Math worksheet on fractions and order of operations with brackets.

Let's solve each of these fraction problems step by step, following the order of operations (PEMDAS):
Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

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Problem 1:

$$
\frac{1}{2} - \frac{1}{6}\left(\frac{3}{2} + \frac{1}{2}\right)
$$

Step 1: Simplify inside the parentheses:
$$
\frac{3}{2} + \frac{1}{2} = \frac{4}{2} = 2
$$

Step 2: Multiply:
$$
\frac{1}{6} \times 2 = \frac{2}{6} = \frac{1}{3}
$$

Step 3: Subtract:
$$
\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}
$$

✔ Answer: $ \frac{1}{6} $

---

Problem 2:

$$
\left(\frac{2}{5} + \frac{1}{6}\right) \times \frac{1}{4} + \frac{3}{5}
$$

Step 1: Add inside parentheses. Find common denominator (LCM of 5 and 6 is 30):
$$
\frac{2}{5} = \frac{12}{30},\quad \frac{1}{6} = \frac{5}{30} \Rightarrow \frac{12}{30} + \frac{5}{30} = \frac{17}{30}
$$

Step 2: Multiply:
$$
\frac{17}{30} \times \frac{1}{4} = \frac{17}{120}
$$

Step 3: Add:
$$
\frac{17}{120} + \frac{3}{5} = \frac{17}{120} + \frac{72}{120} = \frac{89}{120}
$$

✔ Answer: $ \frac{89}{120} $

---

Problem 3:

$$
(20 \div 5 - \frac{1}{2}) \times \frac{1}{4}
$$

Step 1: Division first:
$$
20 \div 5 = 4
$$

Step 2: Subtract:
$$
4 - \frac{1}{2} = \frac{8}{2} - \frac{1}{2} = \frac{7}{2}
$$

Step 3: Multiply:
$$
\frac{7}{2} \times \frac{1}{4} = \frac{7}{8}
$$

✔ Answer: $ \frac{7}{8} $

---

Problem 4:

$$
\left(\frac{1}{6} + \frac{1}{2}\right) \times \frac{1}{3} + \frac{2}{5}
$$

Step 1: Add inside parentheses:
$$
\frac{1}{6} + \frac{1}{2} = \frac{1}{6} + \frac{3}{6} = \frac{4}{6} = \frac{2}{3}
$$

Step 2: Multiply:
$$
\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}
$$

Step 3: Add:
$$
\frac{2}{9} + \frac{2}{5} = \text{LCM of 9 and 5 is 45} \\
\frac{2}{9} = \frac{10}{45},\quad \frac{2}{5} = \frac{18}{45} \Rightarrow \frac{10}{45} + \frac{18}{45} = \frac{28}{45}
$$

✔ Answer: $ \frac{28}{45} $

---

Problem 5:

$$
8\left(\frac{1}{3} + \frac{1}{4}\right) \div 4
$$

Step 1: Add inside parentheses:
$$
\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}
$$

Step 2: Multiply:
$$
8 \times \frac{7}{12} = \frac{56}{12} = \frac{14}{3}
$$

Step 3: Divide by 4:
$$
\frac{14}{3} \div 4 = \frac{14}{3} \times \frac{1}{4} = \frac{14}{12} = \frac{7}{6}
$$

✔ Answer: $ \frac{7}{6} $ or $ 1\frac{1}{6} $

---

Problem 6:

$$
(6 \div 3 - \frac{2}{3}) \times \frac{3}{4}
$$

Step 1: Division:
$$
6 \div 3 = 2
$$

Step 2: Subtract:
$$
2 - \frac{2}{3} = \frac{6}{3} - \frac{2}{3} = \frac{4}{3}
$$

Step 3: Multiply:
$$
\frac{4}{3} \times \frac{3}{4} = \frac{12}{12} = 1
$$

✔ Answer: $ 1 $

---

Problem 7:

$$
\left(\frac{1}{3} + \frac{3}{4}\right) \times \frac{1}{2} + \frac{3}{5}
$$

Step 1: Add inside parentheses:
$$
\frac{1}{3} + \frac{3}{4} = \frac{4}{12} + \frac{9}{12} = \frac{13}{12}
$$

Step 2: Multiply:
$$
\frac{13}{12} \times \frac{1}{2} = \frac{13}{24}
$$

Step 3: Add:
$$
\frac{13}{24} + \frac{3}{5} = \text{LCM of 24 and 5 is 120} \\
\frac{13}{24} = \frac{65}{120},\quad \frac{3}{5} = \frac{72}{120} \Rightarrow \frac{65}{120} + \frac{72}{120} = \frac{137}{120}
$$

✔ Answer: $ \frac{137}{120} $ or $ 1\frac{17}{120} $

---

Problem 8:

$$
28\left(\frac{1}{5} + \frac{1}{3}\right) \div 4
$$

Step 1: Add inside parentheses:
$$
\frac{1}{5} + \frac{1}{3} = \frac{3}{15} + \frac{5}{15} = \frac{8}{15}
$$

Step 2: Multiply:
$$
28 \times \frac{8}{15} = \frac{224}{15}
$$

Step 3: Divide by 4:
$$
\frac{224}{15} \div 4 = \frac{224}{15} \times \frac{1}{4} = \frac{224}{60} = \frac{56}{15}
$$

✔ Answer: $ \frac{56}{15} $ or $ 3\frac{11}{15} $

---

Problem 9:

$$
32\left(\frac{2}{5} + \frac{1}{4}\right) \div 4
$$

Step 1: Add inside parentheses:
$$
\frac{2}{5} + \frac{1}{4} = \frac{8}{20} + \frac{5}{20} = \frac{13}{20}
$$

Step 2: Multiply:
$$
32 \times \frac{13}{20} = \frac{416}{20} = \frac{104}{5}
$$

Step 3: Divide by 4:
$$
\frac{104}{5} \div 4 = \frac{104}{5} \times \frac{1}{4} = \frac{104}{20} = \frac{26}{5}
$$

✔ Answer: $ \frac{26}{5} $ or $ 5\frac{1}{5} $

---

Problem 10:

$$
(36 \div 4 - \frac{1}{2}) \times \frac{2}{3}
$$

Step 1: Division:
$$
36 \div 4 = 9
$$

Step 2: Subtract:
$$
9 - \frac{1}{2} = \frac{18}{2} - \frac{1}{2} = \frac{17}{2}
$$

Step 3: Multiply:
$$
\frac{17}{2} \times \frac{2}{3} = \frac{34}{6} = \frac{17}{3}
$$

✔ Answer: $ \frac{17}{3} $ or $ 5\frac{2}{3} $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $ \frac{1}{6} $ |
| 2 | $ \frac{89}{120} $ |
| 3 | $ \frac{7}{8} $ |
| 4 | $ \frac{28}{45} $ |
| 5 | $ \frac{7}{6} $ |
| 6 | $ 1 $ |
| 7 | $ \frac{137}{120} $ |
| 8 | $ \frac{56}{15} $ |
| 9 | $ \frac{26}{5} $ |
| 10 | $ \frac{17}{3} $ |

Let me know if you'd like these in mixed number form or simplified further!