6th Grade Math Worksheets | Printable PDF Worksheets - Free Printable

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. That 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:

The reciprocal of a number is $ \frac{1}{\text{number}} $. For fractions, flip 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} $ → reciprocal is $ \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 being $ \frac{1}{3} $. Since there are three pieces of $ \frac{1}{3} $ in 1, then:

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

✔ This means: How many $ \frac{1}{3} $s fit into 1? Answer: 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} $

Do left to right:

First: $ \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:

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#### 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} $
→ "Any number multiplied by its reciprocal is equal to 1."

3)
a) $ \frac{11}{6} $
b) $ -\frac{3}{2} $
c) $ \frac{1}{5} $
d) $ 2 $
e) $ \frac{19}{8} $
f) $ \frac{3}{14} $

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#### Section B

1) The diagram shows that 1 is divided into 3 equal parts of $ \frac{1}{3} $, so 3 of them make 1 → $ 1 \div \frac{1}{3} = 3 $

2)
a) 6
b) 3
c) 15
d) 25
e) $ 16\frac{2}{3} $
f) 9

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#### 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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