To find the quotient of [tex]\( -\frac{10}{19} \div \left( -\frac{5}{7} \right) \)[/tex], follow these steps:
1. Understand Division of Fractions: Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore,
[tex]\[
-\frac{10}{19} \div \left( -\frac{5}{7} \right) = -\frac{10}{19} \times \left( -\frac{7}{5} \right)
\][/tex]
2. Find the Reciprocal: The reciprocal of [tex]\( -\frac{5}{7} \)[/tex] is [tex]\( -\frac{7}{5} \)[/tex].
3. Set Up the Multiplication:
[tex]\[
-\frac{10}{19} \times -\frac{7}{5}
\][/tex]
4. Multiply the Numerators: The numerator of the product is:
[tex]\[
(-10) \times (-7) = 70
\][/tex]
5. Multiply the Denominators: The denominator of the product is:
[tex]\[
19 \times 5 = 95
\][/tex]
6. Combine the Results: So the fraction becomes:
[tex]\[
\frac{70}{95}
\][/tex]
7. Simplify the Fraction:
- The greatest common divisor (GCD) of 70 and 95 is 5.
- Hence, [tex]\( 70 \div 5 = 14 \)[/tex].
- And, [tex]\( 95 \div 5 = 19 \)[/tex].
- Therefore, the simplified fraction is:
[tex]\[
\frac{70}{95} = \frac{14}{19}
\][/tex]
Thus, the quotient is:
[tex]\[
\boxed{\frac{14}{19}}
\][/tex]
Let's check which option matches this result:
- [tex]\( \text{A } -\frac{70}{95} \)[/tex]
- [tex]\( \text{B } -\frac{14}{19} \)[/tex]
- [tex]\( \text{C } \frac{14}{19} \)[/tex]
- [tex]\( \text{D } \frac{70}{95} \)[/tex]
The correct answer is clearly:
[tex]\[
\boxed{\text{C}}
\][/tex]
