\[$\frac{8}{6} + \left(-\frac{9}{5}\right) = \square\$\]Add The Fractions.

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Understanding the Problem

When adding fractions with different denominators, it can be challenging to determine the correct sum. In this article, we will explore the process of adding fractions with different denominators, using the given problem as an example: 86+(−95)=□\frac{8}{6} + \left(-\frac{9}{5}\right) = \square. We will break down the solution into manageable steps, making it easier to understand and apply the concept.

The Importance of Finding a Common Denominator

To add fractions with different denominators, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators. In this case, the denominators are 6 and 5. To find the LCM, we can list the multiples of each denominator:

  • Multiples of 6: 6, 12, 18, 24, 30, ...
  • Multiples of 5: 5, 10, 15, 20, 25, 30, ...

As we can see, the first common multiple of 6 and 5 is 30. Therefore, the common denominator is 30.

Converting the Fractions to Have the Same Denominator

Now that we have found the common denominator, we can convert each fraction to have the same denominator. To do this, we multiply the numerator and denominator of each fraction by the necessary factor to obtain the common denominator.

For the first fraction, 86\frac{8}{6}, we need to multiply the numerator and denominator by 5 to obtain a denominator of 30:

86⋅55=4030\frac{8}{6} \cdot \frac{5}{5} = \frac{40}{30}

For the second fraction, (−95)\left(-\frac{9}{5}\right), we need to multiply the numerator and denominator by 6 to obtain a denominator of 30:

(−95)⋅66=(−5430)\left(-\frac{9}{5}\right) \cdot \frac{6}{6} = \left(-\frac{54}{30}\right)

Adding the Fractions

Now that both fractions have the same denominator, we can add them together:

4030+(−5430)=40−5430=−1430\frac{40}{30} + \left(-\frac{54}{30}\right) = \frac{40 - 54}{30} = \frac{-14}{30}

Simplifying the Result

The result, −1430\frac{-14}{30}, can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD). The GCD of 14 and 30 is 2. Therefore, we can simplify the result as follows:

−1430=−14÷230÷2=−715\frac{-14}{30} = \frac{-14 \div 2}{30 \div 2} = \frac{-7}{15}

Conclusion

In this article, we have demonstrated the process of adding fractions with different denominators. We found the common denominator, converted each fraction to have the same denominator, added the fractions, and simplified the result. By following these steps, you can add fractions with different denominators and obtain the correct sum.

Common Mistakes to Avoid

When adding fractions with different denominators, it's essential to avoid common mistakes. Here are a few to watch out for:

  • Not finding the common denominator: Failing to find the common denominator can lead to incorrect results.
  • Not converting the fractions to have the same denominator: Failing to convert the fractions to have the same denominator can also lead to incorrect results.
  • Not simplifying the result: Failing to simplify the result can make it more difficult to work with.

Real-World Applications

Adding fractions with different denominators has numerous real-world applications. Here are a few examples:

  • Cooking: When measuring ingredients, you may need to add fractions with different denominators. For example, a recipe may call for 1/4 cup of sugar and 3/5 cup of flour.
  • Building: When building a structure, you may need to add fractions with different denominators to determine the total amount of materials needed.
  • Science: When conducting experiments, you may need to add fractions with different denominators to determine the total amount of a substance needed.

Conclusion

In conclusion, adding fractions with different denominators requires finding a common denominator, converting each fraction to have the same denominator, adding the fractions, and simplifying the result. By following these steps and avoiding common mistakes, you can add fractions with different denominators and obtain the correct sum. The real-world applications of adding fractions with different denominators are numerous, and it's essential to understand this concept to succeed in various fields.

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Q: What is the first step in adding fractions with different denominators?

A: The first step in adding fractions with different denominators is to find the common denominator. The common denominator is the least common multiple (LCM) of the two denominators.

Q: How do I find the common denominator?

A: To find the common denominator, you can list the multiples of each denominator and find the first common multiple. For example, if the denominators are 6 and 5, you can list the multiples of each denominator and find the first common multiple, which is 30.

Q: What if the denominators are not multiples of each other?

A: If the denominators are not multiples of each other, you can find the least common multiple (LCM) by listing the multiples of each denominator and finding the smallest number that appears in both lists. For example, if the denominators are 8 and 12, you can list the multiples of each denominator and find the LCM, which is 24.

Q: How do I convert the fractions to have the same denominator?

A: To convert the fractions to have the same denominator, you need to multiply the numerator and denominator of each fraction by the necessary factor to obtain the common denominator. For example, if the common denominator is 30, you can multiply the numerator and denominator of each fraction by the necessary factor to obtain a denominator of 30.

Q: What if I have a negative fraction?

A: If you have a negative fraction, you can convert it to a positive fraction by changing the sign of the numerator. For example, if you have the fraction -3/4, you can convert it to a positive fraction by changing the sign of the numerator to get 3/4.

Q: Can I add fractions with different denominators using a calculator?

A: Yes, you can add fractions with different denominators using a calculator. Most calculators have a fraction function that allows you to enter fractions and perform operations on them. However, it's still important to understand the concept of adding fractions with different denominators to ensure that you're using the calculator correctly.

Q: What are some common mistakes to avoid when adding fractions with different denominators?

A: Some common mistakes to avoid when adding fractions with different denominators include:

  • Not finding the common denominator
  • Not converting the fractions to have the same denominator
  • Not simplifying the result
  • Adding the numerators without finding the common denominator

Q: How do I simplify the result of adding fractions with different denominators?

A: To simplify the result of adding fractions with different denominators, you need to divide both the numerator and denominator by their greatest common divisor (GCD). For example, if the result is 12/18, you can simplify it by dividing both the numerator and denominator by their GCD, which is 6, to get 2/3.

Q: What are some real-world applications of adding fractions with different denominators?

A: Some real-world applications of adding fractions with different denominators include:

  • Cooking: When measuring ingredients, you may need to add fractions with different denominators.
  • Building: When building a structure, you may need to add fractions with different denominators to determine the total amount of materials needed.
  • Science: When conducting experiments, you may need to add fractions with different denominators to determine the total amount of a substance needed.

Q: Can I use a formula to add fractions with different denominators?

A: Yes, you can use a formula to add fractions with different denominators. The formula is:

ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

However, it's still important to understand the concept of adding fractions with different denominators to ensure that you're using the formula correctly.