Solve - Exponents and order of operations;complex fractions - Free Printable

Let's solve each of these problems step by step. These are all complex fractions or mixed number operations, involving addition, subtraction, and division of fractions.

We'll go through each one carefully.

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67.

$$
\frac{3}{\frac{4}{5}} \div \frac{6}{\frac{15}{6}}
$$

Wait — the expression is written as:

$$
\frac{3}{\frac{4}{5}} \quad \text{and} \quad \frac{6}{\frac{15}{6}}
$$

But it seems like this might be a division of two complex fractions?

Actually, looking at the formatting, it appears that each problem is a single fraction with a numerator and denominator that are themselves fractions or mixed numbers.

So let’s interpret each problem as:

> Problem 67: $\frac{3}{\frac{4}{5}}$ divided by $\frac{6}{\frac{15}{6}}$? No — actually, from layout, it seems each item (67–82) is a single complex fraction, like:

For example:

- 67: $\frac{3}{\frac{4}{5}}$
- 68: $\frac{5}{\frac{8}{5}}$
- etc.

But then there's more: some have expressions like $3 + \frac{1}{4}$ in the numerator.

Let me re-express them clearly based on standard format.

Looking again:

It looks like each problem is a complex fraction, where both numerator and denominator may be fractions or mixed numbers.

So for example:

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67.

$$
\frac{3}{\frac{4}{5}} = 3 \div \frac{4}{5} = 3 \times \frac{5}{4} = \frac{15}{4}
$$

✔ Answer: $\boxed{\frac{15}{4}}$

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68.

$$
\frac{5}{\frac{8}{5}} = 5 \div \frac{8}{5} = 5 \times \frac{5}{8} = \frac{25}{8}
$$

✔ Answer: $\boxed{\frac{25}{8}}$

---

69.

$$
\frac{\frac{15}{16}}{\frac{5}{6}} = \frac{15}{16} \div \frac{5}{6} = \frac{15}{16} \times \frac{6}{5} = \frac{90}{80} = \frac{9}{8}
$$

✔ Answer: $\boxed{\frac{9}{8}}$

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70.

$$
\frac{\frac{21}{4}}{\frac{4}{7}} = \frac{21}{4} \div \frac{4}{7} = \frac{21}{4} \times \frac{7}{4} = \frac{147}{16}
$$

✔ Answer: $\boxed{\frac{147}{16}}$

---

71.

$$
\frac{3 + \frac{1}{4}}{\frac{2}{3}} = \frac{\frac{12}{4} + \frac{1}{4}}{\frac{2}{3}} = \frac{\frac{13}{4}}{\frac{2}{3}} = \frac{13}{4} \div \frac{2}{3} = \frac{13}{4} \times \frac{3}{2} = \frac{39}{8}
$$

✔ Answer: $\boxed{\frac{39}{8}}$

---

72.

$$
\frac{4 + \frac{1}{2}}{\frac{3}{5}} = \frac{\frac{8}{2} + \frac{1}{2}}{\frac{3}{5}} = \frac{\frac{9}{2}}{\frac{3}{5}} = \frac{9}{2} \div \frac{3}{5} = \frac{9}{2} \times \frac{5}{3} = \frac{45}{6} = \frac{15}{2}
$$

✔ Answer: $\boxed{\frac{15}{2}}$

---

73.

$$
\frac{5 - \frac{3}{4}}{\frac{5}{6}} = \frac{\frac{20}{4} - \frac{3}{4}}{\frac{5}{6}} = \frac{\frac{17}{4}}{\frac{5}{6}} = \frac{17}{4} \div \frac{5}{6} = \frac{17}{4} \times \frac{6}{5} = \frac{102}{20} = \frac{51}{10}
$$

✔ Answer: $\boxed{\frac{51}{10}}$

---

74.

$$
\frac{4 - \frac{3}{5}}{\frac{7}{10}} = \frac{\frac{20}{5} - \frac{3}{5}}{\frac{7}{10}} = \frac{\frac{17}{5}}{\frac{7}{10}} = \frac{17}{5} \div \frac{7}{10} = \frac{17}{5} \times \frac{10}{7} = \frac{170}{35} = \frac{34}{7}
$$

✔ Answer: $\boxed{\frac{34}{7}}$

---

75.

$$
\frac{\frac{2}{3} + \frac{3}{4}}{4 - \frac{1}{3}} = \frac{\frac{8}{12} + \frac{9}{12}}{\frac{12}{3} - \frac{1}{3}} = \frac{\frac{17}{12}}{\frac{11}{3}} = \frac{17}{12} \div \frac{11}{3} = \frac{17}{12} \times \frac{3}{11} = \frac{51}{132} = \frac{17}{44}
$$

✔ Answer: $\boxed{\frac{17}{44}}$

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76.

$$
\frac{\frac{5}{6} + \frac{2}{3}}{3 - \frac{1}{2}} = \frac{\frac{5}{6} + \frac{4}{6}}{\frac{6}{2} - \frac{1}{2}} = \frac{\frac{9}{6}}{\frac{5}{2}} = \frac{3}{2} \div \frac{5}{2} = \frac{3}{2} \times \frac{2}{5} = \frac{6}{10} = \frac{3}{5}
$$

✔ Answer: $\boxed{\frac{3}{5}}$

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77.

$$
\frac{\frac{3}{4} - \frac{1}{3}}{\frac{5}{6} + 2} = \frac{\frac{9}{12} - \frac{4}{12}}{\frac{5}{6} + \frac{12}{6}} = \frac{\frac{5}{12}}{\frac{17}{6}} = \frac{5}{12} \div \frac{17}{6} = \frac{5}{12} \times \frac{6}{17} = \frac{30}{204} = \frac{5}{34}
$$

✔ Answer: $\boxed{\frac{5}{34}}$

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78.

$$
\frac{\frac{2}{3} + \frac{1}{6}}{\frac{3}{4} - \frac{5}{8}} = \frac{\frac{4}{6} + \frac{1}{6}}{\frac{6}{8} - \frac{5}{8}} = \frac{\frac{5}{6}}{\frac{1}{8}} = \frac{5}{6} \div \frac{1}{8} = \frac{5}{6} \times 8 = \frac{40}{6} = \frac{20}{3}
$$

✔ Answer: $\boxed{\frac{20}{3}}$

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79.

$$
\frac{1\frac{1}{2} - 2\frac{2}{3}}{3\frac{1}{4} + 1\frac{1}{6}}
$$

Convert to improper fractions:

- $1\frac{1}{2} = \frac{3}{2}$
- $2\frac{2}{3} = \frac{8}{3}$
- $3\frac{1}{4} = \frac{13}{4}$
- $1\frac{1}{6} = \frac{7}{6}$

Numerator:
$$
\frac{3}{2} - \frac{8}{3} = \frac{9}{6} - \frac{16}{6} = -\frac{7}{6}
$$

Denominator:
$$
\frac{13}{4} + \frac{7}{6} = \frac{39}{12} + \frac{14}{12} = \frac{53}{12}
$$

Now divide:
$$
-\frac{7}{6} \div \frac{53}{12} = -\frac{7}{6} \times \frac{12}{53} = -\frac{84}{318} = -\frac{14}{53}
$$

✔ Answer: $\boxed{-\frac{14}{53}}$

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80.

$$
\frac{1\frac{1}{3} - 4\frac{1}{2}}{2\frac{1}{4} + 3\frac{1}{8}}
$$

Convert:

- $1\frac{1}{3} = \frac{4}{3}$
- $4\frac{1}{2} = \frac{9}{2}$
- $2\frac{1}{4} = \frac{9}{4}$
- $3\frac{1}{8} = \frac{25}{8}$

Numerator:
$$
\frac{4}{3} - \frac{9}{2} = \frac{8}{6} - \frac{27}{6} = -\frac{19}{6}
$$

Denominator:
$$
\frac{9}{4} + \frac{25}{8} = \frac{18}{8} + \frac{25}{8} = \frac{43}{8}
$$

Divide:
$$
-\frac{19}{6} \div \frac{43}{8} = -\frac{19}{6} \times \frac{8}{43} = -\frac{152}{258} = -\frac{76}{129}
$$

✔ Answer: $\boxed{-\frac{76}{129}}$

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81.

$$
\frac{2\frac{1}{6} + 4\frac{2}{3}}{2\frac{1}{3} - 5\frac{1}{4}}
$$

Convert:

- $2\frac{1}{6} = \frac{13}{6}$
- $4\frac{2}{3} = \frac{14}{3}$
- $2\frac{1}{3} = \frac{7}{3}$
- $5\frac{1}{4} = \frac{21}{4}$

Numerator:
$$
\frac{13}{6} + \frac{14}{3} = \frac{13}{6} + \frac{28}{6} = \frac{41}{6}
$$

Denominator:
$$
\frac{7}{3} - \frac{21}{4} = \frac{28}{12} - \frac{63}{12} = -\frac{35}{12}
$$

Divide:
$$
\frac{41}{6} \div \left(-\frac{35}{12}\right) = \frac{41}{6} \times \left(-\frac{12}{35}\right) = -\frac{492}{210} = -\frac{82}{35}
$$

✔ Answer: $\boxed{-\frac{82}{35}}$

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82.

$$
\frac{1\frac{1}{2} + 2\frac{2}{3}}{1\frac{1}{2} - 2\frac{1}{3}}
$$

Convert:

- $1\frac{1}{2} = \frac{3}{2}$
- $2\frac{2}{3} = \frac{8}{3}$
- $2\frac{1}{3} = \frac{7}{3}$

Numerator:
$$
\frac{3}{2} + \frac{8}{3} = \frac{9}{6} + \frac{16}{6} = \frac{25}{6}
$$

Denominator:
$$
\frac{3}{2} - \frac{7}{3} = \frac{9}{6} - \frac{14}{6} = -\frac{5}{6}
$$

Divide:
$$
\frac{25}{6} \div \left(-\frac{5}{6}\right) = \frac{25}{6} \times \left(-\frac{6}{5}\right) = -\frac{150}{30} = -5
$$

✔ Answer: $\boxed{-5}$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 67 | $\frac{15}{4}$ |
| 68 | $\frac{25}{8}$ |
| 69 | $\frac{9}{8}$ |
| 70 | $\frac{147}{16}$ |
| 71 | $\frac{39}{8}$ |
| 72 | $\frac{15}{2}$ |
| 73 | $\frac{51}{10}$ |
| 74 | $\frac{34}{7}$ |
| 75 | $\frac{17}{44}$ |
| 76 | $\frac{3}{5}$ |
| 77 | $\frac{5}{34}$ |
| 78 | $\frac{20}{3}$ |
| 79 | $-\frac{14}{53}$ |
| 80 | $-\frac{76}{129}$ |
| 81 | $-\frac{82}{35}$ |
| 82 | $-5$ |

Let me know if you want any of these converted to mixed numbers!