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Part 1

Perform the indicated operation. Write the sum as a​ fraction, whole​ number, or mixed number in lowest terms.

five sixths56plus+nine fourteenths

To solve this, we need to add the two fractions: \( \frac{5}{6} + \frac{9}{14} \).

### Step 1: Find the Least Common Denominator (LCD)
The denominators are 6 and 14. The least common denominator is the least common multiple (LCM) of these two numbers. 

- The prime factorization of 6 is \( 2 \times 3 \).
- The prime factorization of 14 is \( 2 \times 7 \).

The LCM is \( 2 \times 3 \times 7 = 42 \).

### Step 2: Convert the fractions to have the LCD
Now, we convert both fractions to have the denominator 42.

- For \( \frac{5}{6} \), multiply both the numerator and denominator by 7 to get:
  \[
  \frac{5}{6} = \frac{5 \times 7}{6 \times 7} = \frac{35}{42}
  \]
  
- For \( \frac{9}{14} \), multiply both the numerator and denominator by 3 to get:
  \[
  \frac{9}{14} = \frac{9 \times 3}{14 \times 3} = \frac{27}{42}
  \]

### Step 3: Add the fractions
Now, add the fractions:
\[
\frac{35}{42} + \frac{27}{42} = \frac{35 + 27}{42} = \frac{62}{42}
\]

### Step 4: Simplify the fraction
Simplify \( \frac{62}{42} \) by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 2:
\[
\frac{62}{42} = \frac{62 \div 2}{42 \div 2} = \frac{31}{21}
\]

### Step 5: Write as a mixed number
Since \( 31 \div 21 = 1 \) remainder 10, we can write it as a mixed number:
\[
\frac{31}{21} = 1 \frac{10}{21}
\]

So, \( \frac{5}{6} + \frac{9}{14} = 1 \frac{10}{21} \).

Let me know if you would like further details or clarifications.

### Related Questions:
1. How do you find the least common denominator of two fractions?
2. What are the steps to convert an improper fraction to a mixed number?
3. Can you explain how to simplify a fraction?
4. What is the prime factorization method used to find the LCM?
5. How do you convert a mixed number back to an improper fraction?

### Tip:
When adding fractions, always check if you can simplify the final answer by dividing the numerator and denominator by their greatest common divisor (GCD).

Perform the indicated operation. Write the sum as a fraction, whole number, or mixed number in lowest terms. five sixths (5/6) plus nine fourteenths (9/14).

Learn how to add fractions by finding the least common denominator (LCD) and simplifying the result to a mixed number. This step-by-step solution covers how to add 5/6 and 9/14, converting to common denominators, and simplifying the final answer to 1 10/21. Suitable for students in Grades 6-8.

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