This worksheet guides students through the fundamentals of dividing fractions, starting with reciprocals and moving to complex division problems.

Let's solve each part of this "Dividing Fractions" worksheet step by step, with clear explanations.

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Section A: Reciprocals

#### 1) Prove that $ \frac{3}{4} \times \frac{4}{3} = 1 $

We multiply the two fractions:

$$
\frac{3}{4} \times \frac{4}{3} = \frac{3 \times 4}{4 \times 3} = \frac{12}{12} = 1
$$

✔ So, it is proven.

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#### 2) Fill in the blanks:

We are looking for numbers such that when multiplied, the result is 1. This means we're finding reciprocals.

a) $ \frac{2}{3} \times \boxed{\frac{3}{2}} = 1 $

b) $ \boxed{\frac{7}{5}} \times \frac{5}{7} = 1 $

c) $ 1 = \frac{1}{2} \times \boxed{2} $

d) $ \boxed{\frac{1}{8}} \times 8 = 1 $

> ✔ Any number multiplied by its reciprocal is equal to 1.

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#### 3) Find the reciprocal of each number:

Reciprocal means flipping the numerator and denominator.

a) $ \frac{6}{11} $ → $ \boxed{\frac{11}{6}} $

b) $ -\frac{2}{3} $ → $ \boxed{-\frac{3}{2}} $

c) $ 5 $ → $ \boxed{\frac{1}{5}} $

d) $ \frac{1}{2} $ → $ \boxed{2} $

e) $ \frac{8}{19} $ → $ \boxed{\frac{19}{8}} $

f) $ 4\frac{2}{3} = \frac{14}{3} $ → $ \boxed{\frac{3}{14}} $

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Section B: Dividing integers by fractions

#### 1) Explain how the diagram shows that $ 1 \div \frac{1}{3} = 3 $

The diagram shows a whole (1) divided into three equal parts, each of size $ \frac{1}{3} $. Since there are three such parts in one whole, this means:

$$
1 \div \frac{1}{3} = 3
$$

👉 Explanation:
When you divide 1 by $ \frac{1}{3} $, you're asking: *How many $ \frac{1}{3} $s are in 1?*
There are three thirds in a whole, so the answer is 3.

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#### 2) Calculate the following:

To divide by a fraction, multiply by the reciprocal.

a) $ 2 \div \frac{1}{3} = 2 \times 3 = \boxed{6} $

b) $ 2 \div \frac{2}{3} = 2 \times \frac{3}{2} = \frac{6}{2} = \boxed{3} $

c) $ 10 \div \frac{2}{3} = 10 \times \frac{3}{2} = \frac{30}{2} = \boxed{15} $

d) $ 10 \div \frac{2}{5} = 10 \times \frac{5}{2} = \frac{50}{2} = \boxed{25} $

e) $ 10 \div \frac{3}{5} = 10 \times \frac{5}{3} = \frac{50}{3} = \boxed{16\frac{2}{3}} $

f) $ 21 \div 2\frac{1}{3} = 21 \div \frac{7}{3} = 21 \times \frac{3}{7} = \frac{63}{7} = \boxed{9} $

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Section C: Dividing any pair of fractions

Use the rule:
$$
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
$$

#### 1) Calculate:

a) $ \frac{1}{3} \div \frac{1}{3} = \frac{1}{3} \times \frac{3}{1} = \frac{3}{3} = \boxed{1} $

b) $ \frac{2}{3} \div \frac{1}{2} = \frac{2}{3} \times \frac{2}{1} = \frac{4}{3} = \boxed{1\frac{1}{3}} $

c) $ 4\frac{2}{3} \div \frac{1}{2} = \frac{14}{3} \div \frac{1}{2} = \frac{14}{3} \times 2 = \frac{28}{3} = \boxed{9\frac{1}{3}} $

d) $ \frac{5}{7} \div \frac{5}{12} = \frac{5}{7} \times \frac{12}{5} = \frac{60}{35} = \frac{12}{7} = \boxed{1\frac{5}{7}} $

e) $ -\frac{5}{12} \div \frac{4}{9} = -\frac{5}{12} \times \frac{9}{4} = -\frac{45}{48} = -\frac{15}{16} = \boxed{-\frac{15}{16}} $

f) $ 2\frac{1}{8} \div \frac{9}{10} = \frac{17}{8} \div \frac{9}{10} = \frac{17}{8} \times \frac{10}{9} = \frac{170}{72} = \frac{85}{36} = \boxed{2\frac{13}{36}} $

g) $ \frac{9}{11} \div \frac{9}{11} = \frac{9}{11} \times \frac{11}{9} = \frac{99}{99} = \boxed{1} $

h) $ \frac{7}{12} \div \frac{3}{4} \div \frac{1}{2} $
First do $ \frac{7}{12} \div \frac{3}{4} = \frac{7}{12} \times \frac{4}{3} = \frac{28}{36} = \frac{7}{9} $
Then $ \frac{7}{9} \div \frac{1}{2} = \frac{7}{9} \times 2 = \frac{14}{9} = \boxed{1\frac{5}{9}} $

i) $ 3\frac{1}{7} \div 5\frac{1}{2} = \frac{22}{7} \div \frac{11}{2} = \frac{22}{7} \times \frac{2}{11} = \frac{44}{77} = \frac{4}{7} = \boxed{\frac{4}{7}} $

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✔ Final Answers Summary:

#### Section A
1) $ \frac{3}{4} \times \frac{4}{3} = 1 $ → Proven.
2) a) $ \frac{3}{2} $, b) $ \frac{7}{5} $, c) $ 2 $, d) $ \frac{1}{8} $; reciprocal
3) a) $ \frac{11}{6} $, b) $ -\frac{3}{2} $, c) $ \frac{1}{5} $, d) $ 2 $, e) $ \frac{19}{8} $, f) $ \frac{3}{14} $

#### Section B
1) The diagram shows that 1 whole is made up of 3 parts of $ \frac{1}{3} $, so $ 1 \div \frac{1}{3} = 3 $
2) a) 6, b) 3, c) 15, d) 25, e) $ 16\frac{2}{3} $, f) 9

#### Section C
a) 1
b) $ 1\frac{1}{3} $
c) $ 9\frac{1}{3} $
d) $ 1\frac{5}{7} $
e) $ -\frac{15}{16} $
f) $ 2\frac{13}{36} $
g) 1
h) $ 1\frac{5}{9} $
i) $ \frac{4}{7} $

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