Fraction Multiplication Step By Step Solutions And Guide

When tackling mixed fraction multiplication, like in our first problem, 2 3/9 x 6/7, the initial step involves transforming the mixed fraction into an improper fraction. This transformation is crucial because it simplifies the multiplication process significantly. To convert a mixed fraction to an improper fraction, you multiply the whole number by the denominator and then add the numerator. This result becomes the new numerator, while the denominator remains the same. In our case, 2 3/9 becomes (2 * 9 + 3) / 9, which equals 21/9. This conversion sets the stage for straightforward fraction multiplication.

Now that we've converted the mixed fraction into an improper fraction, the problem transforms into a simpler multiplication of two fractions: 21/9 x 6/7. To multiply fractions, you simply multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator. Thus, (21/9) * (6/7) equals (21 * 6) / (9 * 7). This step is a fundamental aspect of fraction multiplication, allowing us to combine the two fractions into one. The result of this multiplication is 126/63, which is a large fraction that can be simplified.

The final step in solving this problem is to simplify the resulting fraction. Simplifying fractions involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by this GCD. In our case, the GCD of 126 and 63 is 63. Dividing both the numerator and the denominator by 63 gives us 126/63 = 2/1, which simplifies to 2. This simplification not only makes the fraction easier to understand but also provides the answer in its most concise form. Therefore, 2 3/9 x 6/7 equals 2, demonstrating the complete process of multiplying a mixed fraction by a proper fraction and simplifying the result.

When we delve into multiplying proper fractions, like the problem 13/15 x 8/9, the fundamental principle remains consistent: multiply the numerators and the denominators separately. This straightforward approach is a cornerstone of fraction multiplication. In this scenario, we multiply 13 by 8 to obtain the new numerator, and we multiply 15 by 9 to get the new denominator. This process transforms the problem into a single fraction, setting the stage for simplification if necessary. The beauty of multiplying proper fractions lies in its directness, allowing for a clear path to the solution.

Applying the multiplication rule, we find that 13 multiplied by 8 equals 104, and 15 multiplied by 9 equals 135. Therefore, the result of the multiplication is 104/135. This fraction represents the product of the two original fractions. However, it's crucial to examine whether this resulting fraction can be simplified further. Simplification involves identifying common factors between the numerator and the denominator and reducing the fraction to its simplest form. This step ensures that the answer is presented in the most understandable and concise manner.

To determine if 104/135 can be simplified, we look for common factors between 104 and 135. The prime factors of 104 are 2 x 2 x 2 x 13, and the prime factors of 135 are 3 x 3 x 3 x 5. Upon examination, we find that there are no common factors between 104 and 135. This means that the fraction 104/135 is already in its simplest form. Therefore, the final answer to the problem 13/15 x 8/9 is 104/135. This result highlights the importance of checking for simplification after multiplication to ensure the answer is in its most reduced form.

The core concept of fraction multiplication, as exemplified in the problem 4/7 x 2/3, revolves around multiplying the numerators together and the denominators together. This is the fundamental rule that governs how fractions are multiplied, and it's essential to grasp this principle to solve any fraction multiplication problem. In this specific case, we multiply 4 by 2 to find the new numerator, and we multiply 7 by 3 to find the new denominator. This process effectively combines the two fractions into a single fraction, which can then be simplified if necessary. Understanding this basic step is crucial for building confidence in handling more complex fraction multiplication scenarios.

Following the rule of multiplying numerators and denominators, we calculate 4 multiplied by 2 to be 8, and 7 multiplied by 3 to be 21. This results in the fraction 8/21. This fraction represents the direct product of the two original fractions. The next critical step is to assess whether this resulting fraction can be simplified. Simplifying a fraction means reducing it to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). This ensures that the fraction is expressed in its most concise and easily understandable form.

To determine if 8/21 can be simplified, we need to identify the factors of both 8 and 21. The factors of 8 are 1, 2, 4, and 8, while the factors of 21 are 1, 3, 7, and 21. The only common factor between 8 and 21 is 1, which means they do not share any other common divisors. Therefore, the fraction 8/21 is already in its simplest form. As a result, the solution to the problem 4/7 x 2/3 is 8/21. This demonstrates that not all fraction multiplications will result in fractions that can be simplified, and it's important to recognize when a fraction is already in its simplest form.

When we approach multiplying a whole number by a fraction, such as in the problem 3 x 12/20, the first crucial step is to express the whole number as a fraction. This is achieved by placing the whole number over 1. In our case, 3 becomes 3/1. This transformation is essential because it allows us to apply the standard rule of fraction multiplication, which involves multiplying numerators and denominators. By converting the whole number into a fraction, we create a uniform format that makes the multiplication process straightforward and consistent with fraction multiplication principles.

With the whole number now expressed as a fraction, the problem transforms into a standard fraction multiplication: 3/1 x 12/20. To solve this, we multiply the numerators (3 and 12) together and the denominators (1 and 20) together. This process gives us a new fraction that represents the product of the original whole number and fraction. Multiplying numerators and denominators is a fundamental step in fraction multiplication, and it's applied here to combine the two fractional components into a single fraction that represents their product.

Performing the multiplication, we get (3 * 12) / (1 * 20), which equals 36/20. The next critical step is to simplify this resulting fraction. Simplification involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this GCD. This process reduces the fraction to its simplest form, making it easier to understand and work with. In the case of 36/20, simplification is necessary to express the answer in its most concise form.

To simplify 36/20, we need to find the GCD of 36 and 20. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36, while the factors of 20 are 1, 2, 4, 5, 10, and 20. The greatest common divisor is 4. Dividing both the numerator and the denominator by 4, we get 36/4 = 9 and 20/4 = 5. Therefore, the simplified fraction is 9/5. This can also be expressed as a mixed number, which is 1 4/5. Thus, the final answer to the problem 3 x 12/20 is 9/5 or 1 4/5, demonstrating the importance of simplification in fraction multiplication.

In the scenario of multiplying a fraction by a whole number, as seen in the problem 1/4 x 40, the initial step mirrors the previous example: we must express the whole number as a fraction. This is accomplished by placing the whole number over 1, transforming 40 into 40/1. This conversion is crucial because it aligns the whole number with the fractional format, allowing us to apply the standard rules of fraction multiplication. By representing the whole number as a fraction, we set the stage for a straightforward multiplication process that is consistent with the principles of fraction arithmetic.

Now that the whole number is expressed as a fraction, the problem becomes a standard fraction multiplication: 1/4 x 40/1. To solve this, we multiply the numerators (1 and 40) together and the denominators (4 and 1) together. This process yields a new fraction that represents the product of the original fraction and the whole number. Multiplying numerators and denominators is a fundamental operation in fraction multiplication, and it allows us to combine the two numbers into a single fraction that represents their combined value.

Performing the multiplication, we find that (1 * 40) / (4 * 1) equals 40/4. This resulting fraction is a direct product of the original numbers, but it's essential to simplify it to its lowest terms. Simplifying a fraction involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this GCD. This process ensures that the fraction is expressed in its most concise and easily understandable form. Simplification is a critical step in fraction multiplication, as it presents the answer in its most reduced form.

To simplify 40/4, we identify the GCD of 40 and 4. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40, while the factors of 4 are 1, 2, and 4. The greatest common divisor is 4. Dividing both the numerator and the denominator by 4, we get 40/4 = 10 and 4/4 = 1. Therefore, the simplified fraction is 10/1, which is equal to 10. Thus, the final answer to the problem 1/4 x 40 is 10, demonstrating the importance of converting whole numbers to fractions and simplifying the result in fraction multiplication.

In conclusion, mastering fraction multiplication involves understanding the basic principles of multiplying numerators and denominators, converting mixed numbers and whole numbers into fractions, and simplifying the resulting fractions. These steps are crucial for solving a wide range of fraction multiplication problems accurately and efficiently.