Simplifying Ratios How To Reduce 21/49 To Its Lowest Terms

August 5, 2025 by Scholario Team 59 views

Decoding Ratios: Finding the Simplest Form of 21/49

Hey there, math enthusiasts! Today, we're diving into the world of ratios and fractions to simplify the fraction 21/49. Our goal is to express this fraction in its simplest, most reduced form. So, which ratio is the correct representation of 21/49 when reduced to its lowest terms? Is it 1:2, 2:9, 3:7, or 7:9? Let's break it down step by step and discover the answer together.

Understanding Ratios and Fractions

Before we jump into simplifying 21/49, let's quickly recap what ratios and fractions are all about. A ratio is essentially a way to compare two quantities. It shows how much of one thing there is compared to another. For example, if you have 3 apples and 2 oranges, the ratio of apples to oranges is 3:2. A fraction, on the other hand, represents a part of a whole. It consists of two parts: the numerator (the top number) and the denominator (the bottom number). The numerator tells you how many parts you have, and the denominator tells you the total number of parts. For instance, in the fraction 1/4, 1 is the numerator, and 4 is the denominator, indicating one part out of four.

Ratios and fractions are closely related. In fact, a fraction can be seen as a ratio comparing a part to the whole. So, 21/49 can be interpreted as the ratio of 21 to 49. The key to simplifying ratios and fractions lies in finding the greatest common factor (GCF) of the numbers involved. The GCF is the largest number that divides both the numerator and the denominator without leaving a remainder. Once we find the GCF, we can divide both parts of the ratio or fraction by it to reduce it to its simplest form. This process ensures that we're expressing the relationship between the two quantities in the most concise way possible. Now, with this understanding in mind, let's tackle the task of simplifying 21/49 and see which of the given options matches its simplest form.

Finding the Greatest Common Factor (GCF)

To simplify the fraction 21/49, our first step is to identify the greatest common factor (GCF) of 21 and 49. The GCF, as we discussed, is the largest number that divides both 21 and 49 without leaving a remainder. To find it, we can list the factors of each number and see which one is the largest they have in common.

Let's start with 21. The factors of 21 are the numbers that divide evenly into 21, which are 1, 3, 7, and 21. Now, let's consider 49. The factors of 49 are 1, 7, and 49. Comparing the lists of factors for 21 and 49, we can see that the common factors are 1 and 7. However, we're interested in the greatest common factor, which is the largest number they share. In this case, the GCF of 21 and 49 is 7.

Now that we've found the GCF, we can use it to simplify the fraction. This involves dividing both the numerator (21) and the denominator (49) by the GCF. This process ensures that we reduce the fraction to its simplest form, where the numbers are as small as possible while maintaining the same ratio. So, let's proceed with dividing both parts of the fraction by 7 and see what we get. This step is crucial in determining the correct reduced ratio and identifying the answer from the given options.

Simplifying the Fraction

Now that we've identified the greatest common factor (GCF) of 21 and 49 as 7, we can proceed with simplifying the fraction 21/49. The process of simplification involves dividing both the numerator and the denominator by their GCF. This will reduce the fraction to its lowest terms while preserving the proportional relationship between the two numbers.

So, let's divide the numerator, 21, by the GCF, 7: 21 ÷ 7 = 3. This gives us the new numerator for the simplified fraction. Next, we divide the denominator, 49, by the GCF, 7: 49 ÷ 7 = 7. This gives us the new denominator for the simplified fraction.

Therefore, the simplified fraction is 3/7. This means that the ratio 21:49, when reduced to its lowest terms, is equivalent to the ratio 3:7. By dividing both parts of the original fraction by their greatest common factor, we've expressed the relationship between the two quantities in its simplest form. Now, we can compare this simplified ratio with the options provided in the question and identify the correct answer. This step-by-step approach ensures that we not only find the correct answer but also understand the underlying principles of simplifying fractions and ratios.

Identifying the Correct Ratio

After simplifying the fraction 21/49, we arrived at the reduced fraction 3/7. This means that the ratio of 21 to 49, when expressed in its simplest form, is 3:7. Now, let's compare this result with the options provided in the question to identify the correct ratio.

The options given were: 1:2, 2:9, 3:7, and 7:9. Comparing our simplified ratio, 3:7, with these options, we can clearly see that it matches the third option. Therefore, the correct ratio when changing 21/49 to a ratio, reduced to the lowest terms, is 3:7.

This exercise demonstrates the importance of simplifying fractions and ratios to their lowest terms. By finding the greatest common factor and dividing both parts of the fraction or ratio by it, we can express the relationship between the numbers in the most concise and understandable way. In this case, reducing 21/49 to 3/7 allows us to easily see the proportional relationship between the two quantities. So, the next time you encounter a fraction or ratio, remember the power of simplification – it can make complex relationships much clearer!

Conclusion

In conclusion, we successfully determined the correct ratio when changing 21/49 to its simplest form. By understanding the concepts of ratios, fractions, and the greatest common factor, we were able to systematically simplify the fraction and arrive at the answer. We found that the greatest common factor of 21 and 49 is 7, and dividing both the numerator and the denominator by 7 gave us the simplified fraction 3/7. Therefore, the correct ratio is 3:7.

This exercise highlights the importance of mastering fundamental mathematical concepts like simplifying fractions and ratios. These skills are not only crucial for academic success but also have practical applications in everyday life, such as when calculating proportions, scaling recipes, or understanding financial ratios. By breaking down the problem into smaller, manageable steps, we were able to clearly understand the process and confidently arrive at the correct answer. So, keep practicing and honing your math skills, and you'll be well-equipped to tackle any numerical challenge that comes your way!