What Is The Ratio 16 To 40 Written As A Fraction In Lowest Terms?A. \[$\frac{1}{4}\$\] B. \[$\frac{3}{8}\$\] C. \[$\frac{2}{5}\$\] D. \[$\frac{8}{20}\$\]

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What is the Ratio 16 to 40 Written as a Fraction in Lowest Terms?

Understanding the Concept of Fractions

Fractions are a way to represent a part of a whole. They consist of two numbers: a numerator and a denominator. The numerator represents the number of equal parts being considered, while the denominator represents the total number of parts the whole is divided into. In this article, we will explore how to write the ratio 16 to 40 as a fraction in lowest terms.

The Ratio 16 to 40

The ratio 16 to 40 can be written as a fraction by placing the number 16 over the number 40. This is represented as 16/40. However, this fraction is not in its lowest terms, as we can simplify it further.

Simplifying the Fraction

To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In this case, the GCD of 16 and 40 is 8.

Calculating the GCD

To calculate the GCD, we can use the following steps:

  1. List the factors of the numerator (16) and the denominator (40).
  2. Identify the common factors between the two lists.
  3. Choose the greatest common factor from the list of common factors.

Factors of 16

The factors of 16 are: 1, 2, 4, 8, 16

Factors of 40

The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40

Common Factors

The common factors between the two lists are: 1, 2, 4, 8

Greatest Common Factor

The greatest common factor is 8.

Simplifying the Fraction

Now that we have found the GCD, we can simplify the fraction by dividing both the numerator and the denominator by the GCD. In this case, we will divide both 16 and 40 by 8.

Simplified Fraction

The simplified fraction is: 16 ÷ 8 / 40 ÷ 8 = 2/5

Conclusion

In conclusion, the ratio 16 to 40 written as a fraction in lowest terms is 2/5. This is achieved by simplifying the original fraction 16/40 by dividing both the numerator and the denominator by their greatest common divisor, which is 8.

Answer

The correct answer is C. {\frac{2}{5}$}$

Additional Examples

Here are a few more examples of simplifying fractions:

  • The ratio 12 to 18 can be written as a fraction in lowest terms by simplifying 12/18. The GCD of 12 and 18 is 6. Dividing both 12 and 18 by 6 gives us 2/3.
  • The ratio 15 to 25 can be written as a fraction in lowest terms by simplifying 15/25. The GCD of 15 and 25 is 5. Dividing both 15 and 25 by 5 gives us 3/5.

Tips and Tricks

Here are a few tips and tricks for simplifying fractions:

  • Always find the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCD to simplify the fraction.
  • Make sure to simplify the fraction to its lowest terms, as this will make it easier to work with.

Common Mistakes

Here are a few common mistakes to avoid when simplifying fractions:

  • Not finding the greatest common divisor (GCD) of the numerator and the denominator.
  • Not dividing both the numerator and the denominator by the GCD.
  • Not simplifying the fraction to its lowest terms.

Conclusion

In conclusion, simplifying fractions is an important concept in mathematics. By understanding how to simplify fractions, we can make it easier to work with them and solve problems that involve fractions. Remember to always find the greatest common divisor (GCD) of the numerator and the denominator, and divide both the numerator and the denominator by the GCD to simplify the fraction.
Frequently Asked Questions (FAQs) About Simplifying Fractions

Q: What is the greatest common divisor (GCD) and why is it important?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. It is important because it helps us simplify fractions by dividing both the numerator and the denominator by the GCD.

Q: How do I find the greatest common divisor (GCD) of two numbers?

A: To find the GCD of two numbers, you can list the factors of each number and identify the common factors. The greatest common factor is the largest number that appears in both lists.

Q: What is the difference between a simplified fraction and a reduced fraction?

A: A simplified fraction is a fraction that has been reduced to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor. A reduced fraction is a fraction that has been simplified, but it may not be in its lowest terms.

Q: Can I simplify a fraction by dividing only the numerator or only the denominator by the GCD?

A: No, you cannot simplify a fraction by dividing only the numerator or only the denominator by the GCD. To simplify a fraction, you must divide both the numerator and the denominator by the GCD.

Q: What is the purpose of simplifying fractions?

A: The purpose of simplifying fractions is to make them easier to work with and to reduce the complexity of mathematical problems. Simplifying fractions can help you to:

  • Make calculations easier
  • Reduce errors
  • Improve understanding of mathematical concepts
  • Solve problems more efficiently

Q: Can I simplify a fraction with a negative numerator or denominator?

A: Yes, you can simplify a fraction with a negative numerator or denominator. To simplify a fraction with a negative numerator or denominator, you must follow the same steps as simplifying a fraction with positive numerators and denominators.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of representing a part of a whole as a ratio of two numbers. A decimal is a way of representing a number as a sum of powers of 10. While fractions and decimals can be equivalent, they are not the same thing.

Q: Can I convert a fraction to a decimal?

A: Yes, you can convert a fraction to a decimal by dividing the numerator by the denominator. For example, the fraction 1/2 can be converted to a decimal by dividing 1 by 2, which equals 0.5.

Q: Can I convert a decimal to a fraction?

A: Yes, you can convert a decimal to a fraction by finding the greatest common divisor (GCD) of the decimal and the denominator, and then dividing both the decimal and the denominator by the GCD.

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

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

  • Not finding the greatest common divisor (GCD) of the numerator and the denominator
  • Not dividing both the numerator and the denominator by the GCD
  • Not simplifying the fraction to its lowest terms
  • Not checking for common factors between the numerator and the denominator

Q: How can I practice simplifying fractions?

A: You can practice simplifying fractions by:

  • Working through examples and exercises in a textbook or online resource
  • Creating your own examples and exercises
  • Using online tools and calculators to check your work
  • Practicing with real-world applications, such as cooking or finance

Conclusion

In conclusion, simplifying fractions is an important concept in mathematics that can help you to make calculations easier, reduce errors, and improve understanding of mathematical concepts. By following the steps outlined in this article, you can simplify fractions and become more confident in your ability to work with fractions.