help illustrate the concept. We also provided additional resources that can help you learn more about the least common denominator.