Simplifying Fractions A Step-by-Step Guide To 1/5 * 5/6 * 3/7 * 21/26

In the realm of mathematics, simplifying fractions is a foundational skill, particularly crucial when dealing with the multiplication of multiple fractions. This article will provide a comprehensive walkthrough of simplifying the expression $\frac{1}{5} \times \frac{5}{6} \times \frac{3}{7} \times \frac{21}{26}$. We will explore the basic principles of fraction multiplication and simplification, offering a detailed, step-by-step approach to break down the problem into manageable parts. This exercise not only enhances understanding of fraction manipulation but also demonstrates how to efficiently solve similar mathematical problems. Whether you're a student brushing up on your skills or someone looking to refresh your knowledge, this guide will provide clear and concise instructions to master the art of simplifying fractions.

Understanding Fraction Multiplication

Before diving into the specifics of simplifying the given expression, it's essential to grasp the fundamental principle of multiplying fractions. When multiplying fractions, we multiply the numerators (the top numbers) together to get the new numerator, and we multiply the denominators (the bottom numbers) together to get the new denominator. This process can be represented mathematically as follows:

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

This simple rule forms the backbone of fraction multiplication. Applying this to a series of fractions, such as in our case, involves extending this principle across all fractions involved. Understanding this basic rule is the first step in efficiently simplifying complex expressions involving fraction multiplication. By multiplying the numerators and denominators separately, we can consolidate multiple fractions into a single fraction, which can then be simplified. This approach is crucial for tackling problems with numerous fractions, as it provides a clear and structured method for reaching the solution. Mastering this initial step lays the groundwork for more advanced fraction manipulations and problem-solving in mathematics. Keep in mind that while the multiplication itself is straightforward, the key to simplifying lies in identifying opportunities for cancellation before or after the multiplication, a technique we will delve into further in this article. The ability to confidently multiply fractions is not just a procedural skill; it's a foundational element for understanding more complex mathematical concepts.

Step-by-Step Simplification of $\frac{1}{5} \times \frac{5}{6} \times \frac{3}{7} \times \frac{21}{26}$

To simplify the expression $\frac{1}{5} \times \frac{5}{6} \times \frac{3}{7} \times \frac{21}{26}$, we can follow a methodical, step-by-step approach. This ensures accuracy and clarity in our calculations. The initial step involves multiplying all the numerators together and then multiplying all the denominators together. This gives us:

$\frac{1 \times 5 \times 3 \times 21}{5 \times 6 \times 7 \times 26}$

Now, we have a single fraction. Before we perform the actual multiplication, it’s often more efficient to look for opportunities to simplify by canceling out common factors between the numerators and the denominators. This technique reduces the size of the numbers we're dealing with, making the calculation process easier and less prone to errors. In our fraction, we can observe that there are several common factors. For instance, the number 5 appears in both the numerator and the denominator, as does 7 and factors of 3. Identifying and canceling these common factors is a crucial step in simplifying fractions. It not only makes the arithmetic simpler but also provides a clearer understanding of the relationship between the numbers involved. This process of simplification before multiplication is a key strategy in fraction manipulation, and it highlights the importance of recognizing factors and multiples. By adopting this approach, we are setting the stage for a more streamlined and efficient simplification process, which will ultimately lead us to the simplest form of the fraction.

Cancelling Common Factors

After setting up the fraction as $\frac{1 \times 5 \times 3 \times 21}{5 \times 6 \times 7 \times 26}$, the next crucial step in simplifying fractions involves identifying and canceling out common factors. This process is essentially dividing both the numerator and the denominator by the same number, which doesn't change the fraction's value but simplifies its form. Let's break down the cancellation process in our expression:

• We can see a factor of 5 in both the numerator and the denominator. So, we cancel out the 5s:

  $\frac{1 \times \cancel{5} \times 3 \times 21}{\cancel{5} \times 6 \times 7 \times 26}$

• Next, we notice that 21 in the numerator and 7 in the denominator share a common factor of 7. We can divide both by 7 (21 ÷ 7 = 3 and 7 ÷ 7 = 1):

  $\frac{1 \times 3 \times 3 \times \cancel{21}}{\cancel{7} \times 6 \times 26}$

• Now, we have $\frac{1 \times 3 \times 3}{6 \times 26}$. We can simplify further by noticing that 6 in the denominator can be expressed as 2 × 3. This allows us to cancel a factor of 3:

  $\frac{1 \times \cancel{3} \times 3}{2 \times \cancel{3} \times 26}$

By systematically canceling common factors, we significantly reduce the complexity of the fraction. This technique is a cornerstone of fraction simplification and is particularly useful when dealing with large numbers or multiple fractions. The key to effective cancellation lies in recognizing the prime factors of the numbers involved and systematically eliminating the common ones. This not only simplifies the arithmetic but also provides a deeper understanding of the numerical relationships within the fraction. As we proceed with the simplification, each cancellation brings us closer to the simplest form of the fraction, making it easier to manage and interpret. This step-by-step approach ensures that we don't miss any opportunities for simplification and that we maintain accuracy throughout the process. The methodical cancellation of common factors is a testament to the power of breaking down complex problems into smaller, more manageable steps.

Further Simplification and Final Calculation

After canceling common factors, our expression has been simplified to $\frac{1 \times 3}{2 \times 26}$. This fraction is significantly simpler than our original expression, but we can still take it a step further to reach the simplest form. The remaining step involves performing the multiplication in both the numerator and the denominator:

Numerator: 1 × 3 = 3

Denominator: 2 × 26 = 52

So, our fraction now looks like this:

$\frac{3}{52}$

Now, we need to check if there are any more common factors between the numerator (3) and the denominator (52). The factors of 3 are 1 and 3, and the factors of 52 are 1, 2, 4, 13, 26, and 52. By comparing these factors, we can see that 3 and 52 have no common factors other than 1. This means that the fraction $\frac{3}{52}$ is in its simplest form. Reaching this final, simplified fraction is the culmination of our step-by-step process, highlighting the effectiveness of canceling common factors and simplifying along the way. This final calculation not only provides the answer but also reinforces the importance of verifying that the fraction is indeed in its lowest terms. The journey from the original complex expression to this simplified fraction showcases the elegance and efficiency of mathematical simplification techniques. Understanding when a fraction is fully simplified is as important as the simplification process itself, ensuring that we arrive at the most concise and understandable form of the answer. The resulting fraction, $\frac{3}{52}$, is a clear and precise representation of the original expression, demonstrating the power of systematic simplification.

Conclusion

In conclusion, simplifying the expression $\frac{1}{5} \times \frac{5}{6} \times \frac{3}{7} \times \frac{21}{26}$ demonstrates the importance of understanding fraction multiplication and simplification techniques in mathematics. By following a step-by-step approach, we were able to break down a seemingly complex problem into manageable parts. We started by multiplying the numerators and denominators, then we strategically canceled out common factors to reduce the fraction to its simplest form. The final result, $\frac{3}{52}$, showcases the efficiency of these methods. This exercise not only reinforces the fundamental rules of fraction manipulation but also highlights the value of methodical problem-solving. The ability to simplify fractions is a crucial skill in mathematics, applicable in various contexts, from basic arithmetic to more advanced algebraic equations. The process we followed, from initial multiplication to final simplification, underscores the significance of each step and the cumulative effect of these techniques. Furthermore, the practice of identifying and canceling common factors is a skill that extends beyond fraction simplification, aiding in broader mathematical problem-solving. By mastering these techniques, individuals can approach similar mathematical challenges with confidence and precision, ultimately enhancing their overall mathematical proficiency. The journey of simplifying this expression serves as a valuable lesson in the power of methodical simplification and its role in making complex mathematical problems more accessible and solvable.