Class 8 Mathematics worksheet focusing on rational numbers, including properties and calculations.

Let's solve the problems step by step:

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Problem 1: Additive Identity

Given:
\[ 5 + 0 = 5 \]

Solution:
The additive identity is the number that, when added to any number, leaves the number unchanged. Here, adding 0 to 5 results in 5. Therefore, the additive identity is:
\[ \boxed{0} \]

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Problem 2: Multiplicative Identity

Given:
\[ \frac{1}{3} \times 1 = \frac{1}{3} \]

Solution:
The multiplicative identity is the number that, when multiplied by any number, leaves the number unchanged. Here, multiplying \(\frac{1}{3}\) by 1 results in \(\frac{1}{3}\). Therefore, the multiplicative identity is:
\[ \boxed{1} \]

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Problem 3: Additive Inverse

Given:
- Additive inverse of 5 is -5.
- Additive inverse of \(-\frac{3}{7}\) is ______.

Solution:
The additive inverse of a number \(x\) is the number that, when added to \(x\), gives 0. For \(-\frac{3}{7}\), we need a number \(y\) such that:
\[ -\frac{3}{7} + y = 0 \]
Solving for \(y\):
\[ y = \frac{3}{7} \]
Thus, the additive inverse of \(-\frac{3}{7}\) is:
\[ \boxed{\frac{3}{7}} \]

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Problem 4: Multiplicative Inverse

Given:
- Multiplicative inverse of \(\frac{4}{7}\) is \(\frac{7}{4}\).
- Multiplicative inverse of \(\frac{5}{9}\) is ______.

Solution:
The multiplicative inverse of a number \(x\) is the number that, when multiplied by \(x\), gives 1. For \(\frac{5}{9}\), we need a number \(y\) such that:
\[ \frac{5}{9} \times y = 1 \]
Solving for \(y\):
\[ y = \frac{9}{5} \]
Thus, the multiplicative inverse of \(\frac{5}{9}\) is:
\[ \boxed{\frac{9}{5}} \]

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Problem 5: Properties of Rational Numbers

Given:
- \(a \times b = b \times a\) — Commutative Property of multiplication
- \((a + b) + c = a + (b + c)\) — Associative property of addition
- \(a(b + c) = ab + ac\) — Distributive property
- \((\frac{3}{5} \times \frac{2}{7}) \times \frac{4}{5} = \frac{3}{5} \times (\frac{2}{7} \times \frac{4}{5})\) — Name the Property

Solution:
The given expression \((\frac{3}{5} \times \frac{2}{7}) \times \frac{4}{5} = \frac{3}{5} \times (\frac{2}{7} \times \frac{4}{5})\) demonstrates the Associative Property of Multiplication. This property states that the way factors are grouped does not change the product.

Thus, the property is:
\[ \boxed{\text{Associative Property of Multiplication}} \]

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Problem 6: Rational Numbers Greater Than a Given Number

Given:
- Five rational numbers greater than -3 are \(-2, -1, 0, 1, 2\).
- Five rational numbers greater than -2 are ______.

Solution:
To find five rational numbers greater than -2, we can choose any numbers larger than -2. For example:
\[ -1, 0, 1, 2, 3 \]

Thus, five rational numbers greater than -2 are:
\[ \boxed{-1, 0, 1, 2, 3} \]

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Problem 7: Rational Numbers Between \(\frac{1}{3}\) and \(\frac{3}{5}\)

Given:
Find 3 rational numbers between \(\frac{1}{3}\) and \(\frac{3}{5}\).

Solution:
First, convert \(\frac{1}{3}\) and \(\frac{3}{5}\) to fractions with a common denominator. The LCM of 3 and 5 is 15:
\[ \frac{1}{3} = \frac{5}{15}, \quad \frac{3}{5} = \frac{9}{15} \]

Now, we need to find three rational numbers between \(\frac{5}{15}\) and \(\frac{9}{15}\). These could be:
\[ \frac{6}{15}, \frac{7}{15}, \frac{8}{15} \]

Thus, three rational numbers between \(\frac{1}{3}\) and \(\frac{3}{5}\) are:
\[ \boxed{\frac{6}{15}, \frac{7}{15}, \frac{8}{15}} \]

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Problem 8: Rational Numbers Between \(\frac{3}{7}\) and \(\frac{5}{2}\)

Given:
Find 5 rational numbers between \(\frac{3}{7}\) and \(\frac{5}{2}\).

Solution:
First, convert \(\frac{3}{7}\) and \(\frac{5}{2}\) to fractions with a common denominator. The LCM of 7 and 2 is 14:
\[ \frac{3}{7} = \frac{6}{14}, \quad \frac{5}{2} = \frac{35}{14} \]

Now, we need to find five rational numbers between \(\frac{6}{14}\) and \(\frac{35}{14}\). These could be:
\[ \frac{7}{14}, \frac{8}{14}, \frac{9}{14}, \frac{10}{14}, \frac{11}{14} \]

Thus, five rational numbers between \(\frac{3}{7}\) and \(\frac{5}{2}\) are:
\[ \boxed{\frac{7}{14}, \frac{8}{14}, \frac{9}{14}, \frac{10}{14}, \frac{11}{14}} \]

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Final Answers:

1. Additive identity: \(\boxed{0}\)
2. Multiplicative identity: \(\boxed{1}\)
3. Additive inverse of \(-\frac{3}{7}\): \(\boxed{\frac{3}{7}}\)
4. Multiplicative inverse of \(\frac{5}{9}\): \(\boxed{\frac{9}{5}}\)
5. Property: \(\boxed{\text{Associative Property of Multiplication}}\)
6. Five rational numbers greater than -2: \(\boxed{-1, 0, 1, 2, 3}\)
7. Three rational numbers between \(\frac{1}{3}\) and \(\frac{3}{5}\): \(\boxed{\frac{6}{15}, \frac{7}{15}, \frac{8}{15}}\)
8. Five rational numbers between \(\frac{3}{7}\) and \(\frac{5}{2}\): \(\boxed{\frac{7}{14}, \frac{8}{14}, \frac{9}{14}, \frac{10}{14}, \frac{11}{14}}\)