Question

find the quotient. simplify your answer completely. \\(\frac{ - \frac{3}{2} }{ - \frac{5}{6} } = \frac{?}{\square}\\) enter the number that belongs in the green box.

Explanation:

Step1: Recall dividing fractions rule

Dividing by a fraction is multiplying by its reciprocal. So, \(\frac{-\frac{3}{2}}{-\frac{5}{6}} = -\frac{3}{2} \times (-\frac{6}{5})\)

Step2: Multiply the numerators and denominators

Multiply numerators: \(3\times6 = 18\), multiply denominators: \(2\times5 = 10\). The negatives cancel, so we have \(\frac{18}{10}\)

Step3: Simplify the fraction

Divide numerator and denominator by 2: \(\frac{18\div2}{10\div2}=\frac{9}{5}\)? Wait, no, wait. Wait, \(-\frac{3}{2}\times(-\frac{6}{5})=\frac{3\times6}{2\times5}=\frac{18}{10}=\frac{9}{5}\)? Wait, no, 36 is 18, 25 is 10, 18/10 simplifies to 9/5? Wait, no, 18 divided by 2 is 9, 10 divided by 2 is 5. Wait, but let's check again. Wait, the original problem is \(\frac{-\frac{3}{2}}{-\frac{5}{6}}\). The two negatives make a positive. Then, dividing fractions: \(\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}\). So here, \(a = -\frac{3}{2}\), \(b = 1\), \(c = -\frac{5}{6}\), \(d = 1\)? Wait, no, the complex fraction is \(\frac{-\frac{3}{2}}{-\frac{5}{6}}\), which is equivalent to \((-\frac{3}{2})\div(-\frac{5}{6})\). Dividing two negatives gives a positive. Then, dividing by a fraction is multiplying by reciprocal: \((-\frac{3}{2})\times(-\frac{6}{5})=\frac{3\times6}{2\times5}=\frac{18}{10}=\frac{9}{5}\)? Wait, no, 36 is 18, 25 is 10, 18/10 reduces to 9/5? Wait, but 18 divided by 2 is 9, 10 divided by 2 is 5. Wait, but let's do it step by step. Alternatively, \(\frac{-\frac{3}{2}}{-\frac{5}{6}}=\frac{3}{2}\times\frac{6}{5}\) (since negatives cancel). Then, \(\frac{3\times6}{2\times5}=\frac{18}{10}=\frac{9}{5}\)? Wait, no, 36 is 18, 25 is 10, 18/10 is 9/5? Wait, but 9/5 is 1.8. Wait, but maybe I made a mistake. Wait, let's check with another method. Let's convert to decimals. \(-\frac{3}{2} = -1.5\), \(-\frac{5}{6}\approx -0.8333\). Then, \(-1.5\div(-0.8333)\approx1.8\), and 9/5 is 1.8. So that's correct. Wait, but the problem says "Enter the number that belongs in the green box". Wait, maybe the green box is the numerator? Wait, the fraction is \(\frac{[?]}{[?]}\), but the question says "Enter the number that belongs in the green box". Wait, maybe the green box is the numerator, and the gray is the denominator? Wait, the original problem: \(\frac{-\frac{3}{2}}{-\frac{5}{6}}=\frac{[?]}{[?]}\), and we need to find the numerator (green box) and denominator (gray box). Wait, let's re-express the division of fractions correctly. \(\frac{-\frac{3}{2}}{-\frac{5}{6}} = (-\frac{3}{2}) \div (-\frac{5}{6}) = (-\frac{3}{2}) \times (-\frac{6}{5}) = \frac{3\times6}{2\times5} = \frac{18}{10} = \frac{9}{5}\). So the numerator (green box) is 9, denominator is 5? Wait, but the problem says "Enter the number that belongs in the green box", so maybe the green box is the numerator, so 9? Wait, let's check again. Wait, \(\frac{-\frac{3}{2}}{-\frac{5}{6}} = \frac{3}{2} \times \frac{6}{5} = \frac{3\times6}{2\times5} = \frac{18}{10} = \frac{9}{5}\). So the fraction is \(\frac{9}{5}\), so the green box (numerator) is 9.

Answer:

9