Class 8 Mathematics worksheet focusing on rational numbers, including problems on sums, inverses, products, and number line representation.

Let's solve each question from the worksheet step by step and explain the reasoning behind each solution.

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1. The sum of a rational number and its additive inverse is equal to ________

Answer: 0

Explanation:
The additive inverse of a number $ a $ is $ -a $. So, $ a + (-a) = 0 $.
This is true for all rational numbers.

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2. Write the multiplicative inverse of $ \left( -\frac{2}{3} \times \frac{3}{10} \right) $

First, compute the product:
$$
-\frac{2}{3} \times \frac{3}{10} = -\frac{6}{30} = -\frac{1}{5}
$$

Now, the multiplicative inverse of $ -\frac{1}{5} $ is $ -5 $, because:
$$
-\frac{1}{5} \times (-5) = 1
$$

Answer: $ -5 $

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3. Check whether $ 2\frac{1}{4} $ is the multiplicative inverse of 2.2

Convert both numbers to fractions:

- $ 2\frac{1}{4} = \frac{9}{4} $
- $ 2.2 = \frac{22}{10} = \frac{11}{5} $

Now check if their product is 1:
$$
\frac{9}{4} \times \frac{11}{5} = \frac{99}{20} = 4.95 \neq 1
$$

So, no, $ 2\frac{1}{4} $ is not the multiplicative inverse of 2.2.

Answer: No

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4. Find the product of $ \frac{13}{15} $ and the additive inverse of $ -\frac{5}{26} $

Additive inverse of $ -\frac{5}{26} $ is $ \frac{5}{26} $

Now compute:
$$
\frac{13}{15} \times \frac{5}{26} = \frac{13 \times 5}{15 \times 26} = \frac{65}{390}
$$

Simplify:
$$
\frac{65}{390} = \frac{1}{6} \quad \text{(Divide numerator and denominator by 65)}
$$

Answer: $ \frac{1}{6} $

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5. Show $ \left( \frac{2}{5} - \frac{4}{15} \right) $ on the number line

First simplify:
$$
\frac{2}{5} = \frac{6}{15}, \quad \frac{6}{15} - \frac{4}{15} = \frac{2}{15}
$$

So, we need to show $ \frac{2}{15} $ on the number line.

Steps:
- Draw a number line from 0 to 1.
- Divide it into 15 equal parts.
- Mark the point at the 2nd division (i.e., $ \frac{2}{15} $).

Answer: Represent $ \frac{2}{15} $ on the number line.

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6. Write the additive inverse of $ \frac{2}{5} + \frac{1}{3} - \frac{3}{2} + \frac{1}{2} $

First, compute the expression:

Find common denominator (LCM of 5, 3, 2 = 30):

$$
\frac{2}{5} = \frac{12}{30},\quad \frac{1}{3} = \frac{10}{30},\quad -\frac{3}{2} = -\frac{45}{30},\quad \frac{1}{2} = \frac{15}{30}
$$

Now add:
$$
\frac{12 + 10 - 45 + 15}{30} = \frac{-8}{30} = -\frac{4}{15}
$$

Additive inverse of $ -\frac{4}{15} $ is $ \frac{4}{15} $

Answer: $ \frac{4}{15} $

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7. Write four rational numbers between $ -\frac{2}{3} $ and $ \frac{3}{4} $

Convert to decimals:
- $ -\frac{2}{3} \approx -0.666... $
- $ \frac{3}{4} = 0.75 $

We can pick any rational numbers in this range.

Examples:
- $ -0.5 = -\frac{1}{2} $
- $ 0 $
- $ \frac{1}{2} = 0.5 $
- $ \frac{1}{4} = 0.25 $

Or as fractions:
- $ -\frac{1}{2},\ 0,\ \frac{1}{4},\ \frac{1}{2} $

Answer: $ -\frac{1}{2},\ 0,\ \frac{1}{4},\ \frac{1}{2} $ (any four rational numbers in the interval)

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8. Divide the sum of $ 7\frac{2}{5} $ and $ 2\frac{1}{2} $ by their difference

Convert mixed numbers:
- $ 7\frac{2}{5} = \frac{37}{5} $
- $ 2\frac{1}{2} = \frac{5}{2} $

Sum:
$$
\frac{37}{5} + \frac{5}{2} = \frac{74 + 25}{10} = \frac{99}{10}
$$

Difference:
$$
\frac{37}{5} - \frac{5}{2} = \frac{74 - 25}{10} = \frac{49}{10}
$$

Now divide:
$$
\frac{99}{10} \div \frac{49}{10} = \frac{99}{10} \times \frac{10}{49} = \frac{99}{49}
$$

Simplify: $ \frac{99}{49} $ is already simplified.

Answer: $ \frac{99}{49} $

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9. By what number should $ -1\frac{2}{3} $ be divided to get $ \frac{11}{2} $?

Let the number be $ x $. Then:
$$
\frac{-1\frac{2}{3}}{x} = \frac{11}{2}
$$

Convert $ -1\frac{2}{3} = -\frac{5}{3} $

So:
$$
\frac{-5/3}{x} = \frac{11}{2} \Rightarrow \frac{-5}{3x} = \frac{11}{2}
$$

Cross-multiply:
$$
-5 \cdot 2 = 11 \cdot 3x \Rightarrow -10 = 33x \Rightarrow x = -\frac{10}{33}
$$

Answer: $ -\frac{10}{33} $

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10. What should be subtracted from $ \frac{11}{12} $ to get $ -\frac{10}{3} $?

Let the number be $ x $. Then:
$$
\frac{11}{12} - x = -\frac{10}{3}
$$

Solve for $ x $:
$$
-x = -\frac{10}{3} - \frac{11}{12} = -\left(\frac{40}{12} + \frac{11}{12}\right) = -\frac{51}{12}
\Rightarrow x = \frac{51}{12} = \frac{17}{4}
$$

Answer: $ \frac{17}{4} $

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11. Simplify: $ -\frac{6}{7} - \frac{3}{14} - \frac{24}{30} $

Simplify $ \frac{24}{30} = \frac{4}{5} $

Now find LCM of denominators: 7, 14, 5 → LCM = 70

Convert:
- $ -\frac{6}{7} = -\frac{60}{70} $
- $ -\frac{3}{14} = -\frac{15}{70} $
- $ -\frac{4}{5} = -\frac{56}{70} $

Add:
$$
-\frac{60 + 15 + 56}{70} = -\frac{131}{70}
$$

Answer: $ -\frac{131}{70} $

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12. Simplify using suitable property

#### I) $ \frac{3}{4} \times \left( \frac{8}{5} - \frac{16}{15} \right) $

First simplify inside:
$$
\frac{8}{5} = \frac{24}{15},\quad \frac{24}{15} - \frac{16}{15} = \frac{8}{15}
$$

Now:
$$
\frac{3}{4} \times \frac{8}{15} = \frac{24}{60} = \frac{2}{5}
$$

Answer: $ \frac{2}{5} $

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#### II) $ \left( -\frac{3}{5} \times \frac{4}{15} \right) \times \frac{3}{10} \times -\frac{5}{9} $

Use associative property.

Compute step by step:
- $ -\frac{3}{5} \times \frac{4}{15} = -\frac{12}{75} = -\frac{4}{25} $
- $ -\frac{4}{25} \times \frac{3}{10} = -\frac{12}{250} = -\frac{6}{125} $
- $ -\frac{6}{125} \times -\frac{5}{9} = \frac{30}{1125} = \frac{2}{75} $

Answer: $ \frac{2}{75} $

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#### III) $ \frac{3}{2} + \frac{5}{4} - \frac{3}{2} \times \frac{7}{6} $

Follow order of operations (BODMAS): do multiplication first.

$ \frac{3}{2} \times \frac{7}{6} = \frac{21}{12} = \frac{7}{4} $

Now:
$$
\frac{3}{2} + \frac{5}{4} - \frac{7}{4} = \frac{3}{2} + \left( \frac{5 - 7}{4} \right) = \frac{3}{2} - \frac{2}{4} = \frac{3}{2} - \frac{1}{2} = 1
$$

Answer: $ 1 $

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#### IV) $ \frac{3}{7} - \frac{3}{14} + \frac{3}{7} \times \frac{2}{3} $

Multiplication first:
$ \frac{3}{7} \times \frac{2}{3} = \frac{6}{21} = \frac{2}{7} $

Now:
$$
\frac{3}{7} - \frac{3}{14} + \frac{2}{7}
= \left( \frac{3}{7} + \frac{2}{7} \right) - \frac{3}{14} = \frac{5}{7} - \frac{3}{14}
$$

Convert:
$ \frac{5}{7} = \frac{10}{14} $, so:
$ \frac{10}{14} - \frac{3}{14} = \frac{7}{14} = \frac{1}{2} $

Answer: $ \frac{1}{2} $

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13. The sum of two rational numbers is -3. If one of them is $ -\frac{4}{3} $, find the other number.

Let the other number be $ x $. Then:
$$
-\frac{4}{3} + x = -3 \Rightarrow x = -3 + \frac{4}{3} = -\frac{9}{3} + \frac{4}{3} = -\frac{5}{3}
$$

Answer: $ -\frac{5}{3} $

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14. The product of two rational numbers is $ -\frac{28}{27} $. If one of them is $ -\frac{4}{9} $, find the other number.

Let the other number be $ x $. Then:
$$
-\frac{4}{9} \times x = -\frac{28}{27}
\Rightarrow x = \frac{-\frac{28}{27}}{-\frac{4}{9}} = \frac{28}{27} \times \frac{9}{4} = \frac{252}{108} = \frac{7}{3}
$$

Simplify:
$$
\frac{252 \div 36}{108 \div 36} = \frac{7}{3}
$$

Answer: $ \frac{7}{3} $

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15. Find the area of a square park with each side $ 5\frac{1}{2} $ m

Side = $ 5\frac{1}{2} = \frac{11}{2} $ m

Area of square = $ \text{side}^2 = \left( \frac{11}{2} \right)^2 = \frac{121}{4} = 30.25 $ m²

Answer: $ \frac{121}{4} $ m² or $ 30.25 $ m²

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✔ Final Answers Summary:

| Q.No | Answer |
|------|--------|
| 1 | 0 |
| 2 | $ -5 $ |
| 3 | No |
| 4 | $ \frac{1}{6} $ |
| 5 | $ \frac{2}{15} $ on number line |
| 6 | $ \frac{4}{15} $ |
| 7 | $ -\frac{1}{2},\ 0,\ \frac{1}{4},\ \frac{1}{2} $ (example) |
| 8 | $ \frac{99}{49} $ |
| 9 | $ -\frac{10}{33} $ |
| 10 | $ \frac{17}{4} $ |
| 11 | $ -\frac{131}{70} $ |
| 12(I) | $ \frac{2}{5} $ |
| 12(II) | $ \frac{2}{75} $ |
| 12(III) | $ 1 $ |
| 12(IV) | $ \frac{1}{2} $ |
| 13 | $ -\frac{5}{3} $ |
| 14 | $ \frac{7}{3} $ |
| 15 | $ \frac{121}{4} $ m² or $ 30.25 $ m² |

Let me know if you'd like a PDF version or visual help for the number line!