Multiplying Fractions And Expressing Results In Lowest Terms A Comprehensive Guide

Understanding Fraction Multiplication

Fraction multiplication is a fundamental arithmetic operation that involves combining two or more fractions to obtain a new fraction. To multiply fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. The resulting fraction represents the product of the original fractions. In this comprehensive guide, we will delve into the intricacies of fraction multiplication, exploring various examples and techniques for expressing results in their lowest terms. Mastering fraction multiplication is crucial for building a strong foundation in mathematics and tackling more complex arithmetic problems.

Multiplying fractions involves a straightforward process. You begin by multiplying the numerators, which are the numbers above the fraction bar. Subsequently, you multiply the denominators, which are the numbers below the fraction bar. The resulting product of the numerators becomes the new numerator, while the product of the denominators becomes the new denominator. This process yields the product fraction, which represents the result of the multiplication.

To illustrate this process, let's consider the example of multiplying ${\frac{2}{3}}$ and ${\frac{3}{4}}$. First, multiply the numerators: 2 multiplied by 3 equals 6. Then, multiply the denominators: 3 multiplied by 4 equals 12. Consequently, the product fraction is ${\frac{6}{12}}$. This fraction represents the result of multiplying ${\frac{2}{3}}$ and ${\frac{3}{4}}$. However, it is often necessary to express fractions in their simplest form, which leads us to the next crucial step: simplifying fractions.

Simplifying fractions is an essential step in fraction multiplication, allowing us to express the result in its most concise form. A fraction is considered simplified when the numerator and denominator have no common factors other than 1. To simplify a fraction, we identify the greatest common factor (GCF) of the numerator and denominator and divide both by it. This process reduces the fraction to its lowest terms, making it easier to understand and work with. In the following sections, we will explore the simplification process in detail, providing step-by-step guidance and illustrative examples.

Example 1: ${\frac{4}{5} \times \frac{5}{7}}$

In this first example, we will multiply the fractions ${\frac{4}{5}}$ and ${\frac{5}{7}}$. To do this, we first multiply the numerators: 4 multiplied by 5 equals 20. Then, we multiply the denominators: 5 multiplied by 7 equals 35. This gives us the fraction ${\frac{20}{35}}$. Now, we need to simplify this fraction to its lowest terms.

To simplify ${\frac{20}{35}}$, we need to find the greatest common factor (GCF) of 20 and 35. The factors of 20 are 1, 2, 4, 5, 10, and 20. The factors of 35 are 1, 5, 7, and 35. The GCF of 20 and 35 is 5. We divide both the numerator and the denominator by 5: 20 divided by 5 equals 4, and 35 divided by 5 equals 7. Therefore, the simplified fraction is ${\frac{4}{7}}$.

Thus, ${\frac{4}{5} \times \frac{5}{7} = \frac{20}{35} = \frac{4}{7}}$. This demonstrates the process of multiplying two fractions and simplifying the result to its lowest terms. The simplified fraction ${\frac{4}{7}}$ is easier to understand and work with than the original fraction ${\frac{20}{35}}$. Simplifying fractions is a crucial step in fraction multiplication, as it ensures that the result is expressed in its most concise and understandable form. In the following examples, we will continue to practice this process, solidifying your understanding of fraction multiplication and simplification.

Example 2: ${\frac{1}{6} \times \frac{2}{3}}$

In this example, let's multiply the fractions ${\frac{1}{6}}$ and ${\frac{2}{3}}$. As before, we start by multiplying the numerators: 1 multiplied by 2 equals 2. Then, we multiply the denominators: 6 multiplied by 3 equals 18. This gives us the fraction ${\frac{2}{18}}$. Our next step is to simplify this fraction to its lowest terms.

To simplify ${\frac{2}{18}}$, we identify the greatest common factor (GCF) of 2 and 18. The factors of 2 are 1 and 2. The factors of 18 are 1, 2, 3, 6, 9, and 18. The GCF of 2 and 18 is 2. We divide both the numerator and the denominator by 2: 2 divided by 2 equals 1, and 18 divided by 2 equals 9. Therefore, the simplified fraction is ${\frac{1}{9}}$.

Thus, ${\frac{1}{6} \times \frac{2}{3} = \frac{2}{18} = \frac{1}{9}}$. This example further illustrates the importance of simplifying fractions after multiplication. The simplified fraction ${\frac{1}{9}}$ is much simpler and easier to understand than the original fraction ${\frac{2}{18}}$. Simplifying fractions not only makes the result more concise but also makes it easier to compare and work with in further calculations. By consistently simplifying fractions, we ensure that our results are expressed in their most manageable form.

Example 3: ${\frac{5}{9} \times \frac{3}{10}}$

Now, let's tackle the multiplication of ${\frac{5}{9}}$ and ${\frac{3}{10}}$. We begin by multiplying the numerators: 5 multiplied by 3 equals 15. Next, we multiply the denominators: 9 multiplied by 10 equals 90. This results in the fraction ${\frac{15}{90}}$. We then proceed to simplify this fraction to its lowest terms.

To simplify ${\frac{15}{90}}$, we need to find the greatest common factor (GCF) of 15 and 90. The factors of 15 are 1, 3, 5, and 15. The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. The GCF of 15 and 90 is 15. We divide both the numerator and the denominator by 15: 15 divided by 15 equals 1, and 90 divided by 15 equals 6. Therefore, the simplified fraction is ${\frac{1}{6}}$.

Thus, ${\frac{5}{9} \times \frac{3}{10} = \frac{15}{90} = \frac{1}{6}}$. This example reinforces the method of multiplying fractions and simplifying the outcome. The simplified fraction ${\frac{1}{6}}$ is considerably simpler than the initial fraction ${\frac{15}{90}}$. Simplifying fractions not only provides a more concise answer but also aids in understanding the proportional relationship between the numerator and denominator. This practice enhances our ability to interpret and apply fractions effectively in various mathematical contexts.

Example 4: ${\frac{2}{21} \times \frac{7}{10}}$

In this fourth example, we will multiply the fractions ${\frac{2}{21}}$ and ${\frac{7}{10}}$. As before, we start by multiplying the numerators: 2 multiplied by 7 equals 14. Then, we multiply the denominators: 21 multiplied by 10 equals 210. This results in the fraction ${\frac{14}{210}}$. The next step is to simplify this fraction to its lowest terms.

To simplify ${\frac{14}{210}}$, we need to find the greatest common factor (GCF) of 14 and 210. The factors of 14 are 1, 2, 7, and 14. The factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, and 210. The GCF of 14 and 210 is 14. We divide both the numerator and the denominator by 14: 14 divided by 14 equals 1, and 210 divided by 14 equals 15. Therefore, the simplified fraction is ${\frac{1}{15}}$.

Thus, ${\frac{2}{21} \times \frac{7}{10} = \frac{14}{210} = \frac{1}{15}}$. This example further emphasizes the importance of simplification in fraction multiplication. The simplified fraction ${\frac{1}{15}}$ is much easier to work with than the original fraction ${\frac{14}{210}}$. Simplifying fractions not only makes the result more manageable but also aids in visualizing the fractional quantity. By expressing fractions in their simplest form, we gain a clearer understanding of their value and can use them more effectively in problem-solving.

Example 5: ${\frac{4}{15} \times \frac{3}{22}}$

In our final example, we will multiply the fractions ${\frac{4}{15}}$ and ${\frac{3}{22}}$. We begin by multiplying the numerators: 4 multiplied by 3 equals 12. Then, we multiply the denominators: 15 multiplied by 22 equals 330. This gives us the fraction ${\frac{12}{330}}$. Our next task is to simplify this fraction to its lowest terms.

To simplify ${\frac{12}{330}}$, we need to find the greatest common factor (GCF) of 12 and 330. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 330 are 1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165, and 330. The GCF of 12 and 330 is 6. We divide both the numerator and the denominator by 6: 12 divided by 6 equals 2, and 330 divided by 6 equals 55. Therefore, the simplified fraction is ${\frac{2}{55}}$.

Thus, ${\frac{4}{15} \times \frac{3}{22} = \frac{12}{330} = \frac{2}{55}}$. This final example reinforces the importance of simplifying fractions to their lowest terms after multiplication. The simplified fraction ${\frac{2}{55}}$ is much more concise and easier to comprehend than the original fraction ${\frac{12}{330}}$. Simplifying fractions allows us to express the result in its most manageable form, making it easier to use in subsequent calculations and comparisons. This skill is essential for mastering fraction arithmetic and applying it effectively in various mathematical contexts.

Conclusion

In conclusion, mastering fraction multiplication involves a two-step process: multiplying the numerators and denominators, and then simplifying the resulting fraction to its lowest terms. Through the examples provided, we have demonstrated how to effectively multiply fractions and reduce them to their simplest form by identifying and dividing by the greatest common factor. This process is crucial for ensuring that results are expressed clearly and concisely. By consistently applying these techniques, you can confidently tackle fraction multiplication problems and build a strong foundation in arithmetic. Remember, simplifying fractions not only provides a more manageable answer but also enhances your understanding of fractional quantities and their relationships. With practice and a solid grasp of these methods, you will be well-equipped to handle more complex mathematical operations involving fractions.