Mixed Operations involving fractions worksheet - Free Printable

Let's solve each problem step by step, following the order given in the image. We'll break down each operation into clear steps.

---

Problem 1:

\[
\frac{1}{5} + \frac{1}{5} \times \frac{1}{2}
\]

#### Step 1: Perform the multiplication first (following the order of operations).
\[
\frac{1}{5} \times \frac{1}{2} = \frac{1 \times 1}{5 \times 2} = \frac{1}{10}
\]

#### Step 2: Add the result to \(\frac{1}{5}\).
To add fractions, we need a common denominator. The denominators are 5 and 10, so the least common denominator is 10.
\[
\frac{1}{5} = \frac{2}{10}
\]
Now add:
\[
\frac{2}{10} + \frac{1}{10} = \frac{3}{10}
\]

#### Step 3: Final answer.
\[
\boxed{\frac{3}{10}}
\]

---

Problem 2:

\[
\frac{5}{12} - \frac{7}{13} \div \frac{3}{20}
\]

#### Step 1: Perform the division first.
Division of fractions is equivalent to multiplying by the reciprocal:
\[
\frac{7}{13} \div \frac{3}{20} = \frac{7}{13} \times \frac{20}{3} = \frac{7 \times 20}{13 \times 3} = \frac{140}{39}
\]

#### Step 2: Subtract the result from \(\frac{5}{12}\).
To subtract fractions, we need a common denominator. The denominators are 12 and 39. The least common denominator is \(12 \times 39 = 468\).
Convert each fraction:
\[
\frac{5}{12} = \frac{5 \times 39}{12 \times 39} = \frac{195}{468}
\]
\[
\frac{140}{39} = \frac{140 \times 12}{39 \times 12} = \frac{1680}{468}
\]
Now subtract:
\[
\frac{195}{468} - \frac{1680}{468} = \frac{195 - 1680}{468} = \frac{-1485}{468}
\]
Simplify the fraction by finding the greatest common divisor (GCD) of 1485 and 468, which is 9:
\[
\frac{-1485 \div 9}{468 \div 9} = \frac{-165}{52}
\]

#### Step 3: Final answer.
\[
\boxed{-\frac{165}{52}}
\]

---

Problem 3:

\[
\left(1 - \frac{3}{4}\right) \times \frac{3}{7}
\]

#### Step 1: Simplify inside the parentheses.
Convert 1 to a fraction with a denominator of 4:
\[
1 = \frac{4}{4}
\]
Now subtract:
\[
\frac{4}{4} - \frac{3}{4} = \frac{4 - 3}{4} = \frac{1}{4}
\]

#### Step 2: Multiply the result by \(\frac{3}{7}\).
\[
\frac{1}{4} \times \frac{3}{7} = \frac{1 \times 3}{4 \times 7} = \frac{3}{28}
\]

#### Step 3: Final answer.
\[
\boxed{\frac{3}{28}}
\]

---

Problem 4:

\[
\left(3 \frac{7}{10} - \frac{11}{7}\right) \times \frac{8}{5} - 1 \frac{1}{7}
\]

#### Step 1: Convert mixed numbers to improper fractions.
\[
3 \frac{7}{10} = \frac{3 \times 10 + 7}{10} = \frac{37}{10}
\]
\[
1 \frac{1}{7} = \frac{1 \times 7 + 1}{7} = \frac{8}{7}
\]

#### Step 2: Subtract the fractions inside the parentheses.
The denominators are 10 and 7, so the least common denominator is \(10 \times 7 = 70\).
Convert each fraction:
\[
\frac{37}{10} = \frac{37 \times 7}{10 \times 7} = \frac{259}{70}
\]
\[
\frac{11}{7} = \frac{11 \times 10}{7 \times 10} = \frac{110}{70}
\]
Now subtract:
\[
\frac{259}{70} - \frac{110}{70} = \frac{259 - 110}{70} = \frac{149}{70}
\]

#### Step 3: Multiply the result by \(\frac{8}{5}\).
\[
\frac{149}{70} \times \frac{8}{5} = \frac{149 \times 8}{70 \times 5} = \frac{1192}{350}
\]
Simplify the fraction by finding the GCD of 1192 and 350, which is 2:
\[
\frac{1192 \div 2}{350 \div 2} = \frac{596}{175}
\]

#### Step 4: Subtract \(1 \frac{1}{7}\) (which is \(\frac{8}{7}\)) from the result.
Convert \(\frac{8}{7}\) to have a denominator of 175:
\[
\frac{8}{7} = \frac{8 \times 25}{7 \times 25} = \frac{200}{175}
\]
Now subtract:
\[
\frac{596}{175} - \frac{200}{175} = \frac{596 - 200}{175} = \frac{396}{175}
\]

#### Step 5: Final answer.
\[
\boxed{\frac{396}{175}}
\]

---

Problem 5:

\[
\left(\frac{3}{2} \times 3 \frac{1}{2}\right) \div \left(\frac{6}{5} - 1\right)
\]

#### Step 1: Convert the mixed number to an improper fraction.
\[
3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}
\]

#### Step 2: Perform the multiplication inside the first parentheses.
\[
\frac{3}{2} \times \frac{7}{2} = \frac{3 \times 7}{2 \times 2} = \frac{21}{4}
\]

#### Step 3: Simplify the expression inside the second parentheses.
Convert 1 to a fraction with a denominator of 5:
\[
1 = \frac{5}{5}
\]
Now subtract:
\[
\frac{6}{5} - \frac{5}{5} = \frac{6 - 5}{5} = \frac{1}{5}
\]

#### Step 4: Perform the division.
Division of fractions is equivalent to multiplying by the reciprocal:
\[
\frac{21}{4} \div \frac{1}{5} = \frac{21}{4} \times \frac{5}{1} = \frac{21 \times 5}{4 \times 1} = \frac{105}{4}
\]

#### Step 5: Final answer.
\[
\boxed{\frac{105}{4}}
\]

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Final Answers:

1. \(\boxed{\frac{3}{10}}\)
2. \(\boxed{-\frac{165}{52}}\)
3. \(\boxed{\frac{3}{28}}\)
4. \(\boxed{\frac{396}{175}}\)
5. \(\boxed{\frac{105}{4}}\)