Question

Use properties of operations and rules for multiplying and dividing to solve: (-2) x (1/5) divided by (2/3) x (-5/6) divided by (3/4)

Answer 1

To solve the expression \((-2) \times \left(\frac{1}{5}\right) \div \left(\frac{2}{3}\right) \times \left(-\frac{5}{6}\right) \div \left(\frac{3}{4}\right)\), we will follow the order of operations and simplify it step by step.

### Step 1: Simplify the expression by addressing multiplication and division.

We can start by rewriting the division as multiplication by the reciprocal:

\[
(-2) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(-\frac{5}{6}\right) \times \left(\frac{4}{3}\right)
\]

### Step 2: Group the multiplication and multipliers.

Now we'll combine the negative numbers and the fractions together:

1. Multiply \((-2)\) and \(-\frac{5}{6}\) and then the fractions:
- \((-2) \times -\frac{5}{6} = \frac{10}{6} = \frac{5}{3}\)
2. Now combine everything:
- \(\frac{5}{3} \times \frac{1}{5} \times \frac{3}{2} \times \frac{4}{3}\)

### Step 3: Simplify the multiplication of the fractions.

Calculating this step-by-step:

\[
\left(\frac{5}{3} \times \frac{1}{5}\right) = \frac{5 \times 1}{3 \times 5} = \frac{1}{3}
\]

Now we need to multiply that result with \(\frac{3}{2}\):

\[
\frac{1}{3} \times \frac{3}{2} = \frac{1 \times 3}{3 \times 2} = \frac{3}{6} = \frac{1}{2}
\]

Now multiply this result with \(\frac{4}{3}\):

\[
\frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3}
\]

### Final Result

Thus, the final result of \((-2) \times \left(\frac{1}{5}\right) \div \left(\frac{2}{3}\right) \times \left(-\frac{5}{6}\right) \div \left(\frac{3}{4}\right)\) is

\[
\boxed{\frac{2}{3}}
\]