Which rational number has to be multiplied with 64 to get the product `- 49 (3)/(5)` ?

Options

A. `(-35)/(43)`
B. `(-31)/(40)`
C. `(-33)/(40)`
D. `(-37)/(43)`

Answer

To find the rational number that has to be multiplied with 64 to get the product of \(-49 \frac{3}{5}\), we can follow these steps:

### Step 1: Convert the mixed fraction to an improper fraction
The mixed fraction \(-49 \frac{3}{5}\) can be converted to an improper fraction.

To do this:
- Multiply the whole number part (-49) by the denominator (5):
\[ -49 \times 5 = -245 \]
- Add the numerator (3) to this result:
\[ -245 + 3 = -242 \]
- The improper fraction is:
\[ \frac{-242}{5} \]

### Step 2: Set up the equation
We need to find a rational number \(x\) such that:
\[ 64 \times x = \frac{-242}{5} \]

### Step 3: Solve for \(x\)
To isolate \(x\), divide both sides of the equation by 64:
\[ x = \frac{-242}{5} \div 64 \]
This can be rewritten as:
\[ x = \frac{-242}{5} \times \frac{1}{64} \]
\[ x = \frac{-242}{5 \times 64} \]

### Step 4: Calculate the denominator
Now calculate \(5 \times 64\):
\[ 5 \times 64 = 320 \]
So we have:
\[ x = \frac{-242}{320} \]

### Step 5: Simplify the fraction
Next, we simplify \(\frac{-242}{320}\). We can find the greatest common divisor (GCD) of 242 and 320.

The GCD is 2, so we divide both the numerator and denominator by 2:
\[ x = \frac{-121}{160} \]

### Final Answer
Thus, the rational number that needs to be multiplied with 64 to get the product \(-49 \frac{3}{5}\) is:
\[ \boxed{\frac{-121}{160}} \]