By what rational number should `(-8)/(39)` be multiplied to obtain `(1)/(26)`?

Options

A. `(-3)/(8) `
B. `(-3)/(16) `
C. `(-3)/(5) `
D. `(-3)/(4) `

Answer

To solve the problem, we need to find the rational number \( x \) such that:

\[ \left(-\frac{8}{39}\right) \cdot x = \frac{1}{26} \]

**Step 1:** Set up the equation.

We start with the equation:

\[ -\frac{8}{39} \cdot x = \frac{1}{26} \]

**Hint:** The goal is to isolate \( x \) on one side of the equation.

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**Step 2:** Isolate \( x \).

To isolate \( x \), we can divide both sides of the equation by \(-\frac{8}{39}\). Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we rewrite the equation as:

\[ x = \frac{1}{26} \div \left(-\frac{8}{39}\right) \]

This can be rewritten as:

\[ x = \frac{1}{26} \cdot \left(-\frac{39}{8}\right) \]

**Hint:** Remember to multiply by the reciprocal when dividing by a fraction.

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**Step 3:** Perform the multiplication.

Now we multiply the fractions:

\[ x = -\frac{39}{26 \cdot 8} \]

**Hint:** When multiplying fractions, multiply the numerators together and the denominators together.

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**Step 4:** Simplify the expression.

First, calculate the denominator:

\[ 26 \cdot 8 = 208 \]

So we have:

\[ x = -\frac{39}{208} \]

Next, we can check if \( \frac{39}{208} \) can be simplified. The greatest common divisor (GCD) of 39 and 208 is 13. Thus, we can simplify:

\[ x = -\frac{39 \div 13}{208 \div 13} = -\frac{3}{16} \]

**Hint:** Always check if the fraction can be simplified by finding the GCD.

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**Final Answer:**

The rational number that should be multiplied by \(-\frac{8}{39}\) to obtain \(\frac{1}{26}\) is:

\[ x = -\frac{3}{16} \]

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