Practice dividing fractions with this structured worksheet covering reciprocals and integer division.

Absolutely! Let's go through each section of the worksheet step by step, solving and explaining everything clearly.

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## ✔ Section A: Reciprocals

1) Show that \(\frac{3}{4} \times \frac{4}{3} = 1\)

Solution:
Multiply numerator × numerator and denominator × denominator:

\[
\frac{3}{4} \times \frac{4}{3} = \frac{3 \times 4}{4 \times 3} = \frac{12}{12} = 1
\]

✔ So, it’s proven: multiplying a fraction by its reciprocal gives 1.

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2) Fill in the blanks:

We are looking for the reciprocal — the number that when multiplied by the given number gives 1.

> Reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).
> Reciprocal of a whole number \(n\) is \(\frac{1}{n}\).

#### 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.

✔ Final answer: reciprocal

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3) Find the reciprocal of each:

#### a) \(\frac{6}{11}\) → \(\boxed{\frac{11}{6}}\)

#### b) \(-\frac{2}{3}\) → \(\boxed{-\frac{3}{2}}\) (sign stays the same)

#### c) \(5 = \frac{5}{1}\) → \(\boxed{\frac{1}{5}}\)

#### d) \(\frac{1}{2}\) → \(\boxed{2}\)

#### e) \(\frac{8}{19}\) → \(\boxed{\frac{19}{8}}\)

#### f) \(4\frac{2}{3}\) → First convert to improper fraction:

\[
4\frac{2}{3} = \frac{4 \times 3 + 2}{3} = \frac{14}{3} \Rightarrow \text{Reciprocal} = \boxed{\frac{3}{14}}
\]

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## ✔ Section B: Dividing integers by fractions

1) Explain how this diagram shows that \(1 \div \frac{1}{3} = 3\)

Diagram Explanation:

- The top bar represents 1 whole.
- The bottom bar is divided into three equal parts, each labeled \(\frac{1}{3}\).
- Since there are 3 parts of \(\frac{1}{3}\) that make up 1 whole, we can say:

> “How many \(\frac{1}{3}\)s fit into 1?” → Answer: 3

So, \(1 \div \frac{1}{3} = 3\)

✔ This visually demonstrates division as “how many groups of \(\frac{1}{3}\) are in 1?”

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2) Calculate the following:

Remember: To divide by a fraction, multiply by its reciprocal.

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

#### 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) \(10 \div \frac{3}{5} = 10 \times \frac{5}{3} = \frac{50}{3} = \boxed{16\frac{2}{3}}\) (or leave as improper fraction \(\frac{50}{3}\))

#### f) \(21 \div 2\frac{1}{3}\)

First, convert mixed number to improper fraction:

\[
2\frac{1}{3} = \frac{7}{3}
\]

Now divide:

\[
21 \div \frac{7}{3} = 21 \times \frac{3}{7} = \frac{63}{7} = \boxed{9}
\]

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## ✔ Section C: Dividing any pair of fractions

Again, use: Divide = Multiply by reciprocal

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) \(4\frac{2}{3} \div \frac{1}{2}\)

Convert mixed number:

\[
4\frac{2}{3} = \frac{14}{3}
\]

Now divide:

\[
\frac{14}{3} \div \frac{1}{2} = \frac{14}{3} \times \frac{2}{1} = \frac{28}{3} = \boxed{9\frac{1}{3}}
\]

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

(Cancel 5s first: \(\frac{1}{7} \times \frac{12}{1} = \frac{12}{7}\))

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

(Simplify: divide numerator and denominator by 3)

✔ Final answer: \(\boxed{-\frac{15}{16}}\)

#### f) \(2\frac{1}{8} \div \frac{9}{10}\)

Convert mixed number:

\[
2\frac{1}{8} = \frac{17}{8}
\]

Now divide:

\[
\frac{17}{8} \div \frac{9}{10} = \frac{17}{8} \times \frac{10}{9} = \frac{170}{72} = \frac{85}{36} = \boxed{2\frac{13}{36}}
\]

(Simplify by dividing numerator and denominator by 2)

#### 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 (division is left associative):

First: \(\frac{7}{12} \div \frac{3}{4} = \frac{7}{12} \times \frac{4}{3} = \frac{28}{36} = \frac{7}{9}\)

Then: \(\frac{7}{9} \div \frac{1}{2} = \frac{7}{9} \times \frac{2}{1} = \frac{14}{9} = \boxed{1\frac{5}{9}}\)

#### i) \(3\frac{1}{7} \div 5\frac{1}{2}\)

Convert both to improper fractions:

\[
3\frac{1}{7} = \frac{22}{7}, \quad 5\frac{1}{2} = \frac{11}{2}
\]

Now divide:

\[
\frac{22}{7} \div \frac{11}{2} = \frac{22}{7} \times \frac{2}{11} = \frac{44}{77} = \frac{4}{7}
\]

(Cancel 11s: 22 ÷ 11 = 2, 11 ÷ 11 = 1 → \(\frac{2}{7} \times \frac{2}{1} = \frac{4}{7}\))

✔ Final answer: \(\boxed{\frac{4}{7}}\)

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## 🎉 Final Summary of All Answers

Section A

1) Shown: \(\frac{3}{4} \times \frac{4}{3} = 1\)

2)
a) \(\frac{3}{2}\)
b) \(\frac{7}{5}\)
c) \(2\)
d) \(\frac{1}{8}\)
→ Blank: reciprocal

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) Diagram explanation: 1 whole contains 3 pieces of \(\frac{1}{3}\), so \(1 \div \frac{1}{3} = 3\)

2)
a) 6
b) 3
c) 15
d) 25
e) \(16\frac{2}{3}\) or \(\frac{50}{3}\)
f) 9

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Section C

1)
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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Let me know if you’d like a printable version or visual diagrams for any part! 😊