Class 8 Rational Numbers Worksheet with Questions on Inverses and Operations

Here are the step-by-step solutions for the questions in your worksheet.

1. Write the additive inverse of $(\frac{5}{6} - \frac{-4}{21})$
* Step 1: Simplify the expression inside the brackets first. Subtracting a negative is the same as adding.
$\frac{5}{6} + \frac{4}{21}$
* Step 2: Find a common denominator for 6 and 21. The Least Common Multiple (LCM) is 42.
$\frac{5 \times 7}{6 \times 7} + \frac{4 \times 2}{21 \times 2} = \frac{35}{42} + \frac{8}{42}$
* Step 3: Add the numerators.
$\frac{35 + 8}{42} = \frac{43}{42}$
* Step 4: Find the additive inverse. The additive inverse is just the number with the opposite sign.
The additive inverse of $\frac{43}{42}$ is $-\frac{43}{42}$.

2. Write the multiplicative inverse of the sum of $\frac{-8}{15}$ and $\frac{2}{-3}$
* Step 1: Rewrite $\frac{2}{-3}$ as $-\frac{2}{3}$.
* Step 2: Find the sum: $\frac{-8}{15} + (-\frac{2}{3})$.
* Step 3: Common denominator for 15 and 3 is 15.
$\frac{-8}{15} - \frac{2 \times 5}{3 \times 5} = \frac{-8}{15} - \frac{10}{15}$
* Step 4: Add them up: $\frac{-8 - 10}{15} = \frac{-18}{15}$.
* Step 5: Simplify the fraction by dividing top and bottom by 3: $\frac{-6}{5}$.
* Step 6: Find the multiplicative inverse (reciprocal). Flip the fraction upside down.
The reciprocal of $\frac{-6}{5}$ is $\frac{5}{-6}$ or $-\frac{5}{6}$.

3. Find the product of $\frac{3}{8}$ and the additive inverse of $\frac{4}{15}$
* Step 1: Identify the additive inverse of $\frac{4}{15}$, which is $-\frac{4}{15}$.
* Step 2: Multiply $\frac{3}{8}$ by $-\frac{4}{15}$.
$\frac{3}{8} \times \frac{-4}{15}$
* Step 3: Cancel out common factors before multiplying.
3 and 15 cancel to 1 and 5.
4 and 8 cancel to 1 and 2.
* Step 4: Multiply remaining numbers:
Numerator: $1 \times -1 = -1$
Denominator: $2 \times 5 = 10$
Result: $-\frac{1}{10}$

4. Show $(\frac{1}{2} - \frac{3}{4})$ on the number line.
*(Note: Since I cannot draw, here is how you do it)*
* Step 1: Calculate the value: $\frac{1}{2} - \frac{3}{4} = \frac{2}{4} - \frac{3}{4} = -\frac{1}{4}$.
* Step 2: Draw a number line with 0 in the middle, positive numbers to the right, and negative to the left.
* Step 3: Mark points for $0, -1, -2$, etc.
* Step 4: Divide the space between $0$ and $-1$ into 4 equal parts.
* Step 5: Place a dot on the first mark to the left of 0. That point represents $-\frac{1}{4}$.

5. Write four rational numbers between $\frac{-1}{2}$ and $\frac{1}{3}$
* Step 1: Make the denominators the same to compare them easily. LCM of 2 and 3 is 6.
$\frac{-1}{2} = \frac{-3}{6}$
$\frac{1}{3} = \frac{2}{6}$
* Step 2: We need numbers between $\frac{-3}{6}$ and $\frac{2}{6}$.
The integers between -3 and 2 are: -2, -1, 0, 1.
* Step 3: Write them as fractions over 6:
$\frac{-2}{6}, \frac{-1}{6}, \frac{0}{6}, \frac{1}{6}$
*(You can also simplify $\frac{-2}{6}$ to $\frac{-1}{3}$ if you like, but keeping the denominator 6 is fine).*

6. By what number should $\frac{-33}{16}$ be divided to get $\frac{-11}{4}$?
* Step 1: Set up the equation: $\frac{-33}{16} \div x = \frac{-11}{4}$.
* Step 2: Rearrange to solve for $x$:
$x = \frac{-33}{16} \div \frac{-11}{4}$
* Step 3: Change division to multiplication by flipping the second fraction:
$x = \frac{-33}{16} \times \frac{4}{-11}$
* Step 4: Cancel terms.
$-33 \div -11 = 3$
$4 \div 16 = \frac{1}{4}$
* Step 5: Multiply results: $3 \times \frac{1}{4} = \frac{3}{4}$.

7. What should be subtracted from $(\frac{2}{3} - \frac{2}{5})$ to get $\frac{-1}{6}$?
* Step 1: Simplify the first part: $\frac{2}{3} - \frac{2}{5}$.
Common denominator is 15.
$\frac{10}{15} - \frac{6}{15} = \frac{4}{15}$.
* Step 2: Set up equation: $\frac{4}{15} - x = \frac{-1}{6}$.
* Step 3: Solve for $x$:
$x = \frac{4}{15} - (\frac{-1}{6})$
$x = \frac{4}{15} + \frac{1}{6}$
* Step 4: Common denominator for 15 and 6 is 30.
$\frac{4 \times 2}{30} + \frac{1 \times 5}{30} = \frac{8}{30} + \frac{5}{30}$
* Step 5: Add them: $\frac{13}{30}$.

8. Simplify:
*(I will solve one example from each row to show the method, as there are many sub-questions)*

* a) $(\frac{-4}{3} \times \frac{12}{-5}) \times (\frac{3}{7} \times \frac{21}{8})$
First bracket: Negatives cancel. $\frac{4 \times 12}{3 \times 5}$. 12/3=4. So, $\frac{16}{5}$.
Second bracket: $\frac{3 \times 21}{7 \times 8}$. 21/7=3. So, $\frac{9}{8}$.
Multiply results: $\frac{16}{5} \times \frac{9}{8}$. 16/8=2.
Final: $\frac{2 \times 9}{5} = \frac{18}{5}$.

* b) $\frac{3}{5} \times (\frac{-7}{8} - \frac{1}{10})$
Inside bracket: LCD is 40. $\frac{-35}{40} - \frac{4}{40} = \frac{-39}{40}$.
Multiply: $\frac{3}{5} \times \frac{-39}{40} = \frac{-117}{200}$.

* d) $(\frac{-3}{5} - \frac{1}{10}) - (\frac{-4}{5} - \frac{8}{5})$
First bracket: $\frac{-6}{10} - \frac{1}{10} = \frac{-7}{10}$.
Second bracket: $\frac{-4-8}{5} = \frac{-12}{5}$.
Subtract: $\frac{-7}{10} - (\frac{-12}{5}) = \frac{-7}{10} + \frac{24}{10} = \frac{17}{10}$.

9. Simplify by using suitable property:
*(These use Distributive Property: $a(b+c) = ab + ac$)*

* i) $(\frac{-7}{5}) \times (\frac{-3}{5}) + (\frac{-7}{5}) \times (\frac{-2}{5})$
Take out common factor $\frac{-7}{5}$:
$\frac{-7}{5} \times (\frac{-3}{5} + \frac{-2}{5})$
$\frac{-7}{5} \times (\frac{-5}{5})$
$\frac{-7}{5} \times (-1) = \frac{7}{5}$

* ii) $(\frac{-4}{3}) \times \frac{2}{5} + (\frac{-4}{3}) \times \frac{-7}{6}$
Take out $\frac{-4}{3}$:
$\frac{-4}{3} \times (\frac{2}{5} + \frac{-7}{6})$
Inside bracket LCD is 30: $\frac{12}{30} - \frac{35}{30} = \frac{-23}{30}$
Multiply: $\frac{-4}{3} \times \frac{-23}{30} = \frac{92}{90} = \frac{46}{45}$

10. The product of two rational numbers is $\frac{-14}{9}$. If one of them is $\frac{-5}{18}$, find the other number.
* Step 1: Equation: $x \times \frac{-5}{18} = \frac{-14}{9}$.
* Step 2: $x = \frac{-14}{9} \div \frac{-5}{18}$.
* Step 3: $x = \frac{-14}{9} \times \frac{18}{-5}$.
* Step 4: Negatives cancel. 18 divided by 9 is 2.
$x = 14 \times \frac{2}{5} = \frac{28}{5}$.

11. What should be added to $\frac{-4}{15}$ to get additive inverse of $(\frac{-2}{3} + \frac{4}{5})$?
* Step 1: Find the sum inside: $\frac{-2}{3} + \frac{4}{5}$.
LCD is 15. $\frac{-10}{15} + \frac{12}{15} = \frac{2}{15}$.
* Step 2: Find the additive inverse of that sum: $-\frac{2}{15}$.
* Step 3: Set up equation: $\frac{-4}{15} + x = -\frac{2}{15}$.
* Step 4: $x = -\frac{2}{15} - (\frac{-4}{15}) = -\frac{2}{15} + \frac{4}{15}$.
* Step 5: $x = \frac{2}{15}$.

12. What should be multiplied with $(\frac{-5}{9} - \frac{5}{18})$ to get $(\frac{5}{6} - \frac{3}{4})$?
* Step 1: Simplify first bracket: $\frac{-10}{18} - \frac{5}{18} = \frac{-15}{18} = \frac{-5}{6}$.
* Step 2: Simplify second bracket: LCD 12. $\frac{10}{12} - \frac{9}{12} = \frac{1}{12}$.
* Step 3: Equation: $\frac{-5}{6} \times x = \frac{1}{12}$.
* Step 4: $x = \frac{1}{12} \div \frac{-5}{6} = \frac{1}{12} \times \frac{6}{-5}$.
* Step 5: $x = \frac{1}{2 \times -5} = \frac{-1}{10}$.

13. What should be added to $(\frac{-5}{8})$ to get multiplicative inverse of $(\frac{1}{7} - \frac{2}{21})$?
* Step 1: Simplify inside bracket: $\frac{3}{21} - \frac{2}{21} = \frac{1}{21}$.
* Step 2: Multiplicative inverse of $\frac{1}{21}$ is $21$ (or $\frac{21}{1}$).
* Step 3: Equation: $\frac{-5}{8} + x = 21$.
* Step 4: $x = 21 - (\frac{-5}{8}) = 21 + \frac{5}{8}$.
* Step 5: Convert 21 to fraction with denominator 8: $\frac{168}{8}$.
$x = \frac{168 + 5}{8} = \frac{173}{8}$.

14. What should be subtracted from $(\frac{-4}{9} - \frac{-3}{8})$ to get additive inverse of $(\frac{-5}{-4})$?
* Step 1: Simplify first part: $\frac{-4}{9} + \frac{3}{8}$.
LCD is 72. $\frac{-32}{72} + \frac{27}{72} = \frac{-5}{72}$.
* Step 2: Simplify target value: $\frac{-5}{-4} = \frac{5}{4}$.
Additive inverse of $\frac{5}{4}$ is $-\frac{5}{4}$.
* Step 3: Equation: $\frac{-5}{72} - x = -\frac{5}{4}$.
* Step 4: $x = \frac{-5}{72} - (-\frac{5}{4}) = \frac{-5}{72} + \frac{5}{4}$.
* Step 5: LCD is 72. $\frac{5}{4} = \frac{90}{72}$.
$x = \frac{-5 + 90}{72} = \frac{85}{72}$.

Final Answer:
1. $-\frac{43}{42}$
2. $-\frac{5}{6}$
3. $-\frac{1}{10}$
4. Plot $-\frac{1}{4}$ on the number line (between 0 and -1).
5. $\frac{-2}{6}, \frac{-1}{6}, 0, \frac{1}{6}$ (or simplified: $-\frac{1}{3}, -\frac{1}{6}, 0, \frac{1}{6}$)
6. $\frac{3}{4}$
7. $\frac{13}{30}$
8. a) $\frac{18}{5}$, b) $-\frac{117}{200}$, d) $\frac{17}{10}$
9. i) $\frac{7}{5}$, ii) $\frac{46}{45}$
10. $\frac{28}{5}$
11. $\frac{2}{15}$
12. $-\frac{1}{10}$
13. $\frac{173}{8}$
14. $\frac{85}{72}$