Worksheet with mixed operations and exam questions on fractions.

Let's solve each part step by step.

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Mixed Operations

We'll evaluate each expression carefully, following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).

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#### a) $ \frac{1}{5} + \frac{1}{5} \times \frac{1}{2} $

First, do multiplication:

$$
\frac{1}{5} \times \frac{1}{2} = \frac{1}{10}
$$

Now add:

$$
\frac{1}{5} + \frac{1}{10} = \frac{2}{10} + \frac{1}{10} = \frac{3}{10}
$$

✔ Answer: $ \frac{3}{10} $

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#### b) $ \frac{11}{15} \div \frac{4}{5} - \frac{3}{4} $

Division first:

$$
\frac{11}{15} \div \frac{4}{5} = \frac{11}{15} \times \frac{5}{4} = \frac{55}{60} = \frac{11}{12}
$$

Now subtract:

$$
\frac{11}{12} - \frac{3}{4} = \frac{11}{12} - \frac{9}{12} = \frac{2}{12} = \frac{1}{6}
$$

✔ Answer: $ \frac{1}{6} $

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#### c) $ \left( \frac{3}{7} + \frac{1}{14} \right)^2 $

First, add inside parentheses:

$$
\frac{3}{7} = \frac{6}{14}, \quad \frac{6}{14} + \frac{1}{14} = \frac{7}{14} = \frac{1}{2}
$$

Now square:

$$
\left( \frac{1}{2} \right)^2 = \frac{1}{4}
$$

✔ Answer: $ \frac{1}{4} $

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#### d) $ \frac{5}{12} - \frac{7}{13} \div \frac{3}{26} $

Do 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:

$$
\frac{182}{39} = \frac{14}{3} \quad (\text{divide numerator and denominator by } 13)
$$

Now subtract:

$$
\frac{5}{12} - \frac{14}{3} = \frac{5}{12} - \frac{56}{12} = \frac{-51}{12} = -\frac{17}{4}
$$

✔ Answer: $ -\frac{17}{4} $ or $ -4\frac{1}{4} $

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#### e) $ \frac{11}{14} + \frac{3}{4} \times \frac{20}{21} + \frac{15}{8} $

Multiplication first:

$$
\frac{3}{4} \times \frac{20}{21} = \frac{60}{84} = \frac{5}{7}
$$

Now add:

$$
\frac{11}{14} + \frac{5}{7} + \frac{15}{8}
$$

Convert all to common denominator. LCM of 14, 7, 8 is 56.

- $ \frac{11}{14} = \frac{44}{56} $
- $ \frac{5}{7} = \frac{40}{56} $
- $ \frac{15}{8} = \frac{105}{56} $

Add:

$$
\frac{44 + 40 + 105}{56} = \frac{189}{56}
$$

Simplify:

$$
\frac{189}{56} = \frac{27}{8} = 3\frac{3}{8}
$$

✔ Answer: $ 3\frac{3}{8} $

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#### f) $ \frac{12}{5} \div \frac{4}{9} - \frac{29}{20} \div \frac{3}{10} $

Do divisions:

1. $ \frac{12}{5} \div \frac{4}{9} = \frac{12}{5} \times \frac{9}{4} = \frac{108}{20} = \frac{27}{5} $

2. $ \frac{29}{20} \div \frac{3}{10} = \frac{29}{20} \times \frac{10}{3} = \frac{290}{60} = \frac{29}{6} $

Now subtract:

$$
\frac{27}{5} - \frac{29}{6}
$$

LCM of 5 and 6 is 30:

- $ \frac{27}{5} = \frac{162}{30} $
- $ \frac{29}{6} = \frac{145}{30} $

$$
\frac{162 - 145}{30} = \frac{17}{30}
$$

✔ Answer: $ \frac{17}{30} $

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#### g) $ 3\frac{1}{5} + 1\frac{7}{12} \times 3\frac{3}{5} $

Convert mixed numbers to improper fractions:

- $ 3\frac{1}{5} = \frac{16}{5} $
- $ 1\frac{7}{12} = \frac{19}{12} $
- $ 3\frac{3}{5} = \frac{18}{5} $

Now multiply:

$$
\frac{19}{12} \times \frac{18}{5} = \frac{342}{60} = \frac{57}{10}
$$

Now add:

$$
\frac{16}{5} + \frac{57}{10} = \frac{32}{10} + \frac{57}{10} = \frac{89}{10} = 8\frac{9}{10}
$$

✔ Answer: $ 8\frac{9}{10} $

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#### h) $ 4\frac{2}{5} \div (2\frac{1}{5} - \frac{33}{12}) \div 3\frac{1}{5} $

Convert all to improper fractions:

- $ 4\frac{2}{5} = \frac{22}{5} $
- $ 2\frac{1}{5} = \frac{11}{5} $
- $ \frac{33}{12} = \frac{11}{4} $
- $ 3\frac{1}{5} = \frac{16}{5} $

Now compute inside parentheses:

$$
\frac{11}{5} - \frac{11}{4}
$$

LCM of 5 and 4 is 20:

- $ \frac{11}{5} = \frac{44}{20} $
- $ \frac{11}{4} = \frac{55}{20} $

$$
\frac{44}{20} - \frac{55}{20} = \frac{-11}{20}
$$

Now:

$$
\frac{22}{5} \div \left( \frac{-11}{20} \right) \div \frac{16}{5}
$$

First division:

$$
\frac{22}{5} \div \frac{-11}{20} = \frac{22}{5} \times \frac{20}{-11} = \frac{440}{-55} = -8
$$

Now divide by $ \frac{16}{5} $:

$$
-8 \div \frac{16}{5} = -8 \times \frac{5}{16} = -\frac{40}{16} = -\frac{5}{2} = -2\frac{1}{2}
$$

✔ Answer: $ -2\frac{1}{2} $

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Exam Question

> a) There are 24 men in a room.
> $ \frac{1}{2} $ of the men are wearing a red shirt.
> $ \frac{1}{3} $ of the men are wearing a green shirt.
> The rest are wearing a blue shirt.
> Work out the number of men wearing a blue shirt.

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Step-by-step:

- Total men: 24

- Red shirts: $ \frac{1}{2} \times 24 = 12 $

- Green shirts: $ \frac{1}{3} \times 24 = 8 $

- Blue shirts: $ 24 - 12 - 8 = 4 $

✔ Answer: 4 men are wearing blue shirts.

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✔ Final Answers Summary:

Mixed Operations:

a) $ \frac{3}{10} $
b) $ \frac{1}{6} $
c) $ \frac{1}{4} $
d) $ -\frac{17}{4} $
e) $ 3\frac{3}{8} $
f) $ \frac{17}{30} $
g) $ 8\frac{9}{10} $
h) $ -2\frac{1}{2} $

Exam Question:
a) 4 men are wearing blue shirts.

Let me know if you'd like these explained with diagrams or visual steps!