CBSE Class 7 Mathematics Rational Numbers Worksheet Set 01

Question. Solve the following (use BODMAS):
(a) \( \frac{7}{3} + \left(-\frac{5}{3}\right) \)
(b) \( 5\frac{2}{5} + \frac{8}{3} + \left(-3\frac{7}{4}\right) \)
(c) \( \frac{7}{15} \div \left(\frac{2}{3} + \frac{5}{8}\right) \)
(d) \( \left(\frac{2}{5} + \frac{5}{8}\right) \times \left(\frac{7}{3} - \frac{5}{4}\right) \)
(e) \( \left(\frac{4}{7} \div \frac{4}{3}\right) \times 2\frac{4}{7} \)
(f) \( \frac{6}{-5} + \left(-\frac{3}{2}\right) \)
Answer:
(a) \( \frac{7}{3} + \left(-\frac{5}{3}\right) = \frac{7 - 5}{3} = \frac{2}{3} \)

(b) \( 5\frac{2}{5} + \frac{8}{3} + \left(-3\frac{7}{4}\right) \)
Converting mixed fractions to improper fractions:
\( 5\frac{2}{5} = \frac{27}{5} \)
\( -3\frac{7}{4} = -\left(3 + \frac{7}{4}\right) = -\frac{19}{4} \)
LCM of denominators \( 5, 3, 4 \) is \( 60 \):
\( = \frac{27 \times 12}{60} + \frac{8 \times 20}{60} - \frac{19 \times 15}{60} \)
\( = \frac{324 + 160 - 285}{60} = \frac{484 - 285}{60} = \frac{199}{60} = 3\frac{19}{60} \)
(Note: If read as \( -3\frac{1}{4} = -\frac{13}{4} \), then \( \frac{324 + 160 - 195}{60} = \frac{289}{60} = 4\frac{49}{60} \))

(c) \( \frac{7}{15} \div \left(\frac{2}{3} + \frac{5}{8}\right) \)
First solve inside the bracket:
\( \frac{2}{3} + \frac{5}{8} = \frac{2 \times 8 + 5 \times 3}{24} = \frac{16 + 15}{24} = \frac{31}{24} \)
Now divide:
\( \frac{7}{15} \div \frac{31}{24} = \frac{7}{15} \times \frac{24}{31} = \frac{7 \times 8}{5 \times 31} = \frac{56}{155} \)

(d) \( \left(\frac{2}{5} + \frac{5}{8}\right) \times \left(\frac{7}{3} - \frac{5}{4}\right) \)
First bracket: \( \frac{2}{5} + \frac{5}{8} = \frac{16 + 25}{40} = \frac{41}{40} \)
Second bracket: \( \frac{7}{3} - \frac{5}{4} = \frac{28 - 15}{12} = \frac{13}{12} \)
Now multiply:
\( \frac{41}{40} \times \frac{13}{12} = \frac{533}{480} = 1\frac{53}{480} \)

(e) \( \left(\frac{4}{7} \div \frac{4}{3}\right) \times 2\frac{4}{7} \)
First bracket: \( \frac{4}{7} \div \frac{4}{3} = \frac{4}{7} \times \frac{3}{4} = \frac{3}{7} \)
Convert mixed number: \( 2\frac{4}{7} = \frac{18}{7} \)
Now multiply:
\( \frac{3}{7} \times \frac{18}{7} = \frac{54}{49} = 1\frac{5}{49} \)

(f) \( \frac{6}{-5} + \left(-\frac{3}{2}\right) = -\frac{6}{5} - \frac{3}{2} = \frac{-12 - 15}{10} = -\frac{27}{10} = -2\frac{7}{10} \)

Question. Compare:
(a) \( \frac{-2}{4} \) , \( \frac{8}{-16} \)
(b) \( \frac{3}{4} \) , \( -\frac{4}{7} \)
(c) \( -\frac{5}{9} \) , \( -\frac{4}{9} \)
(d) \( \frac{11}{-9} \) , \( -\frac{11}{9} \)
(e) \( \frac{2}{5} \) , \( \frac{5}{4} \)
(f) \( -\frac{2}{5} \) , \( \frac{5}{4} \)
(g) \( \frac{2}{5} \) , \( -\frac{5}{4} \)
(h) \( -\frac{3}{5} \) , \( -\frac{12}{20} \)
Answer:
(a) Simplifying both sides:
\( \frac{-2}{4} = -\frac{1}{2} \) and \( \frac{8}{-16} = -\frac{1}{2} \)
Thus, \( \frac{-2}{4} = \frac{8}{-16} \)

(b) A positive rational number is always greater than a negative rational number.
Thus, \( \frac{3}{4} > -\frac{4}{7} \)

(c) Denominators are the same. Comparing numerators: \( -5 < -4 \).
Thus, \( -\frac{5}{9} < -\frac{4}{9} \)

(d) In standard form, \( \frac{11}{-9} = -\frac{11}{9} \).
Thus, \( \frac{11}{-9} = -\frac{11}{9} \)

(e) Cross multiplying: \( 2 \times 4 = 8 \) and \( 5 \times 5 = 25 \). Since \( 8 < 25 \):
Thus, \( \frac{2}{5} < \frac{5}{4} \)

(f) A negative rational number is always less than a positive rational number.
Thus, \( -\frac{2}{5} < \frac{5}{4} \)

(g) A positive rational number is always greater than a negative rational number.
Thus, \( \frac{2}{5} > -\frac{5}{4} \)

(h) Simplifying the second rational number: \( -\frac{12}{20} = -\frac{12 \div 4}{20 \div 4} = -\frac{3}{5} \).
Thus, \( -\frac{3}{5} = -\frac{12}{20} \)

Question. List the given number of rational numbers between \( -2 \) and \( -1 \):
(a) 3
(b) 4
(c) 5
(d) 6
Answer:
(a) To find 3 rational numbers, write \( -2 \) and \( -1 \) with denominator \( (3 + 1) = 4 \):
\( -2 = -\frac{8}{4} \) and \( -1 = -\frac{4}{4} \)
The 3 rational numbers are: \( -\frac{7}{4}, -\frac{6}{4} \) (or \( -\frac{3}{2} \)), \( -\frac{5}{4} \)

(b) To find 4 rational numbers, write \( -2 \) and \( -1 \) with denominator \( (4 + 1) = 5 \):
\( -2 = -\frac{10}{5} \) and \( -1 = -\frac{5}{5} \)
The 4 rational numbers are: \( -\frac{9}{5}, -\frac{8}{5}, -\frac{7}{5}, -\frac{6}{5} \)

(c) To find 5 rational numbers, write \( -2 \) and \( -1 \) with denominator \( (5 + 1) = 6 \):
\( -2 = -\frac{12}{6} \) and \( -1 = -\frac{6}{6} \)
The 5 rational numbers are: \( -\frac{11}{6}, -\frac{10}{6} \) (or \( -\frac{5}{3} \)), \( -\frac{9}{6} \) (or \( -\frac{3}{2} \)), \( -\frac{8}{6} \) (or \( -\frac{4}{3} \)), \( -\frac{7}{6} \)

(d) To find 6 rational numbers, write \( -2 \) and \( -1 \) with denominator \( (6 + 1) = 7 \):
\( -2 = -\frac{14}{7} \) and \( -1 = -\frac{7}{7} \)
The 6 rational numbers are: \( -\frac{13}{7}, -\frac{12}{7}, -\frac{11}{7}, -\frac{10}{7}, -\frac{9}{7}, -\frac{8}{7} \)

Question. List five rational numbers between:
(a) \( -3 \) and \( -2 \)
(b) \( -2 \) and \( -1 \)
(c) \( -\frac{2}{3} \) and \( -\frac{1}{3} \)
(d) \( -\frac{4}{5} \) and \( -\frac{4}{7} \)
Answer:
(a) Expressing with denominator \( (5 + 1) = 6 \):
\( -3 = -\frac{18}{6} \) and \( -2 = -\frac{12}{6} \)
Five rational numbers are: \( -\frac{17}{6}, -\frac{16}{6}, -\frac{15}{6}, -\frac{14}{6}, -\frac{13}{6} \) (or \( -\frac{17}{6}, -\frac{8}{3}, -\frac{5}{2}, -\frac{7}{3}, -\frac{13}{6} \))

(b) Expressing with denominator \( (5 + 1) = 6 \):
\( -2 = -\frac{12}{6} \) and \( -1 = -\frac{6}{6} \)
Five rational numbers are: \( -\frac{11}{6}, -\frac{10}{6}, -\frac{9}{6}, -\frac{8}{6}, -\frac{7}{6} \) (or \( -\frac{11}{6}, -\frac{5}{3}, -\frac{3}{2}, -\frac{4}{3}, -\frac{7}{6} \))

(c) Multiplying numerator and denominator by 6:
\( -\frac{2}{3} = -\frac{12}{18} \) and \( -\frac{1}{3} = -\frac{6}{18} \)
Five rational numbers are: \( -\frac{11}{18}, -\frac{10}{18}, -\frac{9}{18}, -\frac{8}{18}, -\frac{7}{18} \) (or \( -\frac{11}{18}, -\frac{5}{9}, -\frac{1}{2}, -\frac{4}{9}, -\frac{7}{18} \))

(d) Taking LCM of denominators \( 5 \) and \( 7 \), which is \( 35 \):
\( -\frac{4}{5} = -\frac{28}{35} \) and \( -\frac{4}{7} = -\frac{20}{35} \)
Five rational numbers between them are: \( -\frac{27}{35}, -\frac{26}{35}, -\frac{25}{35}, -\frac{24}{35}, -\frac{23}{35} \) (or \( -\frac{27}{35}, -\frac{26}{35}, -\frac{5}{7}, -\frac{24}{35}, -\frac{23}{35} \))

Question. Rewrite the following rational numbers in their standard form.
(a) \( \frac{-10}{5} \)
(b) \( \frac{25}{-60} \)
(c) \( \frac{-45}{-90} \)
(d) \( \frac{-7}{21} \)
Answer:
(a) Dividing numerator and denominator by their HCF \( 5 \):
\( \frac{-10 \div 5}{5 \div 5} = \frac{-2}{1} = -2 \)

(b) Making the denominator positive and dividing by HCF \( 5 \):
\( \frac{25 \div (-5)}{-60 \div (-5)} = -\frac{5}{12} \)

(c) Dividing numerator and denominator by HCF \( -45 \):
\( \frac{-45 \div (-45)}{-90 \div (-45)} = \frac{1}{2} \)

(d) Dividing numerator and denominator by HCF \( 7 \):
\( \frac{-7 \div 7}{21 \div 7} = -\frac{1}{3} \)

Question. Write true or false. Also explain the reason.
(a) all integers are rational numbers
(b) all fractions are rational numbers
(c) all rational numbers are integers.
(d) all rational numbers are fractions.
(e) 0 is a rational number.
(f) There is only a limited number of rational numbers between two rational numbers.
(g) Some rational numbers are whole numbers.
Answer:
(a) True.
Reason: Any integer \( m \) can be represented as \( \frac{m}{1} \), which is in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \).

(b) True.
Reason: A fraction is expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are whole numbers (\( b \neq 0 \)). Since all whole numbers are integers, every fraction is also a rational number.

(c) False.
Reason: Rational numbers like \( \frac{2}{3} \), \( -\frac{5}{7} \), etc., are not whole numbers or their negatives, and therefore are not integers.

(d) False.
Reason: Fractions cannot be negative, whereas rational numbers include negative values such as \( -\frac{3}{4} \), \( -5 \), etc.

(e) True.
Reason: \( 0 \) can be written as \( \frac{0}{1} \), where both \( 0 \) and \( 1 \) are integers and the denominator is non-zero.

(f) False.
Reason: There are infinitely many (an unlimited number of) rational numbers between any two given rational numbers.

(g) True.
Reason: Whole numbers like \( 0, 1, 2, 3 \dots \) can be written as \( \frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \dots \), which are rational numbers.

Question. Write down the numerator of each of the following rational numbers:
(i) (-7/5)
(ii) (14/-4)
Solution.
(i) Given (-7/5)
Numerator of (-7/5) is -7
(ii) Given (14/-4)
Numerator of (14/-4) is 1

Question. Write down the denominator of each of the following rational numbers:
(i) (-4/5)
(ii) (11/-34)
Solution.
(i) Given (-4/5)
Denominator of (-4/5) is 5
(ii) Given (11/-34)
Denominator of (11/-34) is -43

Question. Write down the rational number whose numerator is (-3) × 4, and whose denominator is (34 – 23) × (7 – 4).
Solution.
Given numerator = (-3) × 4 = -12
Denominator = (34 – 23) × (7 – 4)
= 11 × 3 = 33
Therefore the rational number = (-12/33)

Question. Write down the rational numbers as integers: (7/1), (-12/1), (34/1), (-73/1), (95/1)
Solution.
Given (7/1), (-12/1), (34/1), (-73/1), (95/1)
Integers of (7/1), (-12/1), (34/1), (-73/1), (95/1) are 7, -12, 34, -73, 95

Question. Write the following integers as rational numbers: -15, 17, 85, -100
Solution.
Given -15, 17, 85, -100
The rational numbers of given integers are (-15/1), (17/1), (85/1) and (-100/1)

Question. Write down the rational number whose numerator is the smallest three digit number and denominator is the largest four digit number.
Solution.
Smallest three digit number = 100
Largest four digit number = 9999
Therefore the rational number is = 100/9999

Question. Separate positive and negative rational numbers from the following rational numbers:
(-5/-7), (12/-5), (7/4), (13/-9), 0, (-18/-7), (-95/116), (-1/-9)
Solution.
Given (-5/-7), (12/-5), (7/4), (13/-9), 0, (-18/-7), (-95/116), (-1/-9)
A rational number is said to be positive if its numerator and denominator are either positive integers or both negative integers.
Therefore positive rational numbers are: (-5/-7), (-18/-7), (7/4), (-1/-9)
A rational number is said to be negative integers if its numerator and denominator are such that one of them is positive integer and another one is a negative integer.
Therefore negative rational numbers are: (12/-5), (13/-9), (-95/116)

Question. Which of the following rational numbers are positive:
(i) (-8/7)
(ii) (9/8)
(iii) (-19/-13)
(iv) (-21/13)
Solution.
Given (-8/7), (9/8), (-19/-13), (-21/13)
A rational number is said to be positive if its numerator and denominator are either positive integers or both negative integers.
Therefore the positive rational numbers are (9/8) and (-19/-13)

Question. Which of the following rational numbers are negative:
(i) (-3/7)
(ii) (-5/-8)
(iii) (9/-83)
(iv) (-115/-197)
Solution.
Given (-3/7), (-5/-8), (9/-83), (-115/-197)
A rational number is said to be negative integers if its numerator and denominator are such that one of them is positive integer and another one is a negative integer.
Therefore negative rational numbers are (-3/7) and (9/-83)