Grade VIII Rational Numbers worksheet from International Indian School Riyadh, covering fundamental concepts and operations with rational numbers.

Problem Analysis:

The provided worksheet focuses on rational numbers and their properties. Let's solve each section step by step.

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Section I: Fill in the blanks

1. ________ has no reciprocal.
- Answer: 0
- Explanation: The reciprocal of a number \( x \) is \( \frac{1}{x} \). For \( x = 0 \), \( \frac{1}{0} \) is undefined, so 0 has no reciprocal.

2. There are ___________ numbers of rational numbers between any two numbers.
- Answer: Infinite
- Explanation: Between any two rational numbers, there are infinitely many other rational numbers. This is because rational numbers are dense in the real number line.

3. The product of a number and its multiplicative inverse is ________.
- Answer: 1
- Explanation: The multiplicative inverse of a number \( x \) is \( \frac{1}{x} \). Their product is \( x \times \frac{1}{x} = 1 \).

4. Sum of a number and its negative is ________.
- Answer: 0
- Explanation: The negative of a number \( x \) is \( -x \). Their sum is \( x + (-x) = 0 \).

5. ________ is the multiplicative identity.
- Answer: 1
- Explanation: The multiplicative identity is the number that, when multiplied by any number, leaves it unchanged. This number is 1.

6. ________ is the additive identity.
- Answer: 0
- Explanation: The additive identity is the number that, when added to any number, leaves it unchanged. This number is 0.

7. Additive inverse of \( \frac{3}{7} \) is ________.
- Answer: \( -\frac{3}{7} \)
- Explanation: The additive inverse of a number \( x \) is the number that, when added to \( x \), gives 0. For \( \frac{3}{7} \), the additive inverse is \( -\frac{3}{7} \).

8. Multiplicative inverse of -2 is ________.
- Answer: \( -\frac{1}{2} \)
- Explanation: The multiplicative inverse of a number \( x \) is \( \frac{1}{x} \). For \( -2 \), the multiplicative inverse is \( \frac{1}{-2} = -\frac{1}{2} \).

9. The numbers ________ and ________ are their own reciprocals.
- Answer: 1 and -1
- Explanation: A number is its own reciprocal if \( x = \frac{1}{x} \). Solving this equation gives \( x^2 = 1 \), so \( x = 1 \) or \( x = -1 \).

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Section II: Find the value of the following

#### i) \( \frac{-21}{25} \times \frac{15}{-49} \times \frac{-35}{9} \)
- Simplify step by step:
\[
\frac{-21}{25} \times \frac{15}{-49} \times \frac{-35}{9}
\]
Combine the numerators and denominators:
\[
= \frac{(-21) \times 15 \times (-35)}{25 \times (-49) \times 9}
\]
Simplify the signs:
\[
= \frac{-21 \times 15 \times 35}{25 \times 49 \times 9}
\]
Factorize the numbers:
\[
= \frac{(-3 \times 7) \times (3 \times 5) \times (5 \times 7)}{(5 \times 5) \times (7 \times 7) \times (3 \times 3)}
\]
Cancel out common factors:
\[
= \frac{-3 \times 7 \times 3 \times 5 \times 5 \times 7}{5 \times 5 \times 7 \times 7 \times 3 \times 3}
\]
\[
= \frac{-1}{3 \times 7}
\]
\[
= \frac{-1}{21}
\]

#### ii) \( \frac{-43}{45} \times \left( \frac{-8}{5} + \frac{3}{5} \right) \)
- Simplify the expression inside the parentheses first:
\[
\frac{-8}{5} + \frac{3}{5} = \frac{-8 + 3}{5} = \frac{-5}{5} = -1
\]
- Now multiply:
\[
\frac{-43}{45} \times (-1) = \frac{43}{45}
\]

#### iii) \( \frac{-21}{15} + \frac{7}{25} - \left( \frac{-4}{25} \right) \)
- Simplify the expression:
\[
\frac{-21}{15} + \frac{7}{25} + \frac{4}{25}
\]
- Find a common denominator (LCM of 15 and 25 is 75):
\[
\frac{-21}{15} = \frac{-21 \times 5}{15 \times 5} = \frac{-105}{75}
\]
\[
\frac{7}{25} = \frac{7 \times 3}{25 \times 3} = \frac{21}{75}
\]
\[
\frac{4}{25} = \frac{4 \times 3}{25 \times 3} = \frac{12}{75}
\]
- Add the fractions:
\[
\frac{-105}{75} + \frac{21}{75} + \frac{12}{75} = \frac{-105 + 21 + 12}{75} = \frac{-72}{75}
\]
- Simplify:
\[
\frac{-72}{75} = \frac{-24}{25}
\]

#### iv) \( \frac{-72}{45} \div \frac{32}{25} \)
- Division of fractions is equivalent to multiplying by the reciprocal:
\[
\frac{-72}{45} \div \frac{32}{25} = \frac{-72}{45} \times \frac{25}{32}
\]
- Simplify:
\[
= \frac{-72 \times 25}{45 \times 32}
\]
Factorize the numbers:
\[
= \frac{(-2^3 \times 3^2) \times (5^2)}{(3^2 \times 5) \times (2^5)}
\]
Cancel out common factors:
\[
= \frac{-5}{2^3} = \frac{-5}{8}
\]

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Section III: Verify \( -(-x) = x \) by taking \( x = -2 \)

- Given \( x = -2 \):
\[
-(-x) = -(-(-2)) = -2
\]
- Therefore, \( -(-x) = x \) is verified.

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Section IV: Represent \( -\frac{3}{7} \) and \( \frac{9}{7} \) on a number line

- On a number line:
- \( -\frac{3}{7} \) is located to the left of 0, at a distance of \( \frac{3}{7} \) units.
- \( \frac{9}{7} \) is located to the right of 0, at a distance of \( \frac{9}{7} \) units.

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Section V: State the property used in the following

#### i) \( \frac{-3}{58} \times \frac{7}{8} = \frac{7}{8} \times \frac{-3}{58} \)
- Property: Commutative property of multiplication.
- Explanation: The order of multiplication does not affect the result.

#### ii) \( \left( \frac{-2}{3} + \frac{5}{7} \right) + \left( \frac{3}{5} + \frac{5}{7} \right) = \frac{-2}{3} + \left( \frac{5}{7} + \frac{3}{5} + \frac{5}{7} \right) \)
- Property: Associative property of addition.
- Explanation: The way numbers are grouped during addition does not affect the result.

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Final Answers:

Section I:
1. 0
2. Infinite
3. 1
4. 0
5. 1
6. 0
7. \( -\frac{3}{7} \)
8. \( -\frac{1}{2} \)
9. 1 and -1

Section II:
i) \( \frac{-1}{21} \)
ii) \( \frac{43}{45} \)
iii) \( \frac{-24}{25} \)
iv) \( \frac{-5}{8} \)

Section III:
Verified.

Section IV:
Represented on the number line.

Section V:
i) Commutative property of multiplication
ii) Associative property of addition

Boxed Final Answer:
\[
\boxed{
\begin{array}{l}
\text{Section I:} \\
1. 0 \\
2. \text{Infinite} \\
3. 1 \\
4. 0 \\
5. 1 \\
6. 0 \\
7. -\frac{3}{7} \\
8. -\frac{1}{2} \\
9. 1 \text{ and } -1 \\
\\
\text{Section II:} \\
\text{i) } \frac{-1}{21} \\
\text{ii) } \frac{43}{45} \\
\text{iii) } \frac{-24}{25} \\
\text{iv) } \frac{-5}{8} \\
\\
\text{Section III:} \\
\text{Verified.} \\
\\
\text{Section IV:} \\
\text{Represented on the number line.} \\
\\
\text{Section V:} \\
\text{i) Commutative property of multiplication} \\
\text{ii) Associative property of addition}
\end{array}
}
\]