Understanding the Concept of Reducing Fractions

Oct 13, 2025 by miodeya.com 53 views

Understanding the Concept of Reducing Fractions

In mathematics, reducing fractions to their lowest terms is an essential skill that helps simplify complex fractions and make them easier to work with. When we multiply two fractions together, we need to reduce the resulting fraction to its lowest terms to ensure that it is in its simplest form. In this article, we will explore how to reduce fractions to their lowest terms using the example of the fraction ${$ \frac{7}{8} \times \frac{3}{9}$}$.

What is a Fraction in Lowest Terms?

A fraction is in its lowest terms when the numerator and denominator have no common factors other than 1. In other words, the numerator and denominator are relatively prime. For example, the fraction ${$ \frac{2}{3}$}$ is in its lowest terms because 2 and 3 have no common factors other than 1.

How to Reduce a Fraction to Its Lowest Terms

To reduce a fraction to its lowest terms, we need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both the numerator and denominator without leaving a remainder. Once we have found the GCD, we can divide both the numerator and denominator by the GCD to simplify the fraction.

Reducing the Fraction ${$ \frac{7}{8} \times \frac{3}{9}$

To reduce the fraction [ $\frac{7}{8} \times \frac{3}{9}\$}$ , we need to multiply the two fractions together first.

${$ \frac{7}{8} \times \frac{3}{9} = \frac{7 \times 3}{8 \times 9} = \frac{21}{72}$}$

Now, we need to reduce the fraction ${$ \frac{21}{72}$] to its lowest terms. To do this, we need to find the GCD of 21 and 72.

Finding the GCD of 21 and 72

To find the GCD of 21 and 72, we can use the Euclidean algorithm. The Euclidean algorithm is a method for finding the GCD of two numbers by repeatedly dividing the larger number by the smaller number and taking the remainder.

Step 1: Divide 72 by 21

[ $\frac{72}{21} = 3 \text{ with a remainder of } 9\$}$

Step 2: Divide 21 by 9

${$ \frac{21}{9} = 2 \text{ with a remainder of } 3$}$

Step 3: Divide 9 by 3

${$ \frac{9}{3} = 3 \text{ with a remainder of } 0$}$

Since the remainder is 0, we can stop the algorithm. The GCD of 21 and 72 is 3.

Reducing the Fraction ${$ \frac{21}{72}$]

Now that we have found the GCD of 21 and 72, we can reduce the fraction [$\frac{21}{72}$] to its lowest terms by dividing both the numerator and denominator by the GCD.

[ $\frac{21}{72} = \frac{21 \div 3}{72 \div 3} = \frac{7}{24}\$}$

Therefore, the fraction [ $\frac{7}{8} \times \frac{3}{9}\$] reduced to its lowest terms equals \[$ \frac{7}{24}$].

Conclusion

Reducing fractions to their lowest terms is an essential skill in mathematics that helps simplify complex fractions and make them easier to work with. By following the steps outlined in this article, we can reduce fractions to their lowest terms using the example of the fraction [$\frac{7}{8} \times \frac{3}{9}$]. We can use the Euclidean algorithm to find the GCD of the numerator and denominator, and then divide both the numerator and denominator by the GCD to simplify the fraction.

Answer

The correct answer is:

A. [$\frac{7}{24}$]

Other Options

The other options are incorrect because they do not represent the fraction [$\frac{7}{8} \times \frac{3}{9}$] reduced to its lowest terms.

B. [ $\frac{24}{63}\$] is incorrect because it does not represent the fraction \[$ \frac{7}{8} \times \frac{3}{9}$] reduced to its lowest terms.

C. [ $\frac{8}{21}\$] is incorrect because it does not represent the fraction \[$ \frac{7}{8} \times \frac{3}{9}$] reduced to its lowest terms.

D. [ $\frac{21}{72}\$] is incorrect because it does not represent the fraction \[$ \frac{7}{8} \times \frac{3}{9}$] reduced to its lowest terms.

Final Answer

$The final answer is A. \[$\frac{7}{24}\$].<br/> **Frequently Asked Questions (FAQs) About Reducing Fractions**$

Q: What is the purpose of reducing fractions to their lowest terms?

A: The purpose of reducing fractions to their lowest terms is to simplify complex fractions and make them easier to work with. By reducing a fraction to its lowest terms, we can eliminate any unnecessary factors and make the fraction more manageable.

Q: How do I know if a fraction is in its lowest terms?

A: A fraction is in its lowest terms when the numerator and denominator have no common factors other than 1. In other words, the numerator and denominator are relatively prime.

Q: What is the greatest common divisor (GCD) and how do I find it?

A: The GCD is the largest number that divides both the numerator and denominator without leaving a remainder. To find the GCD, you can use the Euclidean algorithm, which involves repeatedly dividing the larger number by the smaller number and taking the remainder.

Q: How do I reduce a fraction to its lowest terms using the Euclidean algorithm?

A: To reduce a fraction to its lowest terms using the Euclidean algorithm, follow these steps:

Divide the larger number by the smaller number and take the remainder.
Divide the smaller number by the remainder and take the new remainder.
Repeat step 2 until the remainder is 0.
The GCD is the last non-zero remainder.
Divide both the numerator and denominator by the GCD to simplify the fraction.

Q: What if the GCD is 1?

A: If the GCD is 1, then the fraction is already in its lowest terms and cannot be simplified further.

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 built-in function for finding the GCD.

Q: How do I know if a fraction is already in its lowest terms?

A: To determine if a fraction is already in its lowest terms, check if the numerator and denominator have any common factors other than 1. If they do, then the fraction is not in its lowest terms and needs to be reduced.

Q: Can I reduce a fraction to its lowest terms by simply dividing the numerator and denominator by their greatest common factor?

A: Yes, you can reduce a fraction to its lowest terms by simply dividing the numerator and denominator by their greatest common factor. This is a quicker method than using the Euclidean algorithm, but it requires you to know the GCD in advance.

Q: What if I make a mistake when reducing a fraction to its lowest terms?

A: If you make a mistake when reducing a fraction to its lowest terms, you may end up with an incorrect answer. To avoid this, double-check your work and make sure that you have followed the correct steps.

Q: Can I use a computer program to reduce fractions to their lowest terms?

A: Yes, you can use a computer program to reduce fractions to their lowest terms. Many computer algebra systems and math software programs have built-in functions for reducing fractions to their lowest terms.

Q: How do I know if a computer program is reducing a fraction to its lowest terms correctly?

A: To ensure that a computer program is reducing a fraction to its lowest terms correctly, check the program's output against the correct answer. You can also use a calculator or a different computer program to verify the answer.

Q: Can I reduce fractions to their lowest terms in my head?

A: Yes, you can reduce fractions to their lowest terms in your head, but it requires practice and patience. Start by reducing simple fractions and gradually move on to more complex ones. With practice, you will become more comfortable reducing fractions to their lowest terms mentally.