To solve the expression \( \frac{-3}{4} \times \frac{5}{16} \times \frac{7}{12} \div \left( \frac{-3}{16} \times \frac{-5}{7} \right) \), we first simplify both the numerator and the denominator.
Step 1: Simplify the numerator
The numerator is: \[ \frac{-3}{4} \times \frac{5}{16} \times \frac{7}{12} \] We can multiply these fractions: \[ = \frac{-3 \times 5 \times 7}{4 \times 16 \times 12} \]
Calculating the numerator: \[ -3 \times 5 = -15 \] \[ -15 \times 7 = -105 \] So, the numerator becomes: \[ -105 \]
Calculating the denominator for the numerator: \[ 4 \times 16 = 64 \] \[ 64 \times 12 = 768 \] So, the numerator simplifies to: \[ \frac{-105}{768} \]
Step 2: Simplify the denominator
The denominator is: \[ \frac{-3}{16} \times \frac{-5}{7} \] We can multiply these fractions: \[ = \frac{-3 \times -5}{16 \times 7} = \frac{15}{112} \]
Step 3: Division of fractions
Now, we have: \[ \frac{-105}{768} \div \frac{15}{112} \] Dividing by a fraction is equivalent to multiplying by the reciprocal: \[ \frac{-105}{768} \times \frac{112}{15} \]
Step 4: Multiply the fractions
We multiply the numerators and the denominators: \[ = \frac{-105 \times 112}{768 \times 15} \]
Now we simplify:
-
First, simplify \( -105 \) and \( 15 \): \[ -105 \div 15 = -7 \] So we now have: \[ = \frac{-7 \times 112}{768} \]
-
Next, simplify \( 112 \) and \( 768 \): \[ 768 \div 112 = 6.857,... (which isn't an integer, we can use the GCD) \] Finding the GCD of \(112\) and \(768\): \[ 112 = 16 \times 7, \quad 768 = 16 \times 48 \] The GCD is \(16\): \[ 112 \div 16 = 7, \quad 768 \div 16 = 48 \]
So now we have: \[ = \frac{-7 \times 7}{48} = \frac{-49}{48} \]
Final Answer
\[ \frac{-49}{48} \]
This is the simplified result of the original expression.