Fractions - Four Operations Worksheet - Free Printable

Problem Analysis:

We are tasked with solving a series of mixed operations involving fractions and then answering an exam question about the number of men wearing blue shirts in a room.

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Mixed Operations Solutions:

#### a) $\frac{1}{5} + \frac{1}{5} \times \frac{1}{2}$
1. Perform the multiplication first (following the order of operations):
$$
\frac{1}{5} \times \frac{1}{2} = \frac{1 \times 1}{5 \times 2} = \frac{1}{10}
$$
2. Add the result to $\frac{1}{5}$:
$$
\frac{1}{5} + \frac{1}{10}
$$
To add these fractions, find a common denominator. The least common denominator (LCD) of 5 and 10 is 10:
$$
\frac{1}{5} = \frac{2}{10}
$$
So:
$$
\frac{1}{5} + \frac{1}{10} = \frac{2}{10} + \frac{1}{10} = \frac{3}{10}
$$

Answer: $\boxed{\frac{3}{10}}$

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#### b) $\frac{11}{15} \div \frac{4}{5} - \frac{3}{4}$
1. Perform the division first:
$$
\frac{11}{15} \div \frac{4}{5} = \frac{11}{15} \times \frac{5}{4} = \frac{11 \times 5}{15 \times 4} = \frac{55}{60}
$$
Simplify $\frac{55}{60}$ by dividing numerator and denominator by their greatest common divisor (GCD), which is 5:
$$
\frac{55}{60} = \frac{11}{12}
$$
2. Subtract $\frac{3}{4}$ from $\frac{11}{12}$:
$$
\frac{11}{12} - \frac{3}{4}
$$
Find a common denominator. The LCD of 12 and 4 is 12:
$$
\frac{3}{4} = \frac{9}{12}
$$
So:
$$
\frac{11}{12} - \frac{3}{4} = \frac{11}{12} - \frac{9}{12} = \frac{2}{12} = \frac{1}{6}
$$

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

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#### c) $\left( \frac{3}{7} + \frac{1}{14} \right)^2$
1. Add the fractions inside the parentheses:
$$
\frac{3}{7} + \frac{1}{14}
$$
Find a common denominator. The LCD of 7 and 14 is 14:
$$
\frac{3}{7} = \frac{6}{14}
$$
So:
$$
\frac{3}{7} + \frac{1}{14} = \frac{6}{14} + \frac{1}{14} = \frac{7}{14} = \frac{1}{2}
$$
2. Square the result:
$$
\left( \frac{1}{2} \right)^2 = \frac{1}{4}
$$

Answer: $\boxed{\frac{1}{4}}$

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#### d) $\frac{5}{12} - \frac{7}{13} \div \frac{3}{26}$
1. Perform the division first:
$$
\frac{7}{13} \div \frac{3}{26} = \frac{7}{13} \times \frac{26}{3} = \frac{7 \times 26}{13 \times 3} = \frac{182}{39}
$$
Simplify $\frac{182}{39}$ by dividing numerator and denominator by their GCD, which is 13:
$$
\frac{182}{39} = \frac{14}{3}
$$
2. Subtract $\frac{14}{3}$ from $\frac{5}{12}$:
$$
\frac{5}{12} - \frac{14}{3}
$$
Find a common denominator. The LCD of 12 and 3 is 12:
$$
\frac{14}{3} = \frac{56}{12}
$$
So:
$$
\frac{5}{12} - \frac{14}{3} = \frac{5}{12} - \frac{56}{12} = \frac{5 - 56}{12} = \frac{-51}{12}
$$
Simplify $\frac{-51}{12}$ by dividing numerator and denominator by their GCD, which is 3:
$$
\frac{-51}{12} = \frac{-17}{4}
$$

Answer: $\boxed{-\frac{17}{4}}$

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#### e) $\frac{11}{14} + \frac{3}{4} \times \frac{20}{21} + \frac{15}{8}$
1. Perform the multiplication first:
$$
\frac{3}{4} \times \frac{20}{21} = \frac{3 \times 20}{4 \times 21} = \frac{60}{84}
$$
Simplify $\frac{60}{84}$ by dividing numerator and denominator by their GCD, which is 12:
$$
\frac{60}{84} = \frac{5}{7}
$$
2. Add the fractions:
$$
\frac{11}{14} + \frac{5}{7} + \frac{15}{8}
$$
Find a common denominator. The LCD of 14, 7, and 8 is 56:
$$
\frac{11}{14} = \frac{44}{56}, \quad \frac{5}{7} = \frac{40}{56}, \quad \frac{15}{8} = \frac{105}{56}
$$
So:
$$
\frac{11}{14} + \frac{5}{7} + \frac{15}{8} = \frac{44}{56} + \frac{40}{56} + \frac{105}{56} = \frac{44 + 40 + 105}{56} = \frac{189}{56}
$$
Simplify $\frac{189}{56}$ by dividing numerator and denominator by their GCD, which is 7:
$$
\frac{189}{56} = \frac{27}{8}
$$

Answer: $\boxed{\frac{27}{8}}$

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#### f) $\frac{12}{5} \div \frac{4}{9} - \frac{29}{20} \div \frac{3}{10}$
1. Perform the first division:
$$
\frac{12}{5} \div \frac{4}{9} = \frac{12}{5} \times \frac{9}{4} = \frac{12 \times 9}{5 \times 4} = \frac{108}{20}
$$
Simplify $\frac{108}{20}$ by dividing numerator and denominator by their GCD, which is 4:
$$
\frac{108}{20} = \frac{27}{5}
$$
2. Perform the second division:
$$
\frac{29}{20} \div \frac{3}{10} = \frac{29}{20} \times \frac{10}{3} = \frac{29 \times 10}{20 \times 3} = \frac{290}{60}
$$
Simplify $\frac{290}{60}$ by dividing numerator and denominator by their GCD, which is 10:
$$
\frac{290}{60} = \frac{29}{6}
$$
3. Subtract the results:
$$
\frac{27}{5} - \frac{29}{6}
$$
Find a common denominator. The LCD of 5 and 6 is 30:
$$
\frac{27}{5} = \frac{162}{30}, \quad \frac{29}{6} = \frac{145}{30}
$$
So:
$$
\frac{27}{5} - \frac{29}{6} = \frac{162}{30} - \frac{145}{30} = \frac{162 - 145}{30} = \frac{17}{30}
$$

Answer: $\boxed{\frac{17}{30}}$

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#### g) $3 \frac{1}{5} + 1 \frac{7}{12} \times 3 \frac{3}{5}$
1. Convert mixed numbers to improper fractions:
$$
3 \frac{1}{5} = \frac{16}{5}, \quad 1 \frac{7}{12} = \frac{19}{12}, \quad 3 \frac{3}{5} = \frac{18}{5}
$$
2. Perform the multiplication:
$$
\frac{19}{12} \times \frac{18}{5} = \frac{19 \times 18}{12 \times 5} = \frac{342}{60}
$$
Simplify $\frac{342}{60}$ by dividing numerator and denominator by their GCD, which is 6:
$$
\frac{342}{60} = \frac{57}{10}
$$
3. Add the results:
$$
\frac{16}{5} + \frac{57}{10}
$$
Find a common denominator. The LCD of 5 and 10 is 10:
$$
\frac{16}{5} = \frac{32}{10}
$$
So:
$$
\frac{16}{5} + \frac{57}{10} = \frac{32}{10} + \frac{57}{10} = \frac{32 + 57}{10} = \frac{89}{10}
$$

Answer: $\boxed{\frac{89}{10}}$

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#### h) $4 \frac{2}{5} \div \left( 2 \frac{1}{5} - \frac{33}{12} \right) \div 3 \frac{1}{5}$
1. Convert mixed numbers to improper fractions:
$$
4 \frac{2}{5} = \frac{22}{5}, \quad 2 \frac{1}{5} = \frac{11}{5}, \quad 3 \frac{1}{5} = \frac{16}{5}
$$
2. Simplify the expression inside the parentheses:
$$
2 \frac{1}{5} - \frac{33}{12} = \frac{11}{5} - \frac{33}{12}
$$
Find a common denominator. The LCD of 5 and 12 is 60:
$$
\frac{11}{5} = \frac{132}{60}, \quad \frac{33}{12} = \frac{165}{60}
$$
So:
$$
\frac{11}{5} - \frac{33}{12} = \frac{132}{60} - \frac{165}{60} = \frac{132 - 165}{60} = \frac{-33}{60}
$$
Simplify $\frac{-33}{60}$ by dividing numerator and denominator by their GCD, which is 3:
$$
\frac{-33}{60} = \frac{-11}{20}
$$
3. Perform the division:
$$
\frac{22}{5} \div \left( \frac{-11}{20} \right) = \frac{22}{5} \times \frac{20}{-11} = \frac{22 \times 20}{5 \times -11} = \frac{440}{-55} = -8
$$
4. Perform the final division:
$$
-8 \div \frac{16}{5} = -8 \times \frac{5}{16} = \frac{-8 \times 5}{16} = \frac{-40}{16} = -\frac{5}{2}
$$

Answer: $\boxed{-\frac{5}{2}}$

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Exam Question Solution:

#### a) There are 24 men in a room.
- $\frac{1}{2}$ of the men are wearing a red shirt:
$$
\frac{1}{2} \times 24 = 12
$$
- $\frac{1}{3}$ of the men are wearing a green shirt:
$$
\frac{1}{3} \times 24 = 8
$$
- The rest are wearing a blue shirt. The total number of men is 24, so:
$$
\text{Number of men wearing blue shirts} = 24 - (\text{red shirts} + \text{green shirts})
$$
$$
\text{Number of men wearing blue shirts} = 24 - (12 + 8) = 24 - 20 = 4
$$

Answer: $\boxed{4}$

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Final Answers:

1. a) $\boxed{\frac{3}{10}}$
2. b) $\boxed{\frac{1}{6}}$
3. c) $\boxed{\frac{1}{4}}$
4. d) $\boxed{-\frac{17}{4}}$
5. e) $\boxed{\frac{27}{8}}$
6. f) $\boxed{\frac{17}{30}}$
7. g) $\boxed{\frac{89}{10}}$
8. h) $\boxed{-\frac{5}{2}}$
9. Exam Question: $\boxed{4}$