By what rational number should `(-8)/(39)` be multiplied to obtain `(5)/(26)`?

Answer

To solve the problem of finding the rational number \( x \) such that

\[ \frac{-8}{39} \times x = \frac{5}{26}, \]

we can follow these steps:

### Step 1: Set up the equation
We start with the equation:

\[ \frac{-8}{39} \times x = \frac{5}{26}. \]

### Step 2: Isolate \( x \)
To isolate \( x \), we can multiply both sides of the equation by the reciprocal of \(\frac{-8}{39}\). The reciprocal of \(\frac{-8}{39}\) is \(\frac{39}{-8}\). Thus, we multiply both sides by \(\frac{39}{-8}\):

\[ x = \frac{5}{26} \times \frac{39}{-8}. \]

### Step 3: Multiply the fractions
Now we multiply the fractions on the right side:

\[ x = \frac{5 \times 39}{26 \times -8}. \]

### Step 4: Simplify the multiplication
Calculating the numerator and denominator:

- Numerator: \( 5 \times 39 = 195 \)
- Denominator: \( 26 \times -8 = -208 \)

So we have:

\[ x = \frac{195}{-208}. \]

### Step 5: Simplify the fraction
Now we simplify \(\frac{195}{-208}\). We can factor both numbers to see if they have any common factors.

- The prime factorization of \(195\) is \(3 \times 5 \times 13\).
- The prime factorization of \(208\) is \(2^4 \times 13\).

The common factor is \(13\). Thus, we can simplify:

\[ x = \frac{195 \div 13}{-208 \div 13} = \frac{15}{-16}. \]

### Final Answer
Thus, the rational number \( x \) that should be multiplied by \(\frac{-8}{39}\) to obtain \(\frac{5}{26}\) is:

\[ x = \frac{-15}{16}. \]

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