By what rational number should `25/8` be multiplied to get `-20/7` ?

Answer

To find the rational number that should be multiplied by \( \frac{25}{8} \) to get \( -\frac{20}{7} \), we can set up the equation as follows:

Let \( x \) be the unknown rational number. We can write the equation:

\[ \frac{25}{8} \times x = -\frac{20}{7} \]

### Step 1: Isolate \( x \)

To solve for \( x \), we need to isolate it. We can do this by multiplying both sides of the equation by the reciprocal of \( \frac{25}{8} \), which is \( \frac{8}{25} \):

\[ x = -\frac{20}{7} \times \frac{8}{25} \]

### Step 2: Multiply the fractions

Now we will multiply the fractions on the right side:

\[ x = \frac{-20 \times 8}{7 \times 25} \]

### Step 3: Simplify the multiplication

Calculating the numerator and the denominator:

- Numerator: \( -20 \times 8 = -160 \)
- Denominator: \( 7 \times 25 = 175 \)

So we have:

\[ x = \frac{-160}{175} \]

### Step 4: Simplify the fraction

Next, we can simplify \( \frac{-160}{175} \). We can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 5:

\[ x = \frac{-160 \div 5}{175 \div 5} = \frac{-32}{35} \]

### Final Answer

Thus, the rational number that should be multiplied by \( \frac{25}{8} \) to get \( -\frac{20}{7} \) is:

\[ \boxed{-\frac{32}{35}} \]

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