What should be added to `(-5)/(7)` ot get `(-2)/(3)`?

Options

A. `(-29)/(21)`
B. `(29)/(21)`
C. `(1)/(21)`
D. `(-1)/(21)`

Answer

To solve the problem of what should be added to \(-\frac{5}{7}\) to get \(-\frac{2}{3}\), we can follow these steps:

### Step 1: Set up the equation
Let \(x\) be the number that we need to add to \(-\frac{5}{7}\). According to the problem, we can write the equation as:
\[ -\frac{5}{7} + x = -\frac{2}{3} \]

### Step 2: Isolate \(x\)
To find \(x\), we need to isolate it on one side of the equation. We can do this by adding \(\frac{5}{7}\) to both sides:
\[ x = -\frac{2}{3} + \frac{5}{7} \]

### Step 3: Find a common denominator
To add the fractions \(-\frac{2}{3}\) and \(\frac{5}{7}\), we need a common denominator. The least common multiple of 3 and 7 is 21.

### Step 4: Convert the fractions
Now, we convert both fractions to have the common denominator of 21:
\[ -\frac{2}{3} = -\frac{2 \times 7}{3 \times 7} = -\frac{14}{21} \]
\[ \frac{5}{7} = \frac{5 \times 3}{7 \times 3} = \frac{15}{21} \]

### Step 5: Add the fractions
Now we can add the two fractions:
\[ x = -\frac{14}{21} + \frac{15}{21} = \frac{-14 + 15}{21} = \frac{1}{21} \]

### Step 6: Conclusion
Thus, the number that should be added to \(-\frac{5}{7}\) to get \(-\frac{2}{3}\) is:
\[ x = \frac{1}{21} \]