Multiplying Fractions And Whole Numbers A Step By Step Guide

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Multiplying fractions with whole numbers might seem daunting at first, but with a clear understanding of the underlying principles, it becomes a straightforward process. In this comprehensive guide, we will delve into the step-by-step process of multiplying a whole number by a fraction, simplifying the resulting fraction, and expressing the answer as a mixed number. We will use the example of multiplying 12 by 3/5 to illustrate each step in detail, ensuring you grasp the concept thoroughly.

Understanding the Basics

Before we dive into the multiplication process, let's refresh our understanding of fractions and whole numbers. A fraction represents a part of a whole, consisting of two main components: the numerator (the top number) and the denominator (the bottom number). The numerator indicates the number of parts we have, while the denominator indicates the total number of parts the whole is divided into. For example, in the fraction 3/5, 3 is the numerator, and 5 is the denominator, indicating that we have 3 parts out of a total of 5 parts.

A whole number, on the other hand, is a non-negative integer without any fractional or decimal parts. Examples of whole numbers include 0, 1, 2, 3, and so on. To multiply a whole number by a fraction, we need to understand how to represent a whole number as a fraction.

Step 1: Represent the Whole Number as a Fraction

The first step in multiplying a whole number by a fraction is to represent the whole number as a fraction. This is done by simply writing the whole number as the numerator and placing 1 as the denominator. In our example, we have the whole number 12. To represent it as a fraction, we write it as 12/1. This representation doesn't change the value of the number; 12/1 is equivalent to 12 because any number divided by 1 is itself.

This step is crucial because it allows us to perform the multiplication operation between two fractions. Now that we have both numbers represented as fractions (12/1 and 3/5), we can proceed to the next step.

Step 2: Multiply the Numerators and Denominators

Now that we have expressed both the whole number and the fraction as fractions, we can proceed with the multiplication. To multiply two fractions, we simply multiply the numerators together and the denominators together. In our case, we have 12/1 multiplied by 3/5.

To multiply the numerators, we multiply 12 by 3, which gives us 36. This becomes the numerator of our new fraction. Next, we multiply the denominators, which are 1 and 5. Multiplying 1 by 5 gives us 5. This becomes the denominator of our new fraction. Therefore, the result of multiplying 12/1 by 3/5 is 36/5.

So, the multiplication gives us the fraction 36/5. This fraction represents the product of the whole number 12 and the fraction 3/5. However, this fraction is an improper fraction, meaning the numerator is greater than the denominator. To simplify our answer and make it easier to understand, we need to convert it to a mixed number.

Step 3: Simplify the Fraction (If Possible)

Before converting the improper fraction to a mixed number, it's essential to check if the 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 factor (GCF). The GCF is the largest number that divides both the numerator and the denominator without leaving a remainder.

In our case, the fraction is 36/5. To determine if it can be simplified, we need to find the GCF of 36 and 5. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The factors of 5 are 1 and 5. The only common factor between 36 and 5 is 1. This means that the fraction 36/5 is already in its simplest form because the numerator and denominator have no common factors other than 1.

Since the fraction cannot be simplified further, we proceed to the next step, which is converting it to a mixed number.

Step 4: Express the Answer as a Mixed Number

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. A mixed number, on the other hand, is a combination of a whole number and a proper fraction (where the numerator is less than the denominator). To convert an improper fraction to a mixed number, we perform division.

In our case, we have the improper fraction 36/5. To convert it to a mixed number, we divide the numerator (36) by the denominator (5). When we divide 36 by 5, we get a quotient of 7 and a remainder of 1. The quotient becomes the whole number part of the mixed number, and the remainder becomes the numerator of the fractional part. The denominator remains the same.

So, 36 divided by 5 is 7 with a remainder of 1. This means that 36/5 can be expressed as the mixed number 7 1/5. The whole number part is 7, and the fractional part is 1/5. Therefore, the simplified answer expressed as a mixed number is 7 1/5.

Summary and Conclusion

In this guide, we have walked through the process of multiplying a whole number by a fraction, simplifying the resulting fraction, and expressing the answer as a mixed number. We used the example of multiplying 12 by 3/5 to illustrate each step in detail.

Here’s a quick recap of the steps:

  1. Represent the whole number as a fraction by placing it over a denominator of 1.
  2. Multiply the numerators together and the denominators together.
  3. Simplify the resulting fraction if possible by dividing both the numerator and the denominator by their greatest common factor.
  4. Express the improper fraction as a mixed number by dividing the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the numerator of the fractional part.

By following these steps, you can confidently multiply whole numbers by fractions and express the answers in their simplest form. Remember, practice makes perfect, so try solving various examples to reinforce your understanding. Multiplying fractions and whole numbers is a fundamental skill in mathematics, and mastering it will be beneficial in various mathematical contexts.

In conclusion, multiplying 12 by 3/5 gives us 36/5, which simplifies to the mixed number 7 1/5. This detailed explanation should help you grasp the process and apply it to similar problems. Keep practicing, and you'll become proficient in no time!