Simplify the following complex fractions in Exercise Set 5.6.

To solve the given complex fractions, we will simplify each expression step by step. Let's go through each problem systematically.

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Problem 1:

Simplify:
$$
\frac{\frac{3}{4}}{\frac{5}{6}}
$$

#### Solution:
To simplify a fraction divided by another fraction, multiply the numerator by the reciprocal of the denominator:
$$
\frac{\frac{3}{4}}{\frac{5}{6}} = \frac{3}{4} \cdot \frac{6}{5}
$$

Now, multiply the numerators and denominators:
$$
\frac{3 \cdot 6}{4 \cdot 5} = \frac{18}{20}
$$

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2:
$$
\frac{18 \div 2}{20 \div 2} = \frac{9}{10}
$$

Thus, the simplified form is:
$$
\boxed{\frac{9}{10}}
$$

---

Problem 2:

Simplify:
$$
\frac{\frac{7}{8}}{\frac{3}{4}}
$$

#### Solution:
Multiply the numerator by the reciprocal of the denominator:
$$
\frac{\frac{7}{8}}{\frac{3}{4}} = \frac{7}{8} \cdot \frac{4}{3}
$$

Multiply the numerators and denominators:
$$
\frac{7 \cdot 4}{8 \cdot 3} = \frac{28}{24}
$$

Simplify the fraction by dividing both the numerator and the denominator by their GCD, which is 4:
$$
\frac{28 \div 4}{24 \div 4} = \frac{7}{6}
$$

Thus, the simplified form is:
$$
\boxed{\frac{7}{6}}
$$

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Problem 3:

Simplify:
$$
\frac{\frac{5}{6}}{\frac{10}{12}}
$$

#### Solution:
First, simplify the denominator $\frac{10}{12}$:
$$
\frac{10}{12} = \frac{10 \div 2}{12 \div 2} = \frac{5}{6}
$$

Now, the expression becomes:
$$
\frac{\frac{5}{6}}{\frac{5}{6}}
$$

Any number divided by itself is 1:
$$
\frac{\frac{5}{6}}{\frac{5}{6}} = 1
$$

Thus, the simplified form is:
$$
\boxed{1}
$$

---

Problem 4:

Simplify:
$$
\frac{\frac{2}{3}}{\frac{4}{9}}
$$

#### Solution:
Multiply the numerator by the reciprocal of the denominator:
$$
\frac{\frac{2}{3}}{\frac{4}{9}} = \frac{2}{3} \cdot \frac{9}{4}
$$

Multiply the numerators and denominators:
$$
\frac{2 \cdot 9}{3 \cdot 4} = \frac{18}{12}
$$

Simplify the fraction by dividing both the numerator and the denominator by their GCD, which is 6:
$$
\frac{18 \div 6}{12 \div 6} = \frac{3}{2}
$$

Thus, the simplified form is:
$$
\boxed{\frac{3}{2}}
$$

---

Problem 5:

Simplify:
$$
\frac{\frac{3}{5}}{\frac{6}{10}}
$$

#### Solution:
First, simplify the denominator $\frac{6}{10}$:
$$
\frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}
$$

Now, the expression becomes:
$$
\frac{\frac{3}{5}}{\frac{3}{5}}
$$

Any number divided by itself is 1:
$$
\frac{\frac{3}{5}}{\frac{3}{5}} = 1
$$

Thus, the simplified form is:
$$
\boxed{1}
$$

---

Problem 6:

Simplify:
$$
\frac{\frac{7}{8}}{\frac{14}{16}}
$$

#### Solution:
First, simplify the denominator $\frac{14}{16}$:
$$
\frac{14}{16} = \frac{14 \div 2}{16 \div 2} = \frac{7}{8}
$$

Now, the expression becomes:
$$
\frac{\frac{7}{8}}{\frac{7}{8}}
$$

Any number divided by itself is 1:
$$
\frac{\frac{7}{8}}{\frac{7}{8}} = 1
$$

Thus, the simplified form is:
$$
\boxed{1}
$$

---

Problem 7:

Simplify:
$$
\frac{\frac{5}{6}}{\frac{10}{12}}
$$

#### Solution:
This is the same as Problem 3. Simplify the denominator $\frac{10}{12}$:
$$
\frac{10}{12} = \frac{10 \div 2}{12 \div 2} = \frac{5}{6}
$$

Now, the expression becomes:
$$
\frac{\frac{5}{6}}{\frac{5}{6}}
$$

Any number divided by itself is 1:
$$
\frac{\frac{5}{6}}{\frac{5}{6}} = 1
$$

Thus, the simplified form is:
$$
\boxed{1}
$$

---

Problem 8:

Simplify:
$$
\frac{\frac{3}{4}}{\frac{6}{8}}
$$

#### Solution:
First, simplify the denominator $\frac{6}{8}$:
$$
\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}
$$

Now, the expression becomes:
$$
\frac{\frac{3}{4}}{\frac{3}{4}}
$$

Any number divided by itself is 1:
$$
\frac{\frac{3}{4}}{\frac{3}{4}} = 1
$$

Thus, the simplified form is:
$$
\boxed{1}
$$

---

Problem 9:

Simplify:
$$
\frac{\frac{2}{3}}{\frac{4}{6}}
$$

#### Solution:
First, simplify the denominator $\frac{4}{6}$:
$$
\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}
$$

Now, the expression becomes:
$$
\frac{\frac{2}{3}}{\frac{2}{3}}
$$

Any number divided by itself is 1:
$$
\frac{\frac{2}{3}}{\frac{2}{3}} = 1
$$

Thus, the simplified form is:
$$
\boxed{1}
$$

---

Problem 10:

Simplify:
$$
\frac{\frac{5}{7}}{\frac{10}{14}}
$$

#### Solution:
First, simplify the denominator $\frac{10}{14}$:
$$
\frac{10}{14} = \frac{10 \div 2}{14 \div 2} = \frac{5}{7}
$$

Now, the expression becomes:
$$
\frac{\frac{5}{7}}{\frac{5}{7}}
$$

Any number divided by itself is 1:
$$
\frac{\frac{5}{7}}{\frac{5}{7}} = 1
$$

Thus, the simplified form is:
$$
\boxed{1}
$$

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Final Answers:

1. $\boxed{\frac{9}{10}}$
2. $\boxed{\frac{7}{6}}$
3. $\boxed{1}$
4. $\boxed{\frac{3}{2}}$
5. $\boxed{1}$
6. $\boxed{1}$
7. $\boxed{1}$
8. $\boxed{1}$
9. $\boxed{1}$
10. $\boxed{1}$