By what rational number should we multiply `(-15)/(56)` to get `(-5)/(7)` ?

Options

A. `(8)/(3) `
B. `(4)/(3) `
C. `(5)/(3) `
D. `(7)/(3) `

Answer

To solve the problem, we need to find the rational number \( x \) that we should multiply with \( \frac{-15}{56} \) to get \( \frac{-5}{7} \).

### Step-by-step Solution:

1. **Set Up the Equation**:
We start with the equation based on the problem statement:
\[ \frac{-15}{56} \times x = \frac{-5}{7} \]

2. **Isolate \( x \)**:
To find \( x \), we can rearrange the equation by multiplying both sides by the reciprocal of \( \frac{-15}{56} \):
\[ x = \frac{-5}{7} \div \frac{-15}{56} \]
This is equivalent to:
\[ x = \frac{-5}{7} \times \frac{56}{-15} \]

3. **Simplify the Expression**:
Now, we can simplify the expression:
\[ x = \frac{-5 \times 56}{7 \times -15} \]
The negative signs cancel out:
\[ x = \frac{5 \times 56}{7 \times 15} \]

4. **Perform the Multiplication**:
Calculate the numerator and denominator:
- Numerator: \( 5 \times 56 = 280 \)
- Denominator: \( 7 \times 15 = 105 \)
So we have:
\[ x = \frac{280}{105} \]

5. **Reduce the Fraction**:
Now we need to simplify \( \frac{280}{105} \). We can find the greatest common divisor (GCD) of 280 and 105, which is 35:
\[ x = \frac{280 \div 35}{105 \div 35} = \frac{8}{3} \]

6. **Final Answer**:
Therefore, the rational number we need to multiply \( \frac{-15}{56} \) by to get \( \frac{-5}{7} \) is:
\[ x = \frac{8}{3} \]