Fill in blanks: `(27)/(16) divide (........) = (-15)/(8)`.

Options

A. `(-7)/(10) `
B. `(-9)/(10) `
C. `(-11)/(10) `
D. `(-8)/(10) `

Answer

To solve the equation \(\frac{27}{16} \div (\text{...}) = \frac{-15}{8}\), we will follow these steps:

1. **Set up the equation**: Let the blank be represented by \(x\). Thus, we rewrite the equation as:
\[ \frac{27}{16} \div x = \frac{-15}{8} \]

2. **Change division to multiplication**: When we divide by a number, we can multiply by its reciprocal. Therefore, we can rewrite the equation as:
\[ \frac{27}{16} \times \frac{1}{x} = \frac{-15}{8} \]

3. **Cross-multiply**: To eliminate the fractions, we can cross-multiply:
\[ 27 \times 8 = -15 \times 16 \times x \]

4. **Calculate the left side**: Now calculate \(27 \times 8\):
\[ 27 \times 8 = 216 \]

5. **Calculate the right side**: Now calculate \(-15 \times 16\):
\[ -15 \times 16 = -240 \]

6. **Set up the equation**: Now we have:
\[ 216 = -240x \]

7. **Solve for \(x\)**: To find \(x\), divide both sides by \(-240\):
\[ x = \frac{216}{-240} \]

8. **Simplify the fraction**: We can simplify \(\frac{216}{-240}\) by finding the greatest common divisor (GCD) of 216 and 240. The GCD is 24. Thus:
\[ x = \frac{216 \div 24}{-240 \div 24} = \frac{9}{-10} = -\frac{9}{10} \]

9. **Final answer**: Therefore, the blank can be filled with:
\[ x = -\frac{9}{10} \]