Dividing The Sum Of 2/3 And 7/9 By Their Product A Step-by-Step Guide

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In the realm of mathematics, fractions often present a unique challenge, requiring a methodical approach to ensure accurate calculations. This comprehensive guide will walk you through the process of dividing the sum of two fractions, 2/3 and 7/9, by their product. We will break down each step, providing clear explanations and examples to solidify your understanding. Whether you're a student grappling with fractions or simply seeking to refresh your math skills, this guide will equip you with the knowledge and confidence to tackle similar problems.

Understanding Fractions: The Building Blocks of Our Calculation

Before we dive into the calculation itself, let's establish a firm understanding of fractions. A fraction represents a part of a whole, expressed as a ratio between two numbers: the numerator (the top number) and the denominator (the bottom number). The numerator indicates how many parts we have, while the denominator indicates the total number of parts that make up the whole. For example, in the fraction 2/3, the numerator (2) signifies that we have two parts, and the denominator (3) signifies that the whole is divided into three equal parts.

To effectively work with fractions, it's crucial to grasp the concepts of equivalent fractions and common denominators. Equivalent fractions represent the same value, even though they have different numerators and denominators. For instance, 2/3 and 4/6 are equivalent fractions because they both represent the same proportion. Finding a common denominator, a shared multiple of the denominators of two or more fractions, is essential when adding or subtracting fractions. This allows us to combine the fractions accurately.

In our problem, we're dealing with the fractions 2/3 and 7/9. To add these fractions, we'll need to find a common denominator. The least common multiple of 3 and 9 is 9, so we'll convert 2/3 into an equivalent fraction with a denominator of 9. This foundational knowledge will serve as the bedrock for our subsequent calculations.

Step 1: Finding the Sum of 2/3 and 7/9

The first step in solving our problem is to calculate the sum of the two fractions, 2/3 and 7/9. To add fractions, they must have a common denominator, meaning the bottom numbers (denominators) must be the same. As mentioned earlier, the least common multiple (LCM) of 3 and 9 is 9. This means we need to convert 2/3 into an equivalent fraction with a denominator of 9.

To do this, we multiply both the numerator and the denominator of 2/3 by 3. This gives us (2 * 3) / (3 * 3) = 6/9. Now we have two fractions with a common denominator: 6/9 and 7/9.

Adding fractions with a common denominator is straightforward. We simply add the numerators and keep the denominator the same. So, 6/9 + 7/9 = (6 + 7) / 9 = 13/9. Therefore, the sum of 2/3 and 7/9 is 13/9. This fraction is an improper fraction, meaning the numerator is larger than the denominator. We can leave it as an improper fraction for now or convert it to a mixed number (1 4/9), but for the sake of clarity in the following steps, we'll keep it as 13/9.

This step is crucial because it lays the groundwork for the rest of the problem. By finding the sum, we've completed the first part of the calculation and are ready to move on to finding the product.

Step 2: Calculating the Product of 2/3 and 7/9

Now that we've determined the sum of 2/3 and 7/9, the next step is to calculate their product. Multiplying fractions is a simpler process than adding or subtracting them, as we don't need to find a common denominator. To multiply fractions, we simply multiply the numerators together and the denominators together.

In this case, we need to multiply 2/3 by 7/9. Multiplying the numerators, we get 2 * 7 = 14. Multiplying the denominators, we get 3 * 9 = 27. Therefore, the product of 2/3 and 7/9 is 14/27.

Unlike addition and subtraction, multiplication doesn't require a common denominator. We directly multiply the numerators to get the new numerator and the denominators to get the new denominator. This makes multiplication a relatively straightforward operation with fractions.

It's worth noting that before multiplying, we can sometimes simplify the fractions by canceling out common factors between the numerators and denominators. However, in this case, there are no common factors between 2/3 and 7/9, so we proceed directly with the multiplication.

With the product calculated, we're now equipped to perform the final step: dividing the sum by the product.

Step 3: Dividing the Sum (13/9) by the Product (14/27)

The final step in our journey is to divide the sum we calculated in step 1 (13/9) by the product we calculated in step 2 (14/27). Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.

So, to divide 13/9 by 14/27, we multiply 13/9 by the reciprocal of 14/27, which is 27/14. This gives us (13/9) * (27/14).

Before multiplying, we can simplify the fractions by canceling out common factors. We notice that 9 and 27 have a common factor of 9. We can divide both 9 and 27 by 9, which simplifies the expression to (13/1) * (3/14).

Now, we multiply the numerators and the denominators: 13 * 3 = 39 and 1 * 14 = 14. Therefore, (13/1) * (3/14) = 39/14.

The result, 39/14, is an improper fraction. We can leave it in this form, or we can convert it to a mixed number. To convert it to a mixed number, we divide 39 by 14. 39 divided by 14 is 2 with a remainder of 11. So, 39/14 is equal to the mixed number 2 11/14.

Therefore, when we divide the sum of 2/3 and 7/9 by their product, the result is 39/14 or 2 11/14. This completes our step-by-step calculation.

Putting it All Together: A Recap of the Steps

Let's recap the steps we took to solve the problem of dividing the sum of 2/3 and 7/9 by their product:

  1. Find the Sum: We added 2/3 and 7/9. To do this, we found a common denominator (9), converted 2/3 to 6/9, and then added the numerators: 6/9 + 7/9 = 13/9.
  2. Calculate the Product: We multiplied 2/3 and 7/9 by multiplying the numerators and the denominators: (2 * 7) / (3 * 9) = 14/27.
  3. Divide the Sum by the Product: We divided 13/9 by 14/27. Dividing by a fraction is the same as multiplying by its reciprocal. So, we multiplied 13/9 by 27/14. We simplified the fractions by canceling out common factors, resulting in 39/14, which can also be expressed as the mixed number 2 11/14.

By following these steps, we successfully solved the problem. This methodical approach can be applied to similar problems involving fractions.

Practice Makes Perfect: Further Exercises

To solidify your understanding of dividing the sum of fractions by their product, practice is essential. Here are a few additional exercises you can try:

  1. Divide the sum of 1/2 and 3/4 by their product.
  2. Divide the sum of 5/6 and 2/3 by their product.
  3. Divide the sum of 1/4 and 5/8 by their product.
  4. Divide the sum of 3/5 and 1/2 by their product.
  5. Divide the sum of 4/7 and 2/5 by their product.

Working through these exercises will help you internalize the steps involved and develop your confidence in working with fractions. Remember to break down each problem into the same steps we used in this guide: finding the sum, calculating the product, and then dividing the sum by the product.

By mastering these fundamental fraction operations, you'll build a strong foundation for more advanced mathematical concepts. Keep practicing, and you'll become proficient in navigating the world of fractions!