which of the following rational numbers must be added In  the sum of$$ \frac37 $$,$$ \left(\frac{-8}{21}\right)\text{ } $$and $$ \left(\frac{-3}{14}\right) $$to get the result $$ \frac{-5}{14}\text{?} $$

To solve the problem, we need to find the rational number $ x $ that must be added to the sum of $ \frac{3}{7} $, $ \frac{-8}{21} $, and $ \frac{-3}{14} $ to obtain $ \frac{-5}{14} $. 

Let's denote the required rational number by $ x $. The equation we need to solve is:

\[
\frac{3}{7} + \frac{-8}{21} + \frac{-3}{14} + x = \frac{-5}{14}
\]

### Step 1: Find the sum of the fractions on the left side

First, we calculate the sum of the three given fractions $ \frac{3}{7} $, $ \frac{-8}{21} $, and $ \frac{-3}{14} $. 

#### Find a common denominator:
The denominators are 7, 21, and 14. The least common denominator (LCD) for these fractions is 42.

Convert each fraction to have the denominator 42:

\[
\frac{3}{7} = \frac{3 \times 6}{7 \times 6} = \frac{18}{42}
\]
\[
\frac{-8}{21} = \frac{-8 \times 2}{21 \times 2} = \frac{-16}{42}
\]
\[
\frac{-3}{14} = \frac{-3 \times 3}{14 \times 3} = \frac{-9}{42}
\]

#### Sum the fractions:
Now, add the fractions:

\[
\frac{18}{42} + \frac{-16}{42} + \frac{-9}{42} = \frac{18 - 16 - 9}{42} = \frac{-7}{42}
\]

Simplify:

\[
\frac{-7}{42} = \frac{-1}{6}
\]

### Step 2: Add $ x $ to the sum to get the required result

We now have:

\[
\frac{-1}{6} + x = \frac{-5}{14}
\]

### Step 3: Solve for $ x $

To solve for $ x $, subtract $ \frac{-1}{6} $ from both sides:

\[
x = \frac{-5}{14} - \frac{-1}{6}
\]

### Step 4: Find the common denominator and simplify

The least common denominator for 14 and 6 is 42. Convert each fraction:

\[
\frac{-5}{14} = \frac{-5 \times 3}{14 \times 3} = \frac{-15}{42}
\]
\[
\frac{-1}{6} = \frac{-1 \times 7}{6 \times 7} = \frac{-7}{42}
\]

Subtract the fractions:

\[
x = \frac{-15}{42} + \frac{7}{42} = \frac{-15 + 7}{42} = \frac{-8}{42} = \frac{-4}{21}
\]

### Final Answer:
The rational number that must be added is $ \frac{-4}{21} $.

Would you like any further details or have any questions? 

Here are some related questions:
1. How do you simplify rational expressions?
2. What are the steps for finding a common denominator?
3. Can you explain the process of adding fractions with different denominators?
4. What is the importance of the least common denominator?
5. How do you subtract rational numbers?
6. How do you check if two fractions are equivalent?
7. How can rational numbers be compared?
8. What are some common mistakes when adding and subtracting fractions?

**Tip:** Always simplify fractions to their lowest terms to make calculations easier.

Learn how to solve a problem involving adding rational numbers. Find the rational number that must be added to the sum of $ \frac{3}{7}, \frac{-8}{21}, $ and $ \frac{-3}{14} $ to obtain $ \frac{-5}{14} $. Step-by-step solution provided. Suitable for Grades 7-9.

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