Simplify The Fraction $\frac{165}{440}$ Using The Prime Factors Found In The Previous Problem.

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Introduction

Simplifying fractions is an essential skill in mathematics, and it's often used in various mathematical operations, such as adding, subtracting, multiplying, and dividing fractions. In this article, we will simplify the fraction $\frac{165}{440}$ using the prime factors found in the previous problem.

Understanding Prime Factors

Before we dive into simplifying the fraction, let's first understand what prime factors are. Prime factors are the prime numbers that multiply together to give the original number. For example, the prime factors of 12 are 2 and 3, because 2 × 2 × 3 = 12.

Finding Prime Factors of 165 and 440

To simplify the fraction, we need to find the prime factors of both the numerator (165) and the denominator (440).

Prime Factors of 165

The prime factors of 165 are:

  • 3 (because 165 ÷ 3 = 55)
  • 5 (because 55 ÷ 5 = 11)
  • 11 (because 11 is a prime number)

So, the prime factors of 165 are 3, 5, and 11.

Prime Factors of 440

The prime factors of 440 are:

  • 2 (because 440 ÷ 2 = 220)
  • 2 (because 220 ÷ 2 = 110)
  • 2 (because 110 ÷ 2 = 55)
  • 5 (because 55 ÷ 5 = 11)
  • 11 (because 11 is a prime number)

So, the prime factors of 440 are 2, 2, 2, 5, and 11.

Simplifying the Fraction

Now that we have the prime factors of both the numerator and the denominator, we can simplify the fraction.

Canceling Out Common Factors

We can see that both the numerator and the denominator have the prime factors 5 and 11 in common. We can cancel out these common factors to simplify the fraction.

  • Cancel out one factor of 5 from the numerator and the denominator: $\frac{165}{440} = \frac{33}{88}$
  • Cancel out one factor of 11 from the numerator and the denominator: $\frac{33}{88} = \frac{3}{8}$

So, the simplified fraction is $\frac{3}{8}$.

Conclusion

Simplifying fractions is an essential skill in mathematics, and it's often used in various mathematical operations. In this article, we simplified the fraction $\frac{165}{440}$ using the prime factors found in the previous problem. We found the prime factors of both the numerator and the denominator, canceled out common factors, and simplified the fraction to $\frac{3}{8}$.

Frequently Asked Questions

Q: What are prime factors?

A: Prime factors are the prime numbers that multiply together to give the original number.

Q: How do I find the prime factors of a number?

A: To find the prime factors of a number, divide the number by the smallest prime number (2) and continue dividing until you can't divide anymore. Then, divide the result by the next smallest prime number (3) and continue dividing until you can't divide anymore. Repeat this process until you can't divide anymore.

Q: How do I simplify a fraction?

A: To simplify a fraction, find the prime factors of both the numerator and the denominator, cancel out common factors, and simplify the fraction.

Further Reading

If you want to learn more about simplifying fractions, here are some additional resources:

References

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Introduction

Simplifying fractions is an essential skill in mathematics, and it's often used in various mathematical operations. In our previous article, we simplified the fraction $\frac{165}{440}$ using the prime factors found in the previous problem. In this article, we will provide a Q&A guide to help you understand the concept of simplifying fractions better.

Q&A Guide

Q: What are prime factors?

A: Prime factors are the prime numbers that multiply together to give the original number. For example, the prime factors of 12 are 2 and 3, because 2 × 2 × 3 = 12.

Q: How do I find the prime factors of a number?

A: To find the prime factors of a number, divide the number by the smallest prime number (2) and continue dividing until you can't divide anymore. Then, divide the result by the next smallest prime number (3) and continue dividing until you can't divide anymore. Repeat this process until you can't divide anymore.

Q: How do I simplify a fraction?

A: To simplify a fraction, find the prime factors of both the numerator and the denominator, cancel out common factors, and simplify the fraction.

Q: What are common factors?

A: Common factors are the factors that are present in both the numerator and the denominator of a fraction. For example, in the fraction $\frac{165}{440}$, the common factors are 5 and 11.

Q: How do I cancel out common factors?

A: To cancel out common factors, divide both the numerator and the denominator by the common factor. For example, in the fraction $\frac165}{440}$, we can cancel out one factor of 5 from the numerator and the denominator $\frac{165{440} = \frac{33}{88}$

Q: What is the simplified form of a fraction?

A: The simplified form of a fraction is the fraction with the smallest possible numerator and denominator. For example, the simplified form of the fraction $\frac{165}{440}$ is $\frac{3}{8}$.

Q: Why is it important to simplify fractions?

A: Simplifying fractions is important because it makes it easier to perform mathematical operations, such as adding, subtracting, multiplying, and dividing fractions.

Q: Can I simplify a fraction with a variable?

A: Yes, you can simplify a fraction with a variable. To do this, find the prime factors of the variable and the other numbers in the fraction, cancel out common factors, and simplify the fraction.

Q: How do I know if a fraction is already simplified?

A: To determine if a fraction is already simplified, find the prime factors of the numerator and the denominator. If there are no common factors, then the fraction is already simplified.

Examples

Example 1: Simplify the fraction $\frac{24}{36}$

  • Find the prime factors of the numerator and the denominator: 24 = 2 × 2 × 2 × 3, 36 = 2 × 2 × 3 × 3
  • Cancel out common factors: 2 × 2 = 4, 3 × 3 = 9
  • Simplify the fraction: $\frac{24}{36} = \frac{4}{9}$

Example 2: Simplify the fraction $\frac{48}{60}$

  • Find the prime factors of the numerator and the denominator: 48 = 2 × 2 × 2 × 2 × 3, 60 = 2 × 2 × 3 × 5
  • Cancel out common factors: 2 × 2 = 4, 3 = 3
  • Simplify the fraction: $\frac{48}{60} = \frac{4}{5}$

Conclusion

Simplifying fractions is an essential skill in mathematics, and it's often used in various mathematical operations. In this article, we provided a Q&A guide to help you understand the concept of simplifying fractions better. We also provided examples to illustrate the concept.

Frequently Asked Questions

Q: What are some common mistakes to avoid when simplifying fractions?

A: Some common mistakes to avoid when simplifying fractions include:

  • Not finding the prime factors of the numerator and the denominator
  • Not canceling out common factors
  • Not simplifying the fraction to its smallest possible form

Q: How do I practice simplifying fractions?

A: To practice simplifying fractions, try simplifying different fractions using the steps outlined in this article. You can also use online resources, such as math worksheets and practice problems, to help you practice.

Further Reading

If you want to learn more about simplifying fractions, here are some additional resources:

References