Divide the sum of `(-8)/(7) and (5)/(14)` by their product.

Answer

To solve the problem of dividing the sum of \(-\frac{8}{7}\) and \(\frac{5}{14}\) by their product, we can follow these steps:

### Step 1: Find the Sum
We need to calculate the sum of \(-\frac{8}{7}\) and \(\frac{5}{14}\).

To add these fractions, we need a common denominator. The least common multiple (LCM) of 7 and 14 is 14.

We can convert \(-\frac{8}{7}\) to a fraction with a denominator of 14:
\[ -\frac{8}{7} = -\frac{8 \times 2}{7 \times 2} = -\frac{16}{14} \]

Now we can add the two fractions:
\[ -\frac{16}{14} + \frac{5}{14} = \frac{-16 + 5}{14} = \frac{-11}{14} \]

### Step 2: Find the Product
Next, we need to find the product of \(-\frac{8}{7}\) and \(\frac{5}{14}\):
\[ -\frac{8}{7} \times \frac{5}{14} = \frac{-8 \times 5}{7 \times 14} = \frac{-40}{98} \]

We can simplify \(\frac{-40}{98}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2:
\[ \frac{-40 \div 2}{98 \div 2} = \frac{-20}{49} \]

### Step 3: Divide the Sum by the Product
Now we need to divide the sum by the product:
\[ \frac{\frac{-11}{14}}{\frac{-20}{49}} \]

To divide by a fraction, we multiply by its reciprocal:
\[ \frac{-11}{14} \times \frac{49}{-20} = \frac{-11 \times 49}{14 \times -20} \]

The negatives cancel out:
\[ \frac{11 \times 49}{14 \times 20} \]

### Step 4: Simplify the Result
Now we calculate the numerator and denominator:
\[ 11 \times 49 = 539 \]
\[ 14 \times 20 = 280 \]

So we have:
\[ \frac{539}{280} \]

This fraction cannot be simplified further, so the final answer is:
\[ \frac{539}{280} \]