CBSE Class 7 Maths Worksheet - Rational Numbers | PDF - Free Printable

Let's solve each question from the Maths Worksheet for Grade VII on Rational Numbers step by step.

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Fill in the blanks

1. A rational number where both numerator and denominator are positive integers is called a ________ rational number.
✔ Positive
*Explanation:* If both numerator and denominator are positive, the fraction is positive.

2. The number that is neither a positive nor a negative rational number is ________.
✔ Zero (0)
*Explanation:* 0 is neutral; it's not positive or negative.

3. There are ________ number of rational numbers between two rational numbers.
✔ Infinite
*Explanation:* Between any two rational numbers, you can always find more rational numbers (e.g., average of two numbers).

4. $\frac{6}{7} \times \frac{-7}{5} = $______
✔ $\frac{6 \times (-7)}{7 \times 5} = \frac{-42}{35} = \frac{-6}{5}$
*Simplify:* Divide numerator and denominator by 7 → $-\frac{6}{5}$

5. The additive inverse of $\frac{9}{11}$ is ______.
✔ $-\frac{9}{11}$
*Explanation:* Additive inverse means the number that adds to zero: $\frac{9}{11} + (-\frac{9}{11}) = 0$

6. The product of a rational number with its reciprocal is always ________.
✔ 1
*Example:* $\frac{a}{b} \times \frac{b}{a} = 1$, provided $a,b \ne 0$

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Do as directed

7. Write four more numbers in the same pattern: $\frac{2}{5}, \frac{4}{10}, \frac{6}{15}, \frac{8}{20}$
Pattern: Numerator increases by 2, denominator by 5 → So next terms:
- $\frac{10}{25}$, $\frac{12}{30}$, $\frac{14}{35}$, $\frac{16}{40}$
✔ All equivalent to $\frac{2}{5}$

8. Is the number $-\frac{3}{-6}$ rational?
✔ Yes.
Simplify: $-\frac{3}{-6} = \frac{3}{6} = \frac{1}{2}$, which is rational.

9. Is it correct to say that all integers are rational numbers? Justify your answer with example.
✔ Yes.
*Justification:* Any integer $n$ can be written as $\frac{n}{1}$, which is a ratio of two integers.
*Example:* $5 = \frac{5}{1}$, $-3 = \frac{-3}{1}$ → Both rational.

10. All fractions are rational numbers. Is the converse true? Justify.
✔ No, the converse is not true.
*Explanation:* All fractions (like $\frac{p}{q}$, $q \ne 0$) are rational numbers. But not all rational numbers are fractions in the form of p/q with integers only, but actually — wait: by definition, rational numbers are fractions of integers.
So actually:
- Every fraction (with integers) is rational.
- Every rational number can be expressed as a fraction of integers.
So yes, they are equivalent.
But note: "fraction" sometimes refers to proper fractions (numerator < denominator), but mathematically, rational numbers include all such forms.
✔ So: Yes, all fractions are rational numbers. The converse is also true because every rational number can be written as a fraction of integers.
But if “fraction” means only those with integers numerator/denominator, then yes — they are the same set.

However, some people distinguish:
- Fraction: e.g., $\frac{3}{4}$, $\frac{7}{2}$, etc.
- Rational number: includes decimals like $0.333... = \frac{1}{3}$, which may not be written as a simple fraction at first.

But strictly speaking: Every rational number can be written as a fraction of two integers. So the converse is true.

✔ Final Answer: Yes, the converse is true.
*Justification:* Every rational number is expressible as $\frac{p}{q}$, $q \ne 0$, $p,q$ integers → so it is a fraction.

11. List 5 rational numbers between:
a) $-\frac{4}{7}$ and $-\frac{3}{8}$
b) $-1$ and $0$

a) First, find common denominator: LCM of 7 and 8 is 56
$-\frac{4}{7} = -\frac{32}{56}$, $-\frac{3}{8} = -\frac{21}{56}$
We need numbers between $-\frac{32}{56}$ and $-\frac{21}{56}$ → i.e., more than $-\frac{32}{56}$, less than $-\frac{21}{56}$
Examples: $-\frac{31}{56}, -\frac{30}{56}, -\frac{29}{56}, -\frac{28}{56}, -\frac{27}{56}$
✔ Simplified: $-\frac{31}{56}, -\frac{15}{28}, -\frac{29}{56}, -\frac{1}{2}, -\frac{27}{56}$

b) Between $-1$ and $0$:
$-\frac{1}{2}, -\frac{1}{3}, -\frac{1}{4}, -\frac{1}{5}, -\frac{1}{6}$
✔ These are five rational numbers.

12. Reduce to standard form:
a) $\frac{-18}{45}$ → divide numerator and denominator by 9 → $-\frac{2}{5}$
b) $\frac{-3}{-15}$ → $+\frac{3}{15} = \frac{1}{5}$
c) $\frac{28}{56}$ → divide by 28 → $\frac{1}{2}$
d) $\frac{45}{99}$ → divide by 9 → $\frac{5}{11}$

✔ Answers:
a) $-\frac{2}{5}$, b) $\frac{1}{5}$, c) $\frac{1}{2}$, d) $\frac{5}{11}$

13. Find three equivalent fractions for each:
a) $\frac{7}{8}$ → multiply numerator and denominator by 2,3,4:
$\frac{14}{16}, \frac{21}{24}, \frac{28}{32}$
b) $\frac{9}{11}$ → $\frac{18}{22}, \frac{27}{33}, \frac{36}{44}$
c) $\frac{3}{5}$ → $\frac{6}{10}, \frac{9}{15}, \frac{12}{20}$

14. Write the following in descending order:

a) $-\frac{28}{56}, -\frac{2}{3}, -\frac{3}{8}$
Simplify: $-\frac{28}{56} = -\frac{1}{2}$
Compare: $-\frac{1}{2} = -0.5$, $-\frac{2}{3} \approx -0.666$, $-\frac{3}{8} = -0.375$
Descending (largest to smallest): $-\frac{3}{8}, -\frac{1}{2}, -\frac{2}{3}$
✔ So: $-\frac{3}{8}, -\frac{28}{56}, -\frac{2}{3}$

b) $\frac{8}{5}, \frac{2}{6}, \frac{5}{7}$
Simplify: $\frac{2}{6} = \frac{1}{3} \approx 0.333$, $\frac{8}{5} = 1.6$, $\frac{5}{7} \approx 0.714$
Descending: $\frac{8}{5}, \frac{5}{7}, \frac{1}{3}$
✔ So: $\frac{8}{5}, \frac{5}{7}, \frac{2}{6}$

15. Represent the following rational numbers on a number line:
a) $\frac{2}{3}$ → lies between 0 and 1, closer to 1
b) $-\frac{4}{3}$ → $-1.\overline{3}$ → between -2 and -1
c) $-\frac{8}{5} = -1.6$ → between -2 and -1, closer to -2

*(Sketch not possible here, but students should mark positions accordingly)*

16. Write the additive inverse and multiplicative inverse of:

a) $\frac{2}{9}$
- Additive inverse: $-\frac{2}{9}$
- Multiplicative inverse: $\frac{9}{2}$

b) $\frac{8}{7}$
- Additive inverse: $-\frac{8}{7}$
- Multiplicative inverse: $\frac{7}{8}$

c) $-\frac{3}{11}$
- Additive inverse: $\frac{3}{11}$
- Multiplicative inverse: $-\frac{11}{3}$

17. Find the sum of:

a) $3\frac{8}{7} + \frac{2}{5}$ → Convert mixed: $3\frac{8}{7} = \frac{29}{7}$
LCM of 7 and 5 is 35:
$\frac{29}{7} = \frac{145}{35}, \frac{2}{5} = \frac{14}{35}$ → Sum = $\frac{159}{35} = 4\frac{19}{35}$

b) $-2\frac{1}{8} + \frac{6}{10}$ → $-2\frac{1}{8} = -\frac{17}{8}$, $\frac{6}{10} = \frac{3}{5}$
LCM of 8 and 5 is 40:
$-\frac{17}{8} = -\frac{85}{40}, \frac{3}{5} = \frac{24}{40}$ → Sum = $-\frac{61}{40} = -1\frac{21}{40}$

c) $-\frac{7}{13} + (-\frac{8}{15}) = -\left(\frac{7}{13} + \frac{8}{15}\right)$
LCM of 13 and 15 = 195
$\frac{7}{13} = \frac{105}{195}, \frac{8}{15} = \frac{104}{195}$ → Sum = $\frac{209}{195}$ → Negative: $-\frac{209}{195} = -1\frac{14}{195}$

18. Find:

a) $\frac{7}{24} - (-\frac{2}{9}) = \frac{7}{24} + \frac{2}{9}$
LCM of 24 and 9 = 72
$\frac{7}{24} = \frac{21}{72}, \frac{2}{9} = \frac{16}{72}$ → Sum = $\frac{37}{72}$

b) $(-4\frac{1}{9}) - 1\frac{2}{7}$ → Convert:
$-4\frac{1}{9} = -\frac{37}{9}$, $1\frac{2}{7} = \frac{9}{7}$
$-\frac{37}{9} - \frac{9}{7} = -\left(\frac{37}{9} + \frac{9}{7}\right)$
LCM of 9 and 7 = 63
$\frac{37}{9} = \frac{259}{63}, \frac{9}{7} = \frac{81}{63}$ → Sum = $\frac{340}{63}$ → Negative: $-\frac{340}{63} = -5\frac{25}{63}$

c) $8 - 2\frac{11}{14} = 8 - \frac{43}{14} = \frac{112}{14} - \frac{43}{14} = \frac{69}{14} = 4\frac{13}{14}$

d) $\frac{15}{12} - \frac{22}{15}$ → Simplify $\frac{15}{12} = \frac{5}{4}$
LCM of 4 and 15 = 60
$\frac{5}{4} = \frac{75}{60}, \frac{22}{15} = \frac{88}{60}$ → Difference = $\frac{75 - 88}{60} = -\frac{13}{60}$

19. Find the value of:

a) $-4 + \frac{8}{7} = -\frac{28}{7} + \frac{8}{7} = -\frac{20}{7} = -2\frac{6}{7}$

b) $(-\frac{3}{5}) + 7 = -\frac{3}{5} + \frac{35}{5} = \frac{32}{5} = 6\frac{2}{5}$

c) $(-\frac{8}{11}) + \frac{16}{22} = -\frac{8}{11} + \frac{8}{11} = 0$

d) $\frac{3}{13} + (-\frac{9}{2}) = \frac{3}{13} - \frac{9}{2}$
LCM of 13 and 2 = 26
$\frac{3}{13} = \frac{6}{26}, \frac{9}{2} = \frac{117}{26}$ → $\frac{6 - 117}{26} = -\frac{111}{26} = -4\frac{7}{26}$

20. Find the product:

a) $\frac{9}{7} \times (-\frac{8}{3}) = -\frac{72}{21} = -\frac{24}{7} = -3\frac{3}{7}$

b) $(-\frac{2}{3}) \times (\frac{-5}{3}) = \frac{10}{9} = 1\frac{1}{9}$

c) $\frac{7}{15} \times (-\frac{9}{28}) = -\frac{63}{420} = -\frac{3}{20}$

d) $(-\frac{4}{9}) \times \frac{11}{12} = -\frac{44}{108} = -\frac{11}{27}$

21. The sum of two numbers is $-\frac{11}{12}$. One of them is $\frac{3}{2}$. Find the other.
Let the other be $x$:
$x + \frac{3}{2} = -\frac{11}{12}$
$x = -\frac{11}{12} - \frac{3}{2} = -\frac{11}{12} - \frac{18}{12} = -\frac{29}{12}$

22. The product of two numbers is $(-\frac{5}{7})$. One of them is $\frac{3}{4}$. Find the other.
Let the other be $x$:
$x \times \frac{3}{4} = -\frac{5}{7}$
$x = -\frac{5}{7} \div \frac{3}{4} = -\frac{5}{7} \times \frac{4}{3} = -\frac{20}{21}$

23. Simplify: $5\frac{6}{7} + (-2\frac{1}{2}) + 1\frac{5}{6}$
Convert to improper fractions:
$5\frac{6}{7} = \frac{41}{7}$, $-2\frac{1}{2} = -\frac{5}{2}$, $1\frac{5}{6} = \frac{11}{6}$
Now add: $\frac{41}{7} - \frac{5}{2} + \frac{11}{6}$
LCM of 7, 2, 6 = 42
$\frac{41}{7} = \frac{246}{42}, \frac{5}{2} = \frac{105}{42}, \frac{11}{6} = \frac{77}{42}$
Total: $\frac{246 - 105 + 77}{42} = \frac{218}{42} = \frac{109}{21} = 5\frac{4}{21}$

24. Compare:

a) $-3\frac{4}{7}$ and $-3\frac{1}{5}$
Convert to improper:
$-3\frac{4}{7} = -\frac{25}{7} \approx -3.571$, $-3\frac{1}{5} = -\frac{16}{5} = -3.2$
Since $-3.571 < -3.2$, we have:
$-3\frac{4}{7} < -3\frac{1}{5}$

b) $-\frac{7}{5}$ and $-\frac{3}{5}$
Both negative, compare absolute values: $\frac{7}{5} > \frac{3}{5}$ → so $-\frac{7}{5} < -\frac{3}{5}$

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✔ Final Answers Summary:

1. Positive
2. Zero
3. Infinite
4. $-\frac{6}{5}$
5. $-\frac{9}{11}$
6. 1
7. $\frac{10}{25}, \frac{12}{30}, \frac{14}{35}, \frac{16}{40}$
8. Yes
9. Yes, because integers can be written as $\frac{n}{1}$
10. Yes, the converse is true (every rational number is a fraction of integers)
11. a) $-\frac{31}{56}, -\frac{30}{56}, -\frac{29}{56}, -\frac{28}{56}, -\frac{27}{56}$
b) $-\frac{1}{2}, -\frac{1}{3}, -\frac{1}{4}, -\frac{1}{5}, -\frac{1}{6}$
12. a) $-\frac{2}{5}$, b) $\frac{1}{5}$, c) $\frac{1}{2}$, d) $\frac{5}{11}$
13. a) $\frac{14}{16}, \frac{21}{24}, \frac{28}{32}$
b) $\frac{18}{22}, \frac{27}{33}, \frac{36}{44}$
c) $\frac{6}{10}, \frac{9}{15}, \frac{12}{20}$
14. a) $-\frac{3}{8}, -\frac{28}{56}, -\frac{2}{3}$
b) $\frac{8}{5}, \frac{5}{7}, \frac{2}{6}$
15. Represent on number line (as described)
16. a) Additive: $-\frac{2}{9}$, Multiplicative: $\frac{9}{2}$
b) Additive: $-\frac{8}{7}$, Multiplicative: $\frac{7}{8}$
c) Additive: $\frac{3}{11}$, Multiplicative: $-\frac{11}{3}$
17. a) $4\frac{19}{35}$, b) $-1\frac{21}{40}$, c) $-1\frac{14}{195}$
18. a) $\frac{37}{72}$, b) $-5\frac{25}{63}$, c) $4\frac{13}{14}$, d) $-\frac{13}{60}$
19. a) $-2\frac{6}{7}$, b) $6\frac{2}{5}$, c) 0, d) $-4\frac{7}{26}$
20. a) $-3\frac{3}{7}$, b) $1\frac{1}{9}$, c) $-\frac{3}{20}$, d) $-\frac{11}{27}$
21. $-\frac{29}{12}$
22. $-\frac{20}{21}$
23. $5\frac{4}{21}$
24. a) $-3\frac{4}{7} < -3\frac{1}{5}$, b) $-\frac{7}{5} < -\frac{3}{5}$

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