Question

Use the properties of operations and rules for multiplying and dividing to solve:

(−2)×(15)÷(23)×(−56)÷(34)

(1 point)
Responses

−16
negative Start Fraction 1 over 6 End Fraction

23
Start Fraction 2 over 3 End Fraction

−23
negative Start Fraction 2 over 3 End Fraction

Answer 1

To solve the expression \((−2)×(15)÷(23)×(−56)÷(34)\), let's break it down step by step, taking care of the order of operations (which states to perform multiplication and division from left to right).

1. **Step 1: Multiply -2 and 15**
\[
(-2) \times 15 = -30
\]

2. **Step 2: Divide by 23**
\[
-30 \div 23 = -\frac{30}{23}
\]

3. **Step 3: Multiply by -56**
Now, we need to multiply \(-\frac{30}{23}\) by \(-56\):
\[
-\frac{30}{23} \times (-56) = \frac{30 \times 56}{23} = \frac{1680}{23}
\]

4. **Step 4: Divide by 34**
Now perform the final division:
\[
\frac{1680}{23} \div 34 = \frac{1680}{23} \times \frac{1}{34}
\]
This simplifies to:
\[
\frac{1680}{23 \times 34} = \frac{1680}{782}
\]

To simplify \(\frac{1680}{782}\), we need to find the greatest common divisor (GCD). We can perform the prime factorization or use the Euclidean algorithm.

After finding the GCD, we find:
The GCD of 1680 and 782 is 14.

So we simplify:
\[
\frac{1680 \div 14}{782 \div 14} = \frac{120}{56} = \frac{15}{7}
\]

Therefore, the final answer from all calculations is
\[
-\frac{15}{7}
\]

When checking with the options provided:

- **Responses**:
- \(-16\)
- negative \(\frac{1}{6}\)
- \(23\)
- \(\frac{2}{3}\)
- \(-23\)
- negative \(\frac{2}{3}\)

The correct response is not found in the choices based on the calculation from the provided expression.

Please double-check any further interpretations or constraints, as it seems there may be an error in the original potential responses to the question if it was taken directly from that problem statement. Given the computation based on standard rules of arithmetic and fraction simplification, a consistent error in response options could suggest a discrepancy in how the problem was framed or interpreted.