Question

use the properties of operations and rules for multiplying and dividing to solve:
(-3/5)x(5/16)x(7/12)/(-3/16)x(-5/7)

Answer 1

its wrong

Answer 2

To solve the expression \((-3/5) \times (5/16) \times (7/12) / (-3/16) \times (-5/7)\), we will apply the properties of operations, as well as simplifying step-by-step.

1. We first rewrite the expression for clarity:

\[
\frac{(-3/5) \times (5/16) \times (7/12)}{(-3/16) \times (-5/7)}
\]

2. Let's simplify the numerator and the denominator separately.

**Numerator:**
\[
(-3/5) \times (5/16) \times (7/12)
\]
We can multiply the fractions together:

\[
= \frac{-3 \times 5 \times 7}{5 \times 16 \times 12}
\]

Notice that the \(5\) in the numerator cancels with the \(5\) in the denominator:

\[
= \frac{-3 \times 7}{16 \times 12}
\]

Calculating that:

\[
= \frac{-21}{192}
\]

3. **Denominator:**
\[
(-3/16) \times (-5/7)
\]
Here, since both terms are negative, their product will be positive:

\[
= \frac{3 \times 5}{16 \times 7}
\]

Thus:

\[
= \frac{15}{112}
\]

4. Combining our results, we rewrite our original expression as:

\[
\frac{-21/192}{15/112}
\]

Dividing by a fraction is the same as multiplying by its reciprocal:

\[
= \frac{-21}{192} \times \frac{112}{15}
\]

5. Now, we multiply the fractions:

\[
= \frac{-21 \times 112}{192 \times 15}
\]

6. Let's simplify the fraction. We will find common factors. First, we'll calculate each term:

\(-21\) can be factored into \(-3 \times 7\).

\(192\) can be factored (by its prime factorization) as \(2^6 \times 3\).

\(15\) is \(3 \times 5\).

\(112\) is \(2^4 \times 7\).

Substituting these factorizations, we get:

\[
= \frac{-3 \times 7 \times (2^4 \times 7)}{(2^6 \times 3) \times (3 \times 5)}
\]

7. Now cancel common factors:

- The \(3\) in the numerator and denominator.
- The \(7\) in the numerator and denominator.

After canceling the common factors, we simplify:

\[
= \frac{-1 \times 16}{64 \times 5} = \frac{-16}{320}
\]

8. Now we can simplify \(-16/320\):

\[
= \frac{-1}{20} \quad \text{ (by dividing both numerator and denominator by 16)}
\]

Thus, the final solution is:

\[
\boxed{-\frac{1}{20}}
\]

Answer 3

Let’s re-evaluate the expression step by step to ensure we arrive at the correct solution.

We start with the expression:

\[
\frac{(-3/5) \times (5/16) \times (7/12)}{(-3/16) \times (-5/7)}
\]

### Step 1: Simplify the Numerator

The numerator is:

\[
(-3/5) \times (5/16) \times (7/12)
\]

Calculating this step-by-step:

1. **Multiply the first two fractions:**
\[
(-3/5) \times (5/16) = \frac{-3 \times 5}{5 \times 16} = \frac{-3}{16} \quad \text{(the 5 cancels out)}
\]

2. **Now multiply by the third fraction:**
\[
\left(\frac{-3}{16}\right) \times \left(\frac{7}{12}\right) = \frac{-3 \times 7}{16 \times 12} = \frac{-21}{192}
\]

So, our numerator is:

\[
\frac{-21}{192}
\]

### Step 2: Simplify the Denominator

The denominator is:

\[
(-3/16) \times (-5/7)
\]

Since both fractions are negative, their product will be positive:

\[
(-3/16) \times (-5/7) = \frac{3 \times 5}{16 \times 7} = \frac{15}{112}
\]

### Step 3: Structure the Division

Now we can rewrite our expression as:

\[
\frac{-21/192}{15/112}
\]

Dividing by a fraction is equivalent to multiplying by its reciprocal:

\[
= \frac{-21}{192} \times \frac{112}{15}
\]

### Step 4: Multiply the Fractions

Multiplying the fractions:

\[
= \frac{-21 \times 112}{192 \times 15}
\]

### Step 5: Calculate the Numerator and Denominator

1. The numerator:
\[
-21 \times 112 = -2352
\]

2. The denominator:
\[
192 \times 15 = 2880
\]

Thus we have:

\[
\frac{-2352}{2880}
\]

### Step 6: Simplifying the Fraction

Now we can simplify:

1. Find the greatest common divisor (GCD) of 2352 and 2880.

- The GCD is \(48\).

Now we divide the numerator and denominator by \(48\):

\[
\frac{-2352 \div 48}{2880 \div 48} = \frac{-49}{60}
\]

### Conclusion

The final simplified expression is:

\[
\boxed{-\frac{49}{60}}
\]

Thank you for your patience, and I apologize for the earlier error!