To solve the expression \((−3) ÷ (3/4) × (2/7) × (−5/16) ÷ (5/7)\), we can follow the order of operations while applying the properties of multiplication and division.
First, let's rewrite the division by a fraction as multiplication by its reciprocal.
- Changing the division to multiplication: \[ (-3) ÷ (3/4) = (-3) × (4/3) \] \[ (-3) ÷ (5/7) = (-3) × (7/5) \]
Now we can rewrite the entire expression as follows: \[ (-3) × (4/3) × (2/7) × (-5/16) × (7/5) \]
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Next, we can multiply the fractions: \[ = (-3) × \frac{4}{3} × \frac{2}{7} × \frac{-5}{16} × \frac{7}{5} \]
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Now we can combine and simplify:
- Multiply the numerators: \[ = (-3) × 4 × 2 × (-5) × 7 \]
- Multiply the denominators: \[ = 3 × 7 × 16 × 5 \]
So our expression now is: \[ \frac{(-3) × 4 × 2 × (-5) × 7}{3 × 7 × 16 × 5} \]
Now, let's calculate the numerator: \[ (-3) × 4 = -12 \] \[ -12 × 2 = -24 \] \[ -24 × (-5) = 120 \] \[ 120 × 7 = 840 \]
Next, let's compute the denominator: \[ 3 × 7 = 21 \] \[ 21 × 16 = 336 \] \[ 336 × 5 = 1680 \]
So, now our expression is: \[ \frac{840}{1680} \]
- Simplifying \(\frac{840}{1680}\): This can be simplified by dividing both the numerator and denominator by 840: \[ = \frac{840 ÷ 840}{1680 ÷ 840} = \frac{1}{2} \]
Thus, the final answer is: \[ \frac{1}{2} \]