Order of Operations with Positive Fractions and No Exponents (Six ... - Free Printable

Let's solve each expression step by step using the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right), with special attention to fractions.

---

Problem 1:

$$
\left(\frac{1}{9} \times 3\right) + \left(\frac{7}{9} + \frac{1}{8} - \frac{3}{4}\right) \times \left(\frac{3}{8} + \frac{1}{6}\right)
$$

#### Step 1: Simplify inside parentheses.

First group:
$$
\frac{1}{9} \times 3 = \frac{3}{9} = \frac{1}{3}
$$

Second group:
$$
\frac{7}{9} + \frac{1}{8} - \frac{3}{4}
$$

Find a common denominator. LCD of 9, 8, and 4 is 72.

- $\frac{7}{9} = \frac{56}{72}$
- $\frac{1}{8} = \frac{9}{72}$
- $\frac{3}{4} = \frac{54}{72}$

Now:
$$
\frac{56}{72} + \frac{9}{72} - \frac{54}{72} = \frac{11}{72}
$$

Third group:
$$
\frac{3}{8} + \frac{1}{6}
$$

LCD of 8 and 6 is 24.

- $\frac{3}{8} = \frac{9}{24}$
- $\frac{1}{6} = \frac{4}{24}$

So:
$$
\frac{9}{24} + \frac{4}{24} = \frac{13}{24}
$$

Now plug back into original expression:

$$
\frac{1}{3} + \left(\frac{11}{72}\right) \times \left(\frac{13}{24}\right)
$$

#### Step 2: Multiply:
$$
\frac{11}{72} \times \frac{13}{24} = \frac{143}{1728}
$$

#### Step 3: Add:
$$
\frac{1}{3} + \frac{143}{1728}
$$

Convert $\frac{1}{3}$ to denominator 1728:
$$
\frac{1}{3} = \frac{576}{1728}
$$

So:
$$
\frac{576}{1728} + \frac{143}{1728} = \frac{719}{1728}
$$

✔ Answer for Problem 1: $\boxed{\frac{719}{1728}}$

---

Problem 2:

$$
\left(\left(\frac{1}{4} + \frac{1}{2}\right) + \frac{4}{3}\right) \times \frac{8}{9} - \frac{5}{6} + \frac{1}{2} \times \frac{3}{8}
$$

#### Step 1: Simplify inside parentheses.

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

Then:
$$
\frac{3}{4} + \frac{4}{3}
$$

LCD of 4 and 3 is 12:
- $\frac{3}{4} = \frac{9}{12}$
- $\frac{4}{3} = \frac{16}{12}$

So:
$$
\frac{9}{12} + \frac{16}{12} = \frac{25}{12}
$$

Now expression becomes:
$$
\frac{25}{12} \times \frac{8}{9} - \frac{5}{6} + \frac{1}{2} \times \frac{3}{8}
$$

#### Step 2: Perform multiplication (left to right)

First multiplication:
$$
\frac{25}{12} \times \frac{8}{9} = \frac{200}{108} = \frac{50}{27} \quad \text{(simplify by dividing numerator and denominator by 4)}
$$

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

Now expression:
$$
\frac{50}{27} - \frac{5}{6} + \frac{3}{16}
$$

#### Step 3: Perform addition and subtraction from left to right.

We need a common denominator for 27, 6, and 16.

Prime factors:
- 27 = $3^3$
- 6 = $2 \cdot 3$
- 16 = $2^4$

LCM = $2^4 \cdot 3^3 = 16 \cdot 27 = 432$

Convert all fractions:

- $\frac{50}{27} = \frac{50 \times 16}{432} = \frac{800}{432}$
- $\frac{5}{6} = \frac{5 \times 72}{432} = \frac{360}{432}$
- $\frac{3}{16} = \frac{3 \times 27}{432} = \frac{81}{432}$

Now:
$$
\frac{800}{432} - \frac{360}{432} + \frac{81}{432} = \frac{521}{432}
$$

✔ Answer for Problem 2: $\boxed{\frac{521}{432}}$ or $1\frac{89}{432}$

---

Problem 3:

$$
\left(\frac{2}{3} \times \frac{8}{9}\right) \times \left(\frac{4}{5} + \frac{2}{5} - 1\right) \times \left(\frac{1}{6} \div \frac{2}{3}\right)
$$

#### Step 1: Simplify each group.

First group:
$$
\frac{2}{3} \times \frac{8}{9} = \frac{16}{27}
$$

Second group:
$$
\frac{4}{5} + \frac{2}{5} - 1 = \frac{6}{5} - 1 = \frac{1}{5}
$$

Third group:
$$
\frac{1}{6} \div \frac{2}{3} = \frac{1}{6} \times \frac{3}{2} = \frac{3}{12} = \frac{1}{4}
$$

Now expression:
$$
\frac{16}{27} \times \frac{1}{5} \times \frac{1}{4}
$$

Multiply step by step:

$$
\frac{16}{27} \times \frac{1}{5} = \frac{16}{135}
$$

Then:
$$
\frac{16}{135} \times \frac{1}{4} = \frac{16}{540} = \frac{4}{135} \quad \text{(divide numerator and denominator by 4)}
$$

✔ Answer for Problem 3: $\boxed{\frac{4}{135}}$

---

Problem 4:

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

#### Step 1: Simplify inside parentheses.

$$
\frac{4}{5} - \frac{1}{3}
$$

LCD of 5 and 3 is 15:
- $\frac{4}{5} = \frac{12}{15}$
- $\frac{1}{3} = \frac{5}{15}$

So:
$$
\frac{12}{15} - \frac{5}{15} = \frac{7}{15}
$$

Also:
$$
\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}
$$

Now expression:
$$
\frac{3}{5} + \frac{3}{4} \times \frac{7}{15} \div \frac{1}{6} + \frac{1}{8}
$$

#### Step 2: Multiplication and division from left to right.

First: $\frac{3}{4} \times \frac{7}{15} = \frac{21}{60} = \frac{7}{20}$

Then: $\frac{7}{20} \div \frac{1}{6} = \frac{7}{20} \times \frac{6}{1} = \frac{42}{20} = \frac{21}{10}$

Now expression:
$$
\frac{3}{5} + \frac{21}{10} + \frac{1}{8}
$$

#### Step 3: Add all terms.

Find LCD of 5, 10, 8 → LCM of 5, 10, 8.

- 5 = 5
- 10 = 2 × 5
- 8 = 2³

LCM = $2^3 \times 5 = 40$

Convert:

- $\frac{3}{5} = \frac{24}{40}$
- $\frac{21}{10} = \frac{84}{40}$
- $\frac{1}{8} = \frac{5}{40}$

Add:
$$
\frac{24}{40} + \frac{84}{40} + \frac{5}{40} = \frac{113}{40}
$$

✔ Answer for Problem 4: $\boxed{\frac{113}{40}}$ or $2\frac{33}{40}$

---

✔ Final Answers:

1. $\boxed{\frac{719}{1728}}$
2. $\boxed{\frac{521}{432}}$
3. $\boxed{\frac{4}{135}}$
4. $\boxed{\frac{113}{40}}$

Let me know if you'd like these as mixed numbers or decimals!