By what number should `-4(3)/(8)` be divided to obtain `-3(1)/(2)`?

Options

A. `(3)/(4)`
B. `(1)/(4)`
C. `(5)/(4)`
D. `9/7`

Answer

To solve the problem, we need to determine by what number we should divide \(-4\frac{3}{8}\) to obtain \(-3\frac{1}{2}\). Let's go through the steps:

### Step 1: Convert the mixed numbers to improper fractions
1. **Convert \(-4\frac{3}{8}\)**:
- The whole number part is \(-4\) and the fractional part is \(\frac{3}{8}\).
- To convert it to an improper fraction:
\[ -4\frac{3}{8} = \frac{-4 \times 8 + 3}{8} = \frac{-32 + 3}{8} = \frac{-29}{8} \]

2. **Convert \(-3\frac{1}{2}\)**:
- The whole number part is \(-3\) and the fractional part is \(\frac{1}{2}\).
- To convert it to an improper fraction:
\[ -3\frac{1}{2} = \frac{-3 \times 2 + 1}{2} = \frac{-6 + 1}{2} = \frac{-5}{2} \]

### Step 2: Set up the equation
We need to find the number \(X\) such that:
\[ \frac{-29}{8} \div X = \frac{-5}{2} \]
This can be rewritten as:
\[ \frac{-29}{8} = \frac{-5}{2} \times X \]

### Step 3: Solve for \(X\)
To isolate \(X\), we can multiply both sides by the reciprocal of \(\frac{-5}{2}\):
\[ X = \frac{-29}{8} \times \frac{2}{-5} \]
The negative signs cancel out:
\[ X = \frac{29 \times 2}{8 \times 5} = \frac{58}{40} \]

### Step 4: Simplify the fraction
Now we simplify \(\frac{58}{40}\):
- Both the numerator and denominator can be divided by 2:
\[ X = \frac{29}{20} \]

### Final Answer
Thus, the number by which \(-4\frac{3}{8}\) should be divided to obtain \(-3\frac{1}{2}\) is:
\[ \boxed{\frac{29}{20}} \]