Order of Operations with Decimals and Fractions (B) Worksheet for ... - Free Printable

The image you provided contains several problems involving the order of operations with decimals and fractions. Below, I will solve each problem step by step.

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Problem 1:

$$
2 \times \left( \frac{5}{4} - \frac{4}{3} \right) + 2.25
$$

#### Step 1: Simplify inside the parentheses.
$$
\frac{5}{4} - \frac{4}{3}
$$
To subtract these fractions, find a common denominator. The least common denominator (LCD) of 4 and 3 is 12.
$$
\frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}, \quad \frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12}
$$
Now subtract:
$$
\frac{15}{12} - \frac{16}{12} = \frac{15 - 16}{12} = \frac{-1}{12}
$$

#### Step 2: Multiply by 2.
$$
2 \times \left( \frac{-1}{12} \right) = \frac{2 \times (-1)}{12} = \frac{-2}{12} = \frac{-1}{6}
$$

#### Step 3: Add 2.25.
Convert 2.25 to a fraction:
$$
2.25 = \frac{225}{100} = \frac{9}{4}
$$
Now add:
$$
\frac{-1}{6} + \frac{9}{4}
$$
Find a common denominator. The LCD of 6 and 4 is 12.
$$
\frac{-1}{6} = \frac{-1 \times 2}{6 \times 2} = \frac{-2}{12}, \quad \frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12}
$$
Add the fractions:
$$
\frac{-2}{12} + \frac{27}{12} = \frac{-2 + 27}{12} = \frac{25}{12}
$$

#### Final Answer:
$$
\boxed{\frac{25}{12}}
$$

---

Problem 2:

$$
1.75 + 3.6 \times 2 + 1
$$

#### Step 1: Perform multiplication first.
$$
3.6 \times 2 = 7.2
$$

#### Step 2: Add the numbers.
$$
1.75 + 7.2 + 1
$$
Convert all to decimals for ease:
$$
1.75 + 7.2 + 1 = 9.95
$$

#### Final Answer:
$$
\boxed{9.95}
$$

---

Problem 3:

$$
\frac{2}{3} \div \left( 2.25 + \frac{1}{8} \right) \times 5
$$

#### Step 1: Simplify inside the parentheses.
Convert 2.25 to a fraction:
$$
2.25 = \frac{225}{100} = \frac{9}{4}
$$
Now add:
$$
\frac{9}{4} + \frac{1}{8}
$$
Find a common denominator. The LCD of 4 and 8 is 8.
$$
\frac{9}{4} = \frac{9 \times 2}{4 \times 2} = \frac{18}{8}, \quad \frac{1}{8} = \frac{1}{8}
$$
Add the fractions:
$$
\frac{18}{8} + \frac{1}{8} = \frac{18 + 1}{8} = \frac{19}{8}
$$

#### Step 2: Perform the division.
$$
\frac{2}{3} \div \frac{19}{8}
$$
Dividing by a fraction is the same as multiplying by its reciprocal:
$$
\frac{2}{3} \times \frac{8}{19} = \frac{2 \times 8}{3 \times 19} = \frac{16}{57}
$$

#### Step 3: Multiply by 5.
$$
\frac{16}{57} \times 5 = \frac{16 \times 5}{57} = \frac{80}{57}
$$

#### Final Answer:
$$
\boxed{\frac{80}{57}}
$$

---

Problem 4:

$$
0.8 \times \frac{11}{8} \div 2.75 \times \frac{1}{4}
$$

#### Step 1: Convert decimals to fractions.
$$
0.8 = \frac{8}{10} = \frac{4}{5}, \quad 2.75 = \frac{275}{100} = \frac{11}{4}
$$

#### Step 2: Rewrite the expression.
$$
\frac{4}{5} \times \frac{11}{8} \div \frac{11}{4} \times \frac{1}{4}
$$

#### Step 3: Perform the division.
$$
\frac{4}{5} \times \frac{11}{8} \div \frac{11}{4} = \frac{4}{5} \times \frac{11}{8} \times \frac{4}{11}
$$
Simplify:
$$
\frac{4 \times 11 \times 4}{5 \times 8 \times 11} = \frac{4 \times 4}{5 \times 8} = \frac{16}{40} = \frac{2}{5}
$$

#### Step 4: Multiply by \(\frac{1}{4}\).
$$
\frac{2}{5} \times \frac{1}{4} = \frac{2 \times 1}{5 \times 4} = \frac{2}{20} = \frac{1}{10}
$$

#### Final Answer:
$$
\boxed{\frac{1}{10}}
$$

---

Problem 5:

$$
5 \times \frac{3}{4} - 0.25 - \frac{3}{4}
$$

#### Step 1: Perform the multiplication.
$$
5 \times \frac{3}{4} = \frac{5 \times 3}{4} = \frac{15}{4}
$$

#### Step 2: Convert 0.25 to a fraction.
$$
0.25 = \frac{25}{100} = \frac{1}{4}
$$

#### Step 3: Subtract the fractions.
$$
\frac{15}{4} - \frac{1}{4} - \frac{3}{4}
$$
Since the denominators are the same, subtract the numerators:
$$
\frac{15 - 1 - 3}{4} = \frac{11}{4}
$$

#### Final Answer:
$$
\boxed{\frac{11}{4}}
$$

---

Problem 6:

$$
1 + 0.8 \times \left( \frac{3}{5} - 1.8 \right)
$$

#### Step 1: Simplify inside the parentheses.
Convert 1.8 to a fraction:
$$
1.8 = \frac{18}{10} = \frac{9}{5}
$$
Now subtract:
$$
\frac{3}{5} - \frac{9}{5} = \frac{3 - 9}{5} = \frac{-6}{5}
$$

#### Step 2: Multiply by 0.8.
Convert 0.8 to a fraction:
$$
0.8 = \frac{8}{10} = \frac{4}{5}
$$
Now multiply:
$$
\frac{4}{5} \times \frac{-6}{5} = \frac{4 \times (-6)}{5 \times 5} = \frac{-24}{25}
$$

#### Step 3: Add 1.
Convert 1 to a fraction:
$$
1 = \frac{25}{25}
$$
Now add:
$$
\frac{25}{25} + \frac{-24}{25} = \frac{25 - 24}{25} = \frac{1}{25}
$$

#### Final Answer:
$$
\boxed{\frac{1}{25}}
$$

---

Problem 7:

$$
\frac{1}{4} \times \left( \frac{3}{4} + 3.6 \right)
$$

#### Step 1: Simplify inside the parentheses.
Convert 3.6 to a fraction:
$$
3.6 = \frac{36}{10} = \frac{18}{5}
$$
Now add:
$$
\frac{3}{4} + \frac{18}{5}
$$
Find a common denominator. The LCD of 4 and 5 is 20.
$$
\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}, \quad \frac{18}{5} = \frac{18 \times 4}{5 \times 4} = \frac{72}{20}
$$
Add the fractions:
$$
\frac{15}{20} + \frac{72}{20} = \frac{15 + 72}{20} = \frac{87}{20}
$$

#### Step 2: Multiply by \(\frac{1}{4}\).
$$
\frac{1}{4} \times \frac{87}{20} = \frac{1 \times 87}{4 \times 20} = \frac{87}{80}
$$

#### Final Answer:
$$
\boxed{\frac{87}{80}}
$$

---

Problem 8:

$$
\frac{2}{3} \div \left( 4.8 - 1 \frac{3}{4} \right) \times \frac{3}{2}
$$

#### Step 1: Simplify inside the parentheses.
Convert \(1 \frac{3}{4}\) to an improper fraction:
$$
1 \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4}
$$
Convert 4.8 to a fraction:
$$
4.8 = \frac{48}{10} = \frac{24}{5}
$$
Now subtract:
$$
\frac{24}{5} - \frac{7}{4}
$$
Find a common denominator. The LCD of 5 and 4 is 20.
$$
\frac{24}{5} = \frac{24 \times 4}{5 \times 4} = \frac{96}{20}, \quad \frac{7}{4} = \frac{7 \times 5}{4 \times 5} = \frac{35}{20}
$$
Subtract the fractions:
$$
\frac{96}{20} - \frac{35}{20} = \frac{96 - 35}{20} = \frac{61}{20}
$$

#### Step 2: Perform the division.
$$
\frac{2}{3} \div \frac{61}{20}
$$
Dividing by a fraction is the same as multiplying by its reciprocal:
$$
\frac{2}{3} \times \frac{20}{61} = \frac{2 \times 20}{3 \times 61} = \frac{40}{183}
$$

#### Step 3: Multiply by \(\frac{3}{2}\).
$$
\frac{40}{183} \times \frac{3}{2} = \frac{40 \times 3}{183 \times 2} = \frac{120}{366} = \frac{20}{61}
$$

#### Final Answer:
$$
\boxed{\frac{20}{61}}
$$

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Final Answers:

1. \(\boxed{\frac{25}{12}}\)
2. \(\boxed{9.95}\)
3. \(\boxed{\frac{80}{57}}\)
4. \(\boxed{\frac{1}{10}}\)
5. \(\boxed{\frac{11}{4}}\)
6. \(\boxed{\frac{1}{25}}\)
7. \(\boxed{\frac{87}{80}}\)
8. \(\boxed{\frac{20}{61}}\)