To solve the expression \((-\frac{3}{5}) \times (\frac{5}{16}) \times (\frac{7}{12}) \div (-\frac{3}{16}) \times (-\frac{5}{7})\), we can simplify step by step.
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Rewrite the division as multiplication by the reciprocal:
The division of fractions can be rewritten by taking the reciprocal of the divisor: \[ \div (-\frac{3}{16}) \quad \text{becomes} \quad \times (-\frac{16}{3}) \] So the expression now looks like this: \[ (-\frac{3}{5}) \times (\frac{5}{16}) \times (\frac{7}{12}) \times (-\frac{16}{3}) \times (-\frac{5}{7}) \] -
Combine the factors:
We can multiply all the fractions together: \[ (-\frac{3}{5}) \times (\frac{5}{16}) \times (\frac{7}{12}) \times (-\frac{16}{3}) \times (-\frac{5}{7}) = \frac{(-3) \times 5 \times 7 \times (-16) \times (-5)}{5 \times 16 \times 12 \times 3 \times 7} \] -
Calculating the numerator and denominator:
- Numerator:
\((-3) \times 5 \times 7 \times (-16) \times (-5)\) equals: \[ 3 \times 5 \times 7 \times 16 \times 5 = 3 \times 5^2 \times 7 \times 16 = 3 \times 25 \times 7 \times 16 \] - Denominator:
\(5 \times 16 \times 12 \times 3 \times 7\)
- Numerator:
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Cancelling out common factors:
We notice:- The \(5\) in the numerator and denominator cancels.
- The \(7\) in the numerator and denominator cancels.
- The \(3\) cancels with the multiple in the denominator.
- The \(16\) cancels with the multiple in the denominator.
After all cancellations, we can look at the simple ratios remaining.
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Final multiplication:
\(\frac{3 \times 5 \times 7 \times 16 \times 5}{5 \times 16 \times 12 \times 3 \times 7}\) reduces down to a simple multiplication of whatever remains.
Now calculate directly: Using remaining terms leads us to: \[ \frac{1}{12} \]
So, the final answer is: \[ \frac{1}{12} \]
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