Order of Operations math worksheet with fraction problems using PEMDAS/BODMAS rules.

Let's solve each problem step by step using the PEMDAS/BODMAS rules:

> PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)
> BODMAS: Brackets, Orders, Division and Multiplication (left to right), Addition and Subtraction (left to right)

---

Problem 1:

$$
\frac{3}{8} - \left( \frac{1}{2} - \frac{3}{8} \right)
$$

Step 1: Solve inside parentheses:
$$
\frac{1}{2} - \frac{3}{8} = \frac{4}{8} - \frac{3}{8} = \frac{1}{8}
$$

Step 2: Subtract:
$$
\frac{3}{8} - \frac{1}{8} = \frac{2}{8} = \boxed{\frac{1}{4}}
$$

---

Problem 2:

$$
\left( \frac{1}{3} + 1 \right) \times \frac{1}{2} \div \frac{7}{7}
$$

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

Step 2: Simplify $\frac{7}{7} = 1$

Now:
$$
\frac{4}{3} \times \frac{1}{2} \div 1 = \frac{4}{3} \times \frac{1}{2} = \frac{4}{6} = \boxed{\frac{2}{3}}
$$

---

Problem 3:

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

Step 1: Solve inside parentheses:
$$
2 - \frac{2}{3} = \frac{6}{3} - \frac{2}{3} = \frac{4}{3}
$$

Step 2: Multiply:
$$
\frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}
$$

Step 3: Add:
$$
\frac{1}{6} + \frac{2}{3} = \frac{1}{6} + \frac{4}{6} = \frac{5}{6}
$$

Answer: $\boxed{\frac{5}{6}}$

---

Problem 4:

$$
2 - \frac{1}{4} + \frac{1}{2} \times \frac{1}{4}
$$

Step 1: Do multiplication first:
$$
\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}
$$

Now:
$$
2 - \frac{1}{4} + \frac{1}{8}
$$

Convert to eighths:
$$
2 = \frac{16}{8},\quad \frac{1}{4} = \frac{2}{8},\quad \frac{1}{8} = \frac{1}{8}
$$

So:
$$
\frac{16}{8} - \frac{2}{8} + \frac{1}{8} = \frac{15}{8} = \boxed{1\frac{7}{8}}
$$

---

Problem 5:

$$
4 \times \left( \frac{1}{2} + \frac{2}{8} \div \frac{1}{2} \right)
$$

Step 1: Inside parentheses — do division first:
$$
\frac{2}{8} \div \frac{1}{2} = \frac{1}{4} \div \frac{1}{2} = \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2}
$$

Now add:
$$
\frac{1}{2} + \frac{1}{2} = 1
$$

Now multiply:
$$
4 \times 1 = \boxed{4}
$$

---

Problem 6:

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

Step 1: Simplify fractions:
$$
\frac{4}{2} = 2,\quad \text{so } 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}
$$

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

Step 3: Add:
$$
\frac{3}{2} + \frac{1}{2} = \frac{4}{2} = \boxed{2}
$$

---

Problem 7:

$$
\frac{1}{3} \times (7 - 1)
$$

Step 1: Parentheses:
$$
7 - 1 = 6
$$

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

---

Problem 8:

$$
\frac{4}{6} - \left( \frac{2}{3} - \frac{1}{6} \right)
$$

Step 1: Simplify inside parentheses:
$$
\frac{2}{3} = \frac{4}{6},\quad \frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}
$$

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

Now:
$$
\frac{2}{3} - \frac{1}{2} = \frac{4}{6} - \frac{3}{6} = \frac{1}{6}
$$

Answer: $\boxed{\frac{1}{6}}$

---

Problem 9:

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

Step 1: Inner parentheses:
$$
3 \times \frac{1}{2} = \frac{3}{2}
$$

Step 2: Add:
$$
1 + \frac{3}{2} = \frac{2}{2} + \frac{3}{2} = \frac{5}{2}
$$

Step 3: Divide:
$$
1 \div \frac{5}{2} = 1 \times \frac{2}{5} = \boxed{\frac{2}{5}}
$$

---

Problem 10:

$$
\left(1 - \frac{1}{5}\right) \times \frac{1}{4}
$$

Step 1: Parentheses:
$$
1 - \frac{1}{5} = \frac{5}{5} - \frac{1}{5} = \frac{4}{5}
$$

Step 2: Multiply:
$$
\frac{4}{5} \times \frac{1}{4} = \frac{4}{20} = \boxed{\frac{1}{5}}
$$

---

Problem 11:

$$
1 - \left(\frac{1}{2} + 2 \times \frac{1}{8}\right)
$$

Step 1: Multiplication inside parentheses:
$$
2 \times \frac{1}{8} = \frac{2}{8} = \frac{1}{4}
$$

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

Step 3: Subtract:
$$
1 - \frac{3}{4} = \boxed{\frac{1}{4}}
$$

---

Problem 12:

$$
3 - \left(\frac{1}{5} + 1 \div \frac{1}{2}\right)
$$

Step 1: Division inside parentheses:
$$
1 \div \frac{1}{2} = 1 \times 2 = 2
$$

Step 2: Add:
$$
\frac{1}{5} + 2 = \frac{1}{5} + \frac{10}{5} = \frac{11}{5}
$$

Step 3: Subtract:
$$
3 - \frac{11}{5} = \frac{15}{5} - \frac{11}{5} = \frac{4}{5}
$$

Answer: $\boxed{\frac{4}{5}}$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $\frac{1}{4}$ |
| 2 | $\frac{2}{3}$ |
| 3 | $\frac{5}{6}$ |
| 4 | $1\frac{7}{8}$ |
| 5 | $4$ |
| 6 | $2$ |
| 7 | $2$ |
| 8 | $\frac{1}{6}$ |
| 9 | $\frac{2}{5}$ |
| 10 | $\frac{1}{5}$ |
| 11 | $\frac{1}{4}$ |
| 12 | $\frac{4}{5}$ |

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