4. $\frac{6}{9}+\frac{1}{2}=$

$$

Introduction to Fractions

Fractions are a fundamental concept in mathematics, representing a part of a whole. They consist of a numerator and a denominator, where the numerator is the number of equal parts and the denominator is the total number of parts. In this problem, we are given two fractions: $\frac{6}{9}$ and $\frac{1}{2}$. Our goal is to find the sum of these two fractions.

Understanding the Problem

To add fractions, we need to have a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the fractions. In this case, the denominators are 9 and 2. The LCM of 9 and 2 is 18.

Finding the Common Denominator

To find the common denominator, we need to list the multiples of each denominator.

• Multiples of 9: 9, 18, 27, 36, ...
• Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, ...

As we can see, the first common multiple of 9 and 2 is 18. Therefore, the common denominator is 18.

Converting Fractions to Have a Common Denominator

Now that we have the common denominator, we need to convert each fraction to have a denominator of 18.

• $\frac{6}{9}$ can be converted to $\frac{6 \times 2}{9 \times 2} = \frac{12}{18}$.
• $\frac{1}{2}$ can be converted to $\frac{1 \times 9}{2 \times 9} = \frac{9}{18}$.

Adding the Fractions

Now that we have both fractions with a common denominator, we can add them.

$\frac{12}{18} + \frac{9}{18} = \frac{12 + 9}{18} = \frac{21}{18}$

Simplifying the Fraction

The fraction $\frac{21}{18}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 21 and 18 is 3.

$\frac{21}{18} = \frac{21 \div 3}{18 \div 3} = \frac{7}{6}$

Conclusion

In this problem, we added two fractions with different denominators. We found the common denominator, converted each fraction to have the common denominator, added the fractions, and simplified the result. The final answer is $\frac{7}{6}$.

Real-World Applications

Fractions are used in many real-world applications, such as cooking, measuring ingredients, and calculating proportions. For example, if a recipe calls for 1/4 cup of sugar and you want to make half the recipe, you would need to multiply the fraction by 1/2.

Tips and Tricks

• When adding fractions, make sure to have a common denominator.
• Use the least common multiple (LCM) to find the common denominator.
• Convert each fraction to have the common denominator.
• Add the fractions and simplify the result.

Practice Problems

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

Solutions

1. $\frac{3}{4} + \frac{1}{6} = \frac{9}{12} + \frac{2}{12} = \frac{11}{12}$
2. $\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$
3. $\frac{5}{6} + \frac{2}{3} = \frac{10}{12} + \frac{8}{12} = \frac{18}{12} = \frac{3}{2}$
   

$$

Frequently Asked Questions

Q: What is the common denominator for the fractions $\frac{6}{9}$ and $\frac{1}{2}$?

A: The common denominator is the least common multiple (LCM) of the denominators of the fractions. In this case, the denominators are 9 and 2. The LCM of 9 and 2 is 18.

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 the denominator by the same number. For example, to convert $\frac{6}{9}$ to have a denominator of 18, you would multiply the numerator and the denominator by 2, resulting in $\frac{12}{18}$.

Q: What is the greatest common divisor (GCD) of 21 and 18?

A: The GCD of 21 and 18 is 3.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD). For example, to simplify $\frac{21}{18}$, you would divide both the numerator and the denominator by 3, resulting in $\frac{7}{6}$.

Q: What is the final answer to the problem $\frac{6}{9}+\frac{1}{2}=$?

A: The final answer is $\frac{7}{6}$.

Q: What are some real-world applications of fractions?

A: Fractions are used in many real-world applications, such as cooking, measuring ingredients, and calculating proportions. For example, if a recipe calls for 1/4 cup of sugar and you want to make half the recipe, you would need to multiply the fraction by 1/2.

Q: What are some tips and tricks for working with fractions?

A: Some tips and tricks for working with fractions include:

• When adding fractions, make sure to have a common denominator.
• Use the least common multiple (LCM) to find the common denominator.
• Convert each fraction to have the common denominator.
• Add the fractions and simplify the result.

Q: What are some practice problems for working with fractions?

A: Some practice problems for working with fractions include:

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

Q: What are the solutions to the practice problems?

A: The solutions to the practice problems are:

• $\frac{3}{4} + \frac{1}{6} = \frac{11}{12}$
• $\frac{2}{3} + \frac{1}{4} = \frac{11}{12}$
• $\frac{5}{6} + \frac{2}{3} = \frac{3}{2}$

Additional Resources

• For more information on fractions, visit the Khan Academy website.
• For more practice problems, visit the Mathway website.
• For more tips and tricks, visit the Math Open Reference website.

Conclusion

In this article, we have covered the basics of fractions, including finding the common denominator, converting fractions to have a common denominator, adding fractions, and simplifying the result. We have also provided some practice problems and solutions to help you practice working with fractions. We hope this article has been helpful in your understanding of fractions.