Order of Operations with Fractions Worksheets - Math Monks - Free Printable

Let's solve each of these Order of Operations problems step-by-step using PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). We'll work through all 8 problems carefully.

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Problem 1:

$$
\left(\frac{2}{9} \div \frac{1}{3} + \frac{1}{2}\right) \times \frac{2}{5} - \frac{1}{5}
$$

#### Step 1: Simplify inside the parentheses.
- $\frac{2}{9} \div \frac{1}{3} = \frac{2}{9} \times \frac{3}{1} = \frac{6}{9} = \frac{2}{3}$

Now:
$$
\left(\frac{2}{3} + \frac{1}{2}\right)
$$

Find common denominator (6):
$$
\frac{2}{3} = \frac{4}{6},\quad \frac{1}{2} = \frac{3}{6} \Rightarrow \frac{4}{6} + \frac{3}{6} = \frac{7}{6}
$$

So now:
$$
\frac{7}{6} \times \frac{2}{5} - \frac{1}{5}
$$

#### Step 2: Multiply
$$
\frac{7}{6} \times \frac{2}{5} = \frac{14}{30} = \frac{7}{15}
$$

#### Step 3: Subtract
$$
\frac{7}{15} - \frac{1}{5} = \frac{7}{15} - \frac{3}{15} = \frac{4}{15}
$$

✔ Answer: $\boxed{\frac{4}{15}}$

---

Problem 2:

$$
\frac{2}{5} \div \left[\left(\frac{1}{5} - \frac{1}{9} + \frac{4}{9}\right) \times \frac{1}{2}\right]
$$

#### Step 1: Simplify inside the brackets.
First, simplify inside the parentheses:
$$
\frac{1}{5} - \frac{1}{9} + \frac{4}{9}
$$

Combine like terms:
$$
-\frac{1}{9} + \frac{4}{9} = \frac{3}{9} = \frac{1}{3}
$$

Now:
$$
\frac{1}{5} + \frac{1}{3}
$$

Common denominator (15):
$$
\frac{1}{5} = \frac{3}{15},\quad \frac{1}{3} = \frac{5}{15} \Rightarrow \frac{3}{15} + \frac{5}{15} = \frac{8}{15}
$$

Now multiply by $\frac{1}{2}$:
$$
\frac{8}{15} \times \frac{1}{2} = \frac{8}{30} = \frac{4}{15}
$$

Now divide:
$$
\frac{2}{5} \div \frac{4}{15} = \frac{2}{5} \times \frac{15}{4} = \frac{30}{20} = \frac{3}{2}
$$

✔ Answer: $\boxed{\frac{3}{2}}$

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Problem 3:

$$
\frac{1}{2} \div \frac{1}{4} + \frac{3}{8} \times \frac{1}{5} - \frac{1}{8}
$$

#### Step 1: Division and multiplication first (left to right)

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

Now expression becomes:
$$
2 + \frac{3}{40} - \frac{1}{8}
$$

Convert to common denominator (40):
- $2 = \frac{80}{40}$
- $\frac{1}{8} = \frac{5}{40}$

So:
$$
\frac{80}{40} + \frac{3}{40} - \frac{5}{40} = \frac{78}{40} = \frac{39}{20}
$$

✔ Answer: $\boxed{\frac{39}{20}}$

---

Problem 4:

$$
\left(\frac{1}{5} - \frac{1}{8}\right) \times \left(\frac{1}{9} + \frac{1}{6}\right) \div \frac{1}{3}
$$

#### Step 1: Simplify inside parentheses

Left:
$$
\frac{1}{5} - \frac{1}{8} = \frac{8 - 5}{40} = \frac{3}{40}
$$

Right:
$$
\frac{1}{9} + \frac{1}{6} = \frac{2 + 3}{18} = \frac{5}{18}
$$

Now:
$$
\frac{3}{40} \times \frac{5}{18} = \frac{15}{720} = \frac{1}{48}
$$

Now divide by $\frac{1}{3}$:
$$
\frac{1}{48} \div \frac{1}{3} = \frac{1}{48} \times \frac{3}{1} = \frac{3}{48} = \frac{1}{16}
$$

✔ Answer: $\boxed{\frac{1}{16}}$

---

Problem 5:

$$
\left(\frac{2}{3} + \frac{7}{9}\right) \times \left(\frac{7}{8} - \frac{1}{4}\right) \div \frac{1}{2}
$$

#### Step 1: Simplify parentheses

Left:
$$
\frac{2}{3} + \frac{7}{9} = \frac{6}{9} + \frac{7}{9} = \frac{13}{9}
$$

Right:
$$
\frac{7}{8} - \frac{1}{4} = \frac{7}{8} - \frac{2}{8} = \frac{5}{8}
$$

Now:
$$
\frac{13}{9} \times \frac{5}{8} = \frac{65}{72}
$$

Now divide by $\frac{1}{2}$:
$$
\frac{65}{72} \div \frac{1}{2} = \frac{65}{72} \times 2 = \frac{130}{72} = \frac{65}{36}
$$

✔ Answer: $\boxed{\frac{65}{36}}$

---

Problem 6:

$$
\left(\frac{1}{2} \times \frac{3}{8} - \frac{1}{8}\right) \div \frac{3}{4} + \frac{5}{8}
$$

#### Step 1: Inside parentheses

First: $\frac{1}{2} \times \frac{3}{8} = \frac{3}{16}$

Then: $\frac{3}{16} - \frac{1}{8} = \frac{3}{16} - \frac{2}{16} = \frac{1}{16}$

Now divide by $\frac{3}{4}$:
$$
\frac{1}{16} \div \frac{3}{4} = \frac{1}{16} \times \frac{4}{3} = \frac{4}{48} = \frac{1}{12}
$$

Now add $\frac{5}{8}$:
$$
\frac{1}{12} + \frac{5}{8}
$$

LCM of 12 and 8 is 24:
- $\frac{1}{12} = \frac{2}{24}$
- $\frac{5}{8} = \frac{15}{24}$

Sum: $\frac{17}{24}$

✔ Answer: $\boxed{\frac{17}{24}}$

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Problem 7:

$$
\frac{1}{2} \times \left(\frac{7}{9} \div \frac{1}{6} - \frac{5}{6} + \frac{3}{4}\right)
$$

#### Step 1: Inside parentheses

First: $\frac{7}{9} \div \frac{1}{6} = \frac{7}{9} \times 6 = \frac{42}{9} = \frac{14}{3}$

Now:
$$
\frac{14}{3} - \frac{5}{6} + \frac{3}{4}
$$

Convert all to common denominator (LCM of 3,6,4 is 12):

- $\frac{14}{3} = \frac{56}{12}$
- $\frac{5}{6} = \frac{10}{12}$
- $\frac{3}{4} = \frac{9}{12}$

Now:
$$
\frac{56}{12} - \frac{10}{12} + \frac{9}{12} = \frac{55}{12}
$$

Now multiply by $\frac{1}{2}$:
$$
\frac{1}{2} \times \frac{55}{12} = \frac{55}{24}
$$

✔ Answer: $\boxed{\frac{55}{24}}$

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Problem 8:

$$
\left(\frac{3}{8} \times \frac{2}{3}\right) \div \frac{2}{5} + \frac{1}{2} - \frac{3}{4}
$$

#### Step 1: Simplify inside parentheses

$\frac{3}{8} \times \frac{2}{3} = \frac{6}{24} = \frac{1}{4}$

Now divide by $\frac{2}{5}$:
$$
\frac{1}{4} \div \frac{2}{5} = \frac{1}{4} \times \frac{5}{2} = \frac{5}{8}
$$

Now:
$$
\frac{5}{8} + \frac{1}{2} - \frac{3}{4}
$$

Convert to common denominator (8):

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

So:
$$
\frac{5}{8} + \frac{4}{8} - \frac{6}{8} = \frac{3}{8}
$$

✔ Answer: $\boxed{\frac{3}{8}}$

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✔ Final Answers:

1. $\boxed{\frac{4}{15}}$
2. $\boxed{\frac{3}{2}}$
3. $\boxed{\frac{39}{20}}$
4. $\boxed{\frac{1}{16}}$
5. $\boxed{\frac{65}{36}}$
6. $\boxed{\frac{17}{24}}$
7. $\boxed{\frac{55}{24}}$
8. $\boxed{\frac{3}{8}}$

Let me know if you'd like a printable version or explanation for any specific step!