Math worksheet focusing on fractions and order of operations with brackets.

To solve the given problems, we need to carefully follow the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) and handle fractions step by step. Let's solve each problem systematically.

---

Problem 1:

\[
(3 - \frac{1}{3})^2 - \frac{3}{5} \times \frac{1}{5} - 4^2
\]

#### Step 1: Simplify inside the parentheses
\[
3 - \frac{1}{3} = \frac{9}{3} - \frac{1}{3} = \frac{8}{3}
\]

#### Step 2: Square the result
\[
\left(\frac{8}{3}\right)^2 = \frac{64}{9}
\]

#### Step 3: Perform the multiplication
\[
\frac{3}{5} \times \frac{1}{5} = \frac{3}{25}
\]

#### Step 4: Calculate \(4^2\)
\[
4^2 = 16
\]

#### Step 5: Combine all terms
\[
\frac{64}{9} - \frac{3}{25} - 16
\]

#### Step 6: Find a common denominator for \(\frac{64}{9}\) and \(\frac{3}{25}\)
The least common denominator (LCD) of 9 and 25 is 225.
\[
\frac{64}{9} = \frac{64 \times 25}{9 \times 25} = \frac{1600}{225}
\]
\[
\frac{3}{25} = \frac{3 \times 9}{25 \times 9} = \frac{27}{225}
\]

#### Step 7: Subtract the fractions
\[
\frac{1600}{225} - \frac{27}{225} = \frac{1573}{225}
\]

#### Step 8: Subtract 16
Convert 16 to a fraction with denominator 225:
\[
16 = \frac{16 \times 225}{225} = \frac{3600}{225}
\]
\[
\frac{1573}{225} - \frac{3600}{225} = \frac{1573 - 3600}{225} = \frac{-2027}{225}
\]

#### Final Answer:
\[
\boxed{\frac{-2027}{225}}
\]

---

Problem 2:

\[
\left(\frac{2}{5} - \left(\frac{2}{3}\right)^2\right) \times \frac{1}{2} - \left(\frac{3}{2} - \frac{1}{2}\right)^2
\]

#### Step 1: Simplify \(\left(\frac{2}{3}\right)^2\)
\[
\left(\frac{2}{3}\right)^2 = \frac{4}{9}
\]

#### Step 2: Simplify inside the first set of parentheses
\[
\frac{2}{5} - \frac{4}{9}
\]
Find the LCD of 5 and 9, which is 45:
\[
\frac{2}{5} = \frac{2 \times 9}{5 \times 9} = \frac{18}{45}
\]
\[
\frac{4}{9} = \frac{4 \times 5}{9 \times 5} = \frac{20}{45}
\]
\[
\frac{2}{5} - \frac{4}{9} = \frac{18}{45} - \frac{20}{45} = \frac{-2}{45}
\]

#### Step 3: Multiply by \(\frac{1}{2}\)
\[
\left(\frac{-2}{45}\right) \times \frac{1}{2} = \frac{-2 \times 1}{45 \times 2} = \frac{-2}{90} = \frac{-1}{45}
\]

#### Step 4: Simplify inside the second set of parentheses
\[
\frac{3}{2} - \frac{1}{2} = \frac{3 - 1}{2} = \frac{2}{2} = 1
\]

#### Step 5: Square the result
\[
1^2 = 1
\]

#### Step 6: Combine the results
\[
\frac{-1}{45} - 1
\]
Convert 1 to a fraction with denominator 45:
\[
1 = \frac{45}{45}
\]
\[
\frac{-1}{45} - \frac{45}{45} = \frac{-1 - 45}{45} = \frac{-46}{45}
\]

#### Final Answer:
\[
\boxed{\frac{-46}{45}}
\]

---

Problem 3:

\[
(5 + \frac{1}{2})^2 - \frac{1}{3} \times 4^2 \times \frac{3}{4}
\]

#### Step 1: Simplify inside the parentheses
\[
5 + \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = \frac{11}{2}
\]

#### Step 2: Square the result
\[
\left(\frac{11}{2}\right)^2 = \frac{121}{4}
\]

#### Step 3: Calculate \(4^2\)
\[
4^2 = 16
\]

#### Step 4: Perform the multiplication
\[
\frac{1}{3} \times 16 \times \frac{3}{4} = \frac{1 \times 16 \times 3}{3 \times 4} = \frac{48}{12} = 4
\]

#### Step 5: Combine all terms
\[
\frac{121}{4} - 4
\]
Convert 4 to a fraction with denominator 4:
\[
4 = \frac{16}{4}
\]
\[
\frac{121}{4} - \frac{16}{4} = \frac{121 - 16}{4} = \frac{105}{4}
\]

#### Final Answer:
\[
\boxed{\frac{105}{4}}
\]

---

Problem 4:

\[
\left(\frac{3}{2} - \frac{1}{2}\right)^2 + \frac{1}{2} \left(\frac{1}{3} - \left(\frac{3}{2}\right)^2\right)
\]

#### Step 1: Simplify inside the first set of parentheses
\[
\frac{3}{2} - \frac{1}{2} = \frac{3 - 1}{2} = \frac{2}{2} = 1
\]

#### Step 2: Square the result
\[
1^2 = 1
\]

#### Step 3: Simplify inside the second set of parentheses
\[
\left(\frac{3}{2}\right)^2 = \frac{9}{4}
\]
\[
\frac{1}{3} - \frac{9}{4}
\]
Find the LCD of 3 and 4, which is 12:
\[
\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}
\]
\[
\frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12}
\]
\[
\frac{1}{3} - \frac{9}{4} = \frac{4}{12} - \frac{27}{12} = \frac{4 - 27}{12} = \frac{-23}{12}
\]

#### Step 4: Multiply by \(\frac{1}{2}\)
\[
\frac{1}{2} \times \frac{-23}{12} = \frac{1 \times -23}{2 \times 12} = \frac{-23}{24}
\]

#### Step 5: Combine the results
\[
1 + \frac{-23}{24}
\]
Convert 1 to a fraction with denominator 24:
\[
1 = \frac{24}{24}
\]
\[
1 + \frac{-23}{24} = \frac{24}{24} + \frac{-23}{24} = \frac{24 - 23}{24} = \frac{1}{24}
\]

#### Final Answer:
\[
\boxed{\frac{1}{24}}
\]

---

Problem 5:

\[
\left(\frac{2}{3} + \left(\frac{3}{4}\right)^2\right) \times \frac{1}{2} + \left(\frac{1}{3} - \frac{1}{5}\right)^2
\]

#### Step 1: Simplify \(\left(\frac{3}{4}\right)^2\)
\[
\left(\frac{3}{4}\right)^2 = \frac{9}{16}
\]

#### Step 2: Simplify inside the first set of parentheses
\[
\frac{2}{3} + \frac{9}{16}
\]
Find the LCD of 3 and 16, which is 48:
\[
\frac{2}{3} = \frac{2 \times 16}{3 \times 16} = \frac{32}{48}
\]
\[
\frac{9}{16} = \frac{9 \times 3}{16 \times 3} = \frac{27}{48}
\]
\[
\frac{2}{3} + \frac{9}{16} = \frac{32}{48} + \frac{27}{48} = \frac{59}{48}
\]

#### Step 3: Multiply by \(\frac{1}{2}\)
\[
\frac{59}{48} \times \frac{1}{2} = \frac{59 \times 1}{48 \times 2} = \frac{59}{96}
\]

#### Step 4: Simplify inside the second set of parentheses
\[
\frac{1}{3} - \frac{1}{5}
\]
Find the LCD of 3 and 5, which is 15:
\[
\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}
\]
\[
\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}
\]
\[
\frac{1}{3} - \frac{1}{5} = \frac{5}{15} - \frac{3}{15} = \frac{2}{15}
\]

#### Step 5: Square the result
\[
\left(\frac{2}{15}\right)^2 = \frac{4}{225}
\]

#### Step 6: Combine the results
\[
\frac{59}{96} + \frac{4}{225}
\]
Find the LCD of 96 and 225. The prime factorizations are:
\[
96 = 2^5 \times 3, \quad 225 = 3^2 \times 5^2
\]
The LCD is \(2^5 \times 3^2 \times 5^2 = 7200\).

Convert each fraction:
\[
\frac{59}{96} = \frac{59 \times 75}{96 \times 75} = \frac{4425}{7200}
\]
\[
\frac{4}{225} = \frac{4 \times 32}{225 \times 32} = \frac{128}{7200}
\]
\[
\frac{59}{96} + \frac{4}{225} = \frac{4425}{7200} + \frac{128}{7200} = \frac{4425 + 128}{7200} = \frac{4553}{7200}
\]

#### Final Answer:
\[
\boxed{\frac{4553}{7200}}
\]

---

Problem 6:

\[
(3 + \frac{3}{2})^2 - \frac{3}{5} - 3^2 \times \frac{2}{5}
\]

#### Step 1: Simplify inside the parentheses
\[
3 + \frac{3}{2} = \frac{6}{2} + \frac{3}{2} = \frac{9}{2}
\]

#### Step 2: Square the result
\[
\left(\frac{9}{2}\right)^2 = \frac{81}{4}
\]

#### Step 3: Calculate \(3^2\)
\[
3^2 = 9
\]

#### Step 4: Perform the multiplication
\[
9 \times \frac{2}{5} = \frac{9 \times 2}{5} = \frac{18}{5}
\]

#### Step 5: Combine all terms
\[
\frac{81}{4} - \frac{3}{5} - \frac{18}{5}
\]

#### Step 6: Find a common denominator for \(\frac{81}{4}\), \(\frac{3}{5}\), and \(\frac{18}{5}\)
The LCD of 4 and 5 is 20.
\[
\frac{81}{4} = \frac{81 \times 5}{4 \times 5} = \frac{405}{20}
\]
\[
\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}
\]
\[
\frac{18}{5} = \frac{18 \times 4}{5 \times 4} = \frac{72}{20}
\]

#### Step 7: Subtract the fractions
\[
\frac{405}{20} - \frac{12}{20} - \frac{72}{20} = \frac{405 - 12 - 72}{20} = \frac{321}{20}
\]

#### Final Answer:
\[
\boxed{\frac{321}{20}}
\]

---

Problem 7:

\[
\left(5 + \frac{1}{4}\right)^2 - \frac{1}{3} + \frac{3}{4} \times 3^2
\]

#### Step 1: Simplify inside the parentheses
\[
5 + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4}
\]

#### Step 2: Square the result
\[
\left(\frac{21}{4}\right)^2 = \frac{441}{16}
\]

#### Step 3: Calculate \(3^2\)
\[
3^2 = 9
\]

#### Step 4: Perform the multiplication
\[
\frac{3}{4} \times 9 = \frac{3 \times 9}{4} = \frac{27}{4}
\]

#### Step 5: Combine all terms
\[
\frac{441}{16} - \frac{1}{3} + \frac{27}{4}
\]

#### Step 6: Find a common denominator for \(\frac{441}{16}\), \(\frac{1}{3}\), and \(\frac{27}{4}\)
The LCD of 16, 3, and 4 is 48.
\[
\frac{441}{16} = \frac{441 \times 3}{16 \times 3} = \frac{1323}{48}
\]
\[
\frac{1}{3} = \frac{1 \times 16}{3 \times 16} = \frac{16}{48}
\]
\[
\frac{27}{4} = \frac{27 \times 12}{4 \times 12} = \frac{324}{48}
\]

#### Step 7: Combine the fractions
\[
\frac{1323}{48} - \frac{16}{48} + \frac{324}{48} = \frac{1323 - 16 + 324}{48} = \frac{1631}{48}
\]

#### Final Answer:
\[
\boxed{\frac{1631}{48}}
\]

---

Problem 8:

\[
\left(\frac{3}{4}\right)^2 - \frac{1}{2} \times \frac{3}{2} - \left(\frac{3}{2} + \frac{1}{2}\right)^2
\]

#### Step 1: Simplify \(\left(\frac{3}{4}\right)^2\)
\[
\left(\frac{3}{4}\right)^2 = \frac{9}{16}
\]

#### Step 2: Perform the multiplication
\[
\frac{1}{2} \times \frac{3}{2} = \frac{1 \times 3}{2 \times 2} = \frac{3}{4}
\]

#### Step 3: Simplify inside the third set of parentheses
\[
\frac{3}{2} + \frac{1}{2} = \frac{3 + 1}{2} = \frac{4}{2} = 2
\]

#### Step 4: Square the result
\[
2^2 = 4
\]

#### Step 5: Combine all terms
\[
\frac{9}{16} - \frac{3}{4} - 4
\]

#### Step 6: Find a common denominator for \(\frac{9}{16}\), \(\frac{3}{4}\), and 4
The LCD of 16 and 4 is 16.
\[
\frac{3}{4} = \frac{3 \times 4}{4 \times 4} = \frac{12}{16}
\]
\[
4 = \frac{4 \times 16}{16} = \frac{64}{16}
\]

#### Step 7: Subtract the fractions
\[
\frac{9}{16} - \frac{12}{16} - \frac{64}{16} = \frac{9 - 12 - 64}{16} = \frac{-67}{16}
\]

#### Final Answer:
\[
\boxed{\frac{-67}{16}}
\]

---

Problem 9:

\[
\left(\frac{1}{6} + \frac{1}{2}\right)^2 + \frac{1}{3} \left(\frac{3}{2} + \left(\frac{2}{5}\right)^2\right)
\]

#### Step 1: Simplify inside the first set of parentheses
\[
\frac{1}{6} + \frac{1}{2} = \frac{1}{6} + \frac{3}{6} = \frac{4}{6} = \frac{2}{3}
\]

#### Step 2: Square the result
\[
\left(\frac{2}{3}\right)^2 = \frac{4}{9}
\]

#### Step 3: Simplify \(\left(\frac{2}{5}\right)^2\)
\[
\left(\frac{2}{5}\right)^2 = \frac{4}{25}
\]

#### Step 4: Simplify inside the second set of parentheses
\[
\frac{3}{2} + \frac{4}{25}
\]
Find the LCD of 2 and 25, which is 50:
\[
\frac{3}{2} = \frac{3 \times 25}{2 \times 25} = \frac{75}{50}
\]
\[
\frac{4}{25} = \frac{4 \times 2}{25 \times 2} = \frac{8}{50}
\]
\[
\frac{3}{2} + \frac{4}{25} = \frac{75}{50} + \frac{8}{50} = \frac{83}{50}
\]

#### Step 5: Multiply by \(\frac{1}{3}\)
\[
\frac{1}{3} \times \frac{83}{50} = \frac{1 \times 83}{3 \times 50} = \frac{83}{150}
\]

#### Step 6: Combine the results
\[
\frac{4}{9} + \frac{83}{150}
\]

#### Step 7: Find a common denominator for \(\frac{4}{9}\) and \(\frac{83}{150}\)
The LCD of 9 and 150 is 450.
\[
\frac{4}{9} = \frac{4 \times 50}{9 \times 50} = \frac{200}{450}
\]
\[
\frac{83}{150} = \frac{83 \times 3}{150 \times 3} = \frac{249}{450}
\]

#### Step 8: Add the fractions
\[
\frac{200}{450} + \frac{249}{450} = \frac{200 + 249}{450} = \frac{449}{450}
\]

#### Final Answer:
\[
\boxed{\frac{449}{450}}
\]

---

Problem 10:

\[
\left(\frac{1}{3} + \frac{1}{3}\right)^2 - \frac{1}{5} \left(\frac{1}{5} + \left(\frac{3}{4}\right)^2\right)
\]

#### Step 1: Simplify inside the first set of parentheses
\[
\frac{1}{3} + \frac{1}{3} = \frac{2}{3}
\]

#### Step 2: Square the result
\[
\left(\frac{2}{3}\right)^2 = \frac{4}{9}
\]

#### Step 3: Simplify \(\left(\frac{3}{4}\right)^2\)
\[
\left(\frac{3}{4}\right)^2 = \frac{9}{16}
\]

#### Step 4: Simplify inside the second set of parentheses
\[
\frac{1}{5} + \frac{9}{16}
\]
Find the LCD of 5 and 16, which is 80:
\[
\frac{1}{5} = \frac{1 \times 16}{5 \times 16} = \frac{16}{80}
\]
\[
\frac{9}{16} = \frac{9 \times 5}{16 \times 5} = \frac{45}{80}
\]
\[
\frac{1}{5} + \frac{9}{16} = \frac{16}{80} + \frac{45}{80} = \frac{61}{80}
\]

#### Step 5: Multiply by \(\frac{1}{5}\)
\[
\frac{1}{5} \times \frac{61}{80} = \frac{1 \times 61}{5 \times 80} = \frac{61}{400}
\]

#### Step 6: Combine the results
\[
\frac{4}{9} - \frac{61}{400}
\]

#### Step 7: Find a common denominator for \(\frac{4}{9}\) and \(\frac{61}{400}\)
The LCD of 9 and 400 is 3600.
\[
\frac{4}{9} = \frac{4 \times 400}{9 \times 400} = \frac{1600}{3600}
\]
\[
\frac{61}{400} = \frac{61 \times 9}{400 \times 9} = \frac{549}{3600}
\]

#### Step 8: Subtract the fractions
\[
\frac{1600}{3600} - \frac{549}{3600} = \frac{1600 - 549}{3600} = \frac{1051}{3600}
\]

#### Final Answer:
\[
\boxed{\frac{1051}{3600}}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ \frac{-2027}{225} \\
2. & \ \frac{-46}{45} \\
3. & \ \frac{105}{4} \\
4. & \ \frac{1}{24} \\
5. & \ \frac{4553}{7200} \\
6. & \ \frac{321}{20} \\
7. & \ \frac{1631}{48} \\
8. & \ \frac{-67}{16} \\
9. & \ \frac{449}{450} \\
10. & \ \frac{1051}{3600}
\end{aligned}
}
\]