What Is The Sum?$\frac{4}{7}+\frac{2}{8}$A. $\frac{23}{28}$ B. $\frac{17}{28}$ C. $\frac{11}{14}$ D. $\frac{3}{28}$

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Understanding the Problem

When dealing with fractions, adding them together can be a bit tricky. However, with the right approach, it can be a straightforward process. In this article, we will explore how to add two fractions, specifically $\frac{4}{7}$ and $\frac{2}{8}$, and find their sum.

The Importance of Finding a Common Denominator

To add two fractions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the two fractions. In this case, the denominators are 7 and 8. To find the LCM, we can list the multiples of each number and find the smallest number that appears in both lists.

Finding the Least Common Multiple (LCM)

To find the LCM of 7 and 8, we can list the multiples of each number:

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84 Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88

As we can see, the smallest number that appears in both lists is 56. Therefore, the LCM of 7 and 8 is 56.

Converting the Fractions to Have a Common Denominator

Now that we have found the LCM, we can convert both fractions to have a denominator of 56. To do this, we need to multiply the numerator and denominator of each fraction by the necessary factor.

For the first fraction, $\frac{4}{7}$, we need to multiply the numerator and denominator by 8:

$\frac{4}{7} \cdot \frac{8}{8} = \frac{32}{56}$

For the second fraction, $\frac{2}{8}$, we need to multiply the numerator and denominator by 7:

$\frac{2}{8} \cdot \frac{7}{7} = \frac{14}{56}$

Adding the Fractions

Now that both fractions have a common denominator, we can add them together:

$\frac{32}{56} + \frac{14}{56} = \frac{46}{56}$

Simplifying the Fraction

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 46 and 56 is 2.

$\frac{46}{56} = \frac{23}{28}$

Conclusion

In conclusion, the sum of $\frac{4}{7}$ and $\frac{2}{8}$ is $\frac{23}{28}$. This is the correct answer, and it can be verified by adding the fractions together and simplifying the result.

Common Mistakes to Avoid

When adding fractions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not finding a common denominator: If the fractions do not have a common denominator, it's impossible to add them together.
• Not converting the fractions to have a common denominator: If the fractions are not converted to have a common denominator, it's impossible to add them together.
• Not simplifying the fraction: If the fraction is not simplified, it may not be in its simplest form.

Real-World Applications

Adding fractions is an important skill that has many real-world applications. Here are a few examples:

• Cooking: When cooking, you may need to add fractions of ingredients together. For example, if a recipe calls for 1/4 cup of sugar and 1/8 cup of sugar, you would need to add these fractions together to get the total amount of sugar needed.
• Building: When building a structure, you may need to add fractions of materials together. For example, if you need to add 1/2 inch of wood and 1/4 inch of wood, you would need to add these fractions together to get the total amount of wood needed.
• Science: When conducting scientific experiments, you may need to add fractions of chemicals together. For example, if you need to add 1/4 cup of acid and 1/8 cup of acid, you would need to add these fractions together to get the total amount of acid needed.

Practice Problems

Here are a few practice problems to help you practice adding fractions:

• $\frac{3}{4} + \frac{1}{8} = ?$
• $\frac{2}{3} + \frac{1}{6} = ?$
• $\frac{5}{6} + \frac{1}{4} = ?$

Conclusion

In conclusion, adding fractions is an important skill that has many real-world applications. By following the steps outlined in this article, you can add fractions together and simplify the result. Remember to find a common denominator, convert the fractions to have a common denominator, and simplify the fraction. With practice, you will become proficient in adding fractions and be able to apply this skill in many different situations.

Q: What is the first step in adding fractions?

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

Q: How do I find the least common multiple (LCM) of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest number that appears in both lists.

Q: What if the fractions do not have a common denominator?

A: If the fractions do not have a common denominator, it is impossible to add them together. You will need to find a common denominator before you can add the fractions.

Q: How do I convert a fraction to have a common denominator?

A: To convert a fraction to have a common denominator, you need to multiply the numerator and denominator of the fraction by the necessary factor.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator of a fraction.

Q: Why is it important to simplify a fraction?

A: Simplifying a fraction is important because it ensures that the fraction is in its simplest form. This makes it easier to work with the fraction and to compare it to other fractions.

Q: Can I add fractions with different signs?

A: Yes, you can add fractions with different signs. When adding fractions with different signs, you need to follow the rules of addition, which state that a + (-b) = a - b.

Q: Can I subtract fractions?

A: Yes, you can subtract fractions. To subtract fractions, you need to follow the same steps as adding fractions, but with a negative sign.

Q: What is the difference between adding and subtracting fractions?

A: The main difference between adding and subtracting fractions is the sign of the second fraction. When adding fractions, the second fraction has a positive sign, while when subtracting fractions, the second fraction has a negative sign.

Q: Can I add fractions with unlike denominators?

A: Yes, you can add fractions with unlike denominators. To add fractions with unlike denominators, you need to find a common denominator and then add the fractions.

Q: Can I add fractions with decimals?

A: Yes, you can add fractions with decimals. To add fractions with decimals, you need to convert the decimals to fractions and then add the fractions.

Q: Can I add fractions with mixed numbers?

A: Yes, you can add fractions with mixed numbers. To add fractions with mixed numbers, you need to convert the mixed numbers to improper fractions and then add the fractions.

Q: What is the final step in adding fractions?

A: The final step in adding fractions is to simplify the result. Simplifying the result ensures that the fraction is in its simplest form.

Q: Why is it important to practice adding fractions?

A: Practicing adding fractions is important because it helps you to develop your skills and to become proficient in adding fractions. With practice, you will become more confident and accurate in adding fractions.

Q: Can I use a calculator to add fractions?

A: Yes, you can use a calculator to add fractions. However, it is still important to understand the steps involved in adding fractions, as this will help you to check your work and to understand the concept of adding fractions.

Q: Can I add fractions with variables?

A: Yes, you can add fractions with variables. To add fractions with variables, you need to follow the same steps as adding fractions with constants, but with variables.

Q: Can I add fractions with exponents?

A: Yes, you can add fractions with exponents. To add fractions with exponents, you need to follow the same steps as adding fractions with constants, but with exponents.

Q: Can I add fractions with radicals?

A: Yes, you can add fractions with radicals. To add fractions with radicals, you need to follow the same steps as adding fractions with constants, but with radicals.

Q: Can I add fractions with complex numbers?

A: Yes, you can add fractions with complex numbers. To add fractions with complex numbers, you need to follow the same steps as adding fractions with constants, but with complex numbers.

Q: Can I add fractions with matrices?

A: Yes, you can add fractions with matrices. To add fractions with matrices, you need to follow the same steps as adding fractions with constants, but with matrices.

Q: Can I add fractions with vectors?

A: Yes, you can add fractions with vectors. To add fractions with vectors, you need to follow the same steps as adding fractions with constants, but with vectors.

Q: Can I add fractions with tensors?

A: Yes, you can add fractions with tensors. To add fractions with tensors, you need to follow the same steps as adding fractions with constants, but with tensors.

Q: Can I add fractions with other mathematical objects?

A: Yes, you can add fractions with other mathematical objects, such as polynomials, rational functions, and trigonometric functions. To add fractions with other mathematical objects, you need to follow the same steps as adding fractions with constants, but with the specific mathematical object.