Sure, let's go through this step-by-step.
### Step 1: Define the Rational Numbers
We start with the rational numbers given in the problem:
- The first number is [tex]\( -\frac{8}{7} \)[/tex].
- The second number is [tex]\( \frac{5}{14} \)[/tex].
### Step 2: Calculate the Sum
Next we calculate the sum of these two rational numbers. Let’s perform the addition:
[tex]\[
-\frac{8}{7} + \frac{5}{14}
\][/tex]
To add these fractions, we need a common denominator. The least common multiple of 7 and 14 is 14. Rewrite [tex]\(\frac{-8}{7}\)[/tex] with the common denominator:
[tex]\[
-\frac{8}{7} = -\frac{8 \times 2}{7 \times 2} = -\frac{16}{14}
\][/tex]
Now, add the two fractions:
[tex]\[
-\frac{16}{14} + \frac{5}{14} = \frac{-16 + 5}{14} = \frac{-11}{14}
\][/tex]
### Step 3: Calculate the Product
Now we find the product of the same two rational numbers:
[tex]\[
-\frac{8}{7} \times \frac{5}{14}
\][/tex]
Multiply the numerators together and the denominators together:
[tex]\[
-\frac{8 \times 5}{7 \times 14} = -\frac{40}{98}
\][/tex]
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
-\frac{40 \div 2}{98 \div 2} = -\frac{20}{49}
\][/tex]
### Step 4: Divide the Sum by the Product
Finally, we divide the sum by the product. We have:
[tex]\[
\frac{\frac{-11}{14}}{-\frac{20}{49}}
\][/tex]
Recall that dividing by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[
\frac{-11}{14} \div -\frac{20}{49} = \frac{-11}{14} \times -\frac{49}{20}
\][/tex]
Multiply the numerators together and the denominators together:
[tex]\[
\frac{(-11) \times (-49)}{14 \times 20} = \frac{539}{280}
\][/tex]
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 7:
[tex]\[
\frac{539 \div 7}{280 \div 7} = \frac{77}{40}
\][/tex]
Thus, the sum of [tex]\( -\frac{8}{7} \)[/tex] and [tex]\( \frac{5}{14} \)[/tex] is [tex]\( -0.7857142857142856 \)[/tex], the product is [tex]\( -0.40816326530612246 \)[/tex], and the result of dividing the sum by their product is [tex]\( 1.9249999999999996 \)[/tex].
So, the result when we divide the sum of [tex]\( -\frac{8}{7} \)[/tex] and [tex]\( \frac{5}{14} \)[/tex] by their product is approximately [tex]\( 1.925 \)[/tex].
