Add And Simplify:26. \[$\begin{array}{r}\frac{1}{8} \\ \frac{3}{8} \\ +\quad \frac{5}{8} \\ \hline\end{array}\$\]27. \[$\frac{1}{2} + \frac{1}{4}\$\]28. \[$\frac{2}{3} + \frac{1}{4}\$\]29. \[$15 \frac{3}{4} + 12
Introduction
Fractions are an essential part of mathematics, and adding and simplifying them is a crucial skill to master. In this article, we will explore the process of adding and simplifying fractions, including examples and step-by-step solutions. We will also discuss the importance of finding common denominators and simplifying fractions to their lowest terms.
Adding Fractions with the Same Denominator
When adding fractions with the same denominator, we can simply add the numerators and keep the denominator the same. Let's consider the following example:
Example 1: Adding Fractions with the Same Denominator
{\begin{array}{r}\frac{1}{8} \ \frac{3}{8} \ +\quad \frac{5}{8} \ \hline\end{array}$}$
To add these fractions, we can simply add the numerators (1 + 3 + 5) and keep the denominator (8) the same.
{\frac{1+3+5}{8} = \frac{9}{8}$}$
Therefore, the sum of the fractions is {\frac{9}{8}$}$.
Adding Fractions with Different Denominators
When adding fractions with different denominators, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators. Let's consider the following example:
Example 2: Adding Fractions with Different Denominators
{\frac{1}{2} + \frac{1}{4}$}$
To add these fractions, we need to find a common denominator. The LCM of 2 and 4 is 4. Therefore, we can rewrite the fractions with a common denominator of 4.
{\frac{1}{2} = \frac{2}{4}$ and [$\frac{1}{4}$ remains the same]
Now we can add the fractions:
[\frac{2}{4} + \frac{1}{4} = \frac{3}{4}\$}
Therefore, the sum of the fractions is {\frac{3}{4}$}$.
Adding Mixed Numbers
When adding mixed numbers, we need to add the whole numbers and the fractions separately. Let's consider the following example:
Example 3: Adding Mixed Numbers
${15 \frac{3}{4} + 12 \frac{2}{3}\$}
To add these mixed numbers, we need to add the whole numbers and the fractions separately.
${\frac{3}{4} + \frac{2}{3}$}$
To add the fractions, we need to find a common denominator. The LCM of 4 and 3 is 12. Therefore, we can rewrite the fractions with a common denominator of 12.
{\frac{3}{4} = \frac{9}{12}$ and [\frac{2}{3} = \frac{8}{12}\$}
Now we can add the fractions:
{\frac{9}{12} + \frac{8}{12} = \frac{17}{12}$}$
Therefore, the sum of the fractions is {\frac{17}{12}$.
Simplifying Fractions
When simplifying fractions, we need to find the greatest common divisor (GCD) of the numerator and the denominator. Let's consider the following example:
Example 4: Simplifying Fractions
[\frac{6}{8}\$}
To simplify this fraction, we need to find the GCD of 6 and 8. The GCD of 6 and 8 is 2. Therefore, we can divide both the numerator and the denominator by 2.
{\frac{6}{8} = \frac{3}{4}$}$
Therefore, the simplified fraction is {\frac{3}{4}$}$.
Conclusion
Adding and simplifying fractions is an essential skill in mathematics. By following the steps outlined in this article, you can master the process of adding and simplifying fractions. Remember to find common denominators and simplify fractions to their lowest terms. With practice and patience, you can become proficient in adding and simplifying fractions.
Common Denominators and Simplifying Fractions: A Summary
- When adding fractions with the same denominator, simply add the numerators and keep the denominator the same.
- When adding fractions with different denominators, find a common denominator and rewrite the fractions with a common denominator.
- When simplifying fractions, find the GCD of the numerator and the denominator and divide both the numerator and the denominator by the GCD.
- When adding mixed numbers, add the whole numbers and the fractions separately.
Introduction
Adding and simplifying fractions can be a challenging task, but with practice and patience, you can master it. In this article, we will answer some frequently asked questions about adding and simplifying fractions.
Q: What is the difference between adding fractions with the same denominator and adding fractions with different denominators?
A: When adding fractions with the same denominator, you can simply add the numerators and keep the denominator the same. However, when adding fractions with different denominators, you need to find a common denominator and rewrite the fractions with a common denominator.
Q: How do I find a common denominator?
A: To find a common denominator, you need to find the least common multiple (LCM) of the two denominators. You can use a calculator or a formula to find the LCM.
Q: What is the least common multiple (LCM)?
A: The LCM is the smallest number that is a multiple of both numbers. For example, the LCM of 2 and 4 is 4, because 4 is the smallest number that is a multiple of both 2 and 4.
Q: How do I simplify a fraction?
A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both the numerator and the denominator by the GCD.
Q: What is the greatest common divisor (GCD)?
A: The GCD is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 6 and 8 is 2, because 2 is the largest number that divides both 6 and 8 without leaving a remainder.
Q: Can I simplify a fraction by dividing both the numerator and the denominator by a number other than the GCD?
A: No, you cannot simplify a fraction by dividing both the numerator and the denominator by a number other than the GCD. This will result in an incorrect answer.
Q: How do I add mixed numbers?
A: To add mixed numbers, you need to add the whole numbers and the fractions separately. You can then add the fractions by finding a common denominator and rewriting the fractions with a common denominator.
Q: Can I add a fraction to a whole number?
A: Yes, you can add a fraction to a whole number by converting the whole number to a fraction with the same denominator as the fraction. You can then add the fractions.
Q: How do I subtract fractions?
A: To subtract fractions, you need to find a common denominator and rewrite the fractions with a common denominator. You can then subtract the numerators and keep the denominator the same.
Q: Can I subtract a fraction from a whole number?
A: Yes, you can subtract a fraction from a whole number by converting the whole number to a fraction with the same denominator as the fraction. You can then subtract the fractions.
Conclusion
Adding and simplifying fractions can be a challenging task, but with practice and patience, you can master it. By following the steps outlined in this article, you can answer frequently asked questions about adding and simplifying fractions.
Common Denominators and Simplifying Fractions: A Summary
- When adding fractions with the same denominator, simply add the numerators and keep the denominator the same.
- When adding fractions with different denominators, find a common denominator and rewrite the fractions with a common denominator.
- When simplifying fractions, find the GCD of the numerator and the denominator and divide both the numerator and the denominator by the GCD.
- When adding mixed numbers, add the whole numbers and the fractions separately.
- When subtracting fractions, find a common denominator and rewrite the fractions with a common denominator.
By following these steps and practicing regularly, you can become proficient in adding and simplifying fractions.