By what rational number should `-9/35` be multiplied to get `3/5` ?

Answer

To solve the problem of finding the rational number \( x \) that should be multiplied by \( -\frac{9}{35} \) to get \( \frac{3}{5} \), we can set up the equation as follows:

### Step 1: Set up the equation
We want to find \( x \) such that:
\[ -\frac{9}{35} \times x = \frac{3}{5} \]

### Step 2: Isolate \( x \)
To isolate \( x \), we can divide both sides of the equation by \( -\frac{9}{35} \):
\[ x = \frac{\frac{3}{5}}{-\frac{9}{35}} \]

### Step 3: Simplify the right side
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the equation as:
\[ x = \frac{3}{5} \times -\frac{35}{9} \]

### Step 4: Multiply the fractions
Now, we multiply the fractions:
\[ x = \frac{3 \times -35}{5 \times 9} \]
Calculating the numerator and denominator:
\[ x = \frac{-105}{45} \]

### Step 5: Simplify the fraction
Now, we simplify \( \frac{-105}{45} \). We can divide both the numerator and the denominator by their greatest common divisor, which is 15:
\[ x = \frac{-105 \div 15}{45 \div 15} = \frac{-7}{3} \]

### Final Answer
Thus, the rational number that should be multiplied by \( -\frac{9}{35} \) to get \( \frac{3}{5} \) is:
\[ \boxed{-\frac{7}{3}} \]

---