Identifying Common Factors

Oct 13, 2025 by miodeya.com 3 views

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Introduction

Reducing a fraction to its lowest terms is an essential skill in mathematics, particularly in algebra and calculus. It involves simplifying a fraction by canceling out any common factors in the numerator and denominator. In this article, we will discuss how to reduce a fraction to its lowest terms, with a focus on the given problem: $\frac{-5jk}{35j^2k^2}$ .

Understanding the Problem

The given problem is a fraction with variables in the numerator and denominator. The numerator is $-5jk$ , and the denominator is $35j^2k^2$ . To reduce this fraction to its lowest terms, we need to identify any common factors in the numerator and denominator.

Identifying Common Factors

To identify common factors, we need to look for any variables or constants that appear in both the numerator and denominator. In this case, we can see that both the numerator and denominator contain the variables $j$ and $k$ . We can also see that the constant $5$ appears in the numerator, and the constant $35$ appears in the denominator.

Canceling Out Common Factors

Now that we have identified the common factors, we can cancel them out to simplify the fraction. We can start by canceling out the variable $j$ . Since $j$ appears in both the numerator and denominator, we can cancel it out by dividing both the numerator and denominator by $j$ . This gives us:

\frac{-5k}{35jk}

Next, we can cancel out the variable $k$ . Since $k$ appears in both the numerator and denominator, we can cancel it out by dividing both the numerator and denominator by $k$ . This gives us:

\frac{-5}{35j}

Simplifying the Fraction

Now that we have canceled out the common factors, we can simplify the fraction further. We can see that the constant $5$ appears in the numerator, and the constant $35$ appears in the denominator. We can simplify the fraction by dividing both the numerator and denominator by $5$ . This gives us:

\frac{-1}{7j}

Conclusion

In conclusion, we have reduced the given fraction to its lowest terms. We started by identifying the common factors in the numerator and denominator, and then canceled them out to simplify the fraction. Finally, we simplified the fraction further by dividing both the numerator and denominator by the common factor $5$ . The final answer is $\frac{-1}{7j}$ .

Example Problems

Here are a few example problems to help you practice reducing fractions to their lowest terms:

$\frac{12x^2y}{24xy^2} =$
$\frac{15z^3}{45z^2} =$
$\frac{-9a^2b}{-27a^3b^2} =$

Step-by-Step Solution

Here is a step-by-step solution to the given problem:

Identify the common factors in the numerator and denominator.
Cancel out the common factors by dividing both the numerator and denominator by the common factor.
Simplify the fraction further by dividing both the numerator and denominator by any remaining common factors.

Tips and Tricks

Here are a few tips and tricks to help you reduce fractions to their lowest terms:

Make sure to identify all the common factors in the numerator and denominator.
Cancel out the common factors by dividing both the numerator and denominator by the common factor.
Simplify the fraction further by dividing both the numerator and denominator by any remaining common factors.

Common Mistakes

Here are a few common mistakes to avoid when reducing fractions to their lowest terms:

Failing to identify all the common factors in the numerator and denominator.
Not canceling out the common factors by dividing both the numerator and denominator by the common factor.
Not simplifying the fraction further by dividing both the numerator and denominator by any remaining common factors.

Real-World Applications

Reducing fractions to their lowest terms has many real-world applications. Here are a few examples:

In algebra, reducing fractions to their lowest terms is essential for solving equations and inequalities.
In calculus, reducing fractions to their lowest terms is essential for finding derivatives and integrals.
In physics, reducing fractions to their lowest terms is essential for solving problems involving motion and energy.

Conclusion

In conclusion, reducing a fraction to its lowest terms is an essential skill in mathematics. It involves identifying common factors in the numerator and denominator, canceling them out, and simplifying the fraction further. By following the steps outlined in this article, you can reduce any fraction to its lowest terms.