CBSE Class 7 Mathematics Rational Numbers Worksheet Set 02

Question 1. The expression which best describes the rational no. \(\frac{0}{3}\) is:
(a) 0
(b) 1
(c) 3
(d) none of the options
Answer: (a) 0
Numerator = 0 and denominator = 3, and since the numerator is 0 and denominator is non-zero, the value of the rational number is 0.
In simple words: Zero divided by any non-zero number is always zero.
Exam Tip: Remember that \(\frac{0}{a} = 0\) where \(a \neq 0\), but \(\frac{a}{0}\) is undefined.

Question 2. The additive inverse of \(\frac{-3}{7}\) is:
(a) \(\frac{7}{-3}\)
(b) \(\frac{4}{-3}\)
(c) \(\frac{-4}{3}\)
(d) \(\frac{-24}{18}\)
Answer: (none of the options) Correct value is \(\frac{3}{7}\)
The additive inverse of a number \(x\) is what must be added to \(x\) to get 0, which is \(-x\).
Additive inverse of \(\frac{-3}{7}\) = \(-\left(\frac{-3}{7}\right) = \frac{3}{7}\).
In simple words: To find the additive inverse, simply change the sign of the number from negative to positive or positive to negative.

Exam Tip: The sum of a number and its additive inverse is always zero.

Question 3. The product of rational no. \(\frac{3}{4}\) and its multiplicative inverse is:
(a) 1
(b) 0
(c) \(\frac{3}{4}\)
(d) \(\frac{9}{16}\)
Answer: (a) 1
The multiplicative inverse of \(\frac{3}{4}\) is \(\frac{4}{3}\).
Product = \(\frac{3}{4} \times \frac{4}{3} = 1\).
In simple words: Multiplying any non-zero number by its reciprocal always results in 1.

Exam Tip: Multiplicative inverse is also called the reciprocal.

Question 4. The value of \(\left|-2 - \left(\frac{-3}{4}\right)\right|\) is:
(a) \(\frac{-5}{4}\)
(b) \(\frac{5}{4}\)
(c) \(\frac{11}{4}\)
(d) none of the options
Answer: (b) \(\frac{5}{4}\)
First, simplify inside the absolute value:
\(-2 - \left(\frac{-3}{4}\right) = -2 + \frac{3}{4}\)
\(= \frac{-8}{4} + \frac{3}{4} = \frac{-5}{4}\)
Now, take the absolute value:
\(\left|\frac{-5}{4}\right| = \frac{5}{4}\)
In simple words: The absolute value bars make any negative number inside them positive.

Exam Tip: Always simplify the expression completely inside the modulus before removing the absolute value bars.

Question 5. The sum of rational no. \(\frac{3}{7}\) and its reciprocal is:
(a) 0
(b) 1
(c) \(\frac{7}{12}\)
(d) \(\frac{58}{21}\)
Answer: (d) \(\frac{58}{21}\)
The reciprocal of \(\frac{3}{7}\) is \(\frac{7}{3}\).
Sum = \(\frac{3}{7} + \frac{7}{3}\)
LCM of 7 and 3 is 21.
\(= \frac{3 \times 3}{7 \times 3} + \frac{7 \times 7}{3 \times 7}\)
\(= \frac{9}{21} + \frac{49}{21} = \frac{58}{21}\)
In simple words: Find the reciprocal, make the denominators equal using LCM, and add the numerators.

Exam Tip: Do not confuse reciprocal (multiplicative inverse) with additive inverse.

Question 6. Express \(\frac{7}{4}\) as a rational number with denominator: (a) -16 (b) 24
Answer:
(a) To make the denominator -16, multiply both numerator and denominator by \(-4\):
\(\frac{7 \times (-4)}{4 \times (-4)} = \frac{-28}{-16}\)
(b) To make the denominator 24, multiply both numerator and denominator by 6:
\(\frac{7 \times 6}{4 \times 6} = \frac{42}{24}\)
In simple words: Multiply the numerator and denominator by the same number needed to change the old denominator into the target denominator.

Exam Tip: Multiplying numerator and denominator by the same non-zero integer gives an equivalent rational number.

Question 7. Express \(\frac{5}{11}\) as a rational number with numerator: (a) -25 (b) 625
Answer:
(a) To make the numerator -25, multiply both numerator and denominator by \(-5\):
\(\frac{5 \times (-5)}{11 \times (-5)} = \frac{-25}{-55}\)
(b) To make the numerator 625, multiply both numerator and denominator by 125 (since \(625 \div 5 = 125\)):
\(\frac{5 \times 125}{11 \times 125} = \frac{625}{1375}\)
In simple words: Find what number multiplies the given numerator to reach the target numerator, then multiply the denominator by that same number.

Exam Tip: Always apply the multiplier to both top and bottom numbers to keep the fraction's value unchanged.

Question 8. Reduce the following rational numbers to the standard form. (a) \(\frac{14}{-32}\) (b) \(\frac{63}{135}\) (c) \(-\frac{52}{143}\) (d) \(\frac{-48}{72}\)
Answer:
(a) \(\frac{14}{-32} = \frac{14 \div 2}{-32 \div 2} = \frac{7}{-16} = \frac{-7}{16}\)
(b) \(\frac{63}{135} = \frac{63 \div 9}{135 \div 9} = \frac{7}{15}\)
(c) \(-\frac{52}{143} = \frac{-52 \div 13}{143 \div 13} = \frac{-4}{11}\)
(d) \(\frac{-48}{72} = \frac{-48 \div 24}{72 \div 24} = \frac{-2}{3}\)
In simple words: Divide the numerator and denominator by their HCF until no common factor other than 1 remains, and keep the denominator positive.

Exam Tip: In standard form, the denominator must always be positive and the numbers must be in lowest terms.

Question 9. Write 3 equivalent rational numbers of: a) \(\frac{-7}{8}\) b) \(\frac{-3}{-4}\)
Answer:
(a) For \(\frac{-7}{8}\), multiply numerator and denominator by 2, 3, and 4:
\(\frac{-7 \times 2}{8 \times 2} = \frac{-14}{16}\)
\(\frac{-7 \times 3}{8 \times 3} = \frac{-21}{24}\)
\(\frac{-7 \times 4}{8 \times 4} = \frac{-28}{32}\)
Three equivalent numbers are \(\frac{-14}{16}\), \(\frac{-21}{24}\), and \(\frac{-28}{32}\).
(b) First simplify \(\frac{-3}{-4} = \frac{3}{4}\). Now multiply by 2, 3, and 4:
\(\frac{3 \times 2}{4 \times 2} = \frac{6}{8}\)
\(\frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)
\(\frac{3 \times 4}{4 \times 4} = \frac{12}{16}\)
Three equivalent numbers are \(\frac{6}{8}\), \(\frac{9}{12}\), and \(\frac{12}{16}\).
In simple words: Multiply both numerator and denominator by 2, 3, 4, etc., to get equivalent fractions.

Exam Tip: You can generate infinite equivalent rational numbers by multiplying or dividing by any non-zero integer.

Question 10. Represent the following rational numbers on the number line (a) \(\frac{-4}{5}\) (b) \(\frac{9}{4}\). Also write the nearest integers between which each of them lies.
Answer:
(a) \(\frac{-4}{5}\) lies between 0 and \(-1\).
Nearest integers: \(-1\) and 0.
(b) \(\frac{9}{4} = 2\frac{1}{4}\) lies between 2 and 3.
Nearest integers: 2 and 3.
In simple words: Convert improper fractions to mixed fractions to easily spot which two whole numbers they fall between.

Exam Tip: Divide the unit space on the number line according to the denominator of the fraction.

Question 11. Arrange each of the following in descending order: \(\frac{4}{7}, \frac{4}{9}, \frac{8}{5}, \frac{-18}{5}\)
Answer:
First, find the LCM of denominators 7, 9, 5, 5 which is \(7 \times 9 \times 5 = 315\).
Convert each rational number to have the common denominator 315:
\(\frac{4}{7} = \frac{4 \times 45}{7 \times 45} = \frac{180}{315}\)
\(\frac{4}{9} = \frac{4 \times 35}{9 \times 35} = \frac{140}{315}\)
\(\frac{8}{5} = \frac{8 \times 63}{5 \times 63} = \frac{504}{315}\)
\(\frac{-18}{5} = \frac{-18 \times 63}{5 \times 63} = \frac{-1134}{315}\)
Comparing numerators: \(504 > 180 > 140 > -1134\)
Descending order: \(\frac{8}{5}, \frac{4}{7}, \frac{4}{9}, \frac{-18}{5}\)
In simple words: Make all denominators equal using LCM, then arrange from largest numerator to smallest numerator.

Exam Tip: Positive numbers are always greater than negative numbers.

Question 12. Simplify:
(a) \(\frac{15}{11} + \frac{-25}{11}\)
(b) \(\frac{-1}{2} + \frac{5}{12}\)
(c) \(\frac{-5}{7} + \frac{-4}{9}\)
(d) \(\frac{13}{4} + \frac{25}{9}\)
(e) \(\frac{21}{11} + \frac{-13}{8}\)
(f) \(\frac{3}{4} + \frac{5}{6} + \frac{7}{8}\)
(g) \(\frac{2}{3} + \frac{3}{4} - \frac{5}{12} - \frac{1}{8}\)
(h) \(\frac{3}{8} \times \frac{16}{27}\)
(i) \(\frac{-13}{19} \times \frac{-4}{13}\)
(j) \(3\frac{5}{13} - 1\frac{1}{2}\)
Answer:
(a) \(\frac{15 + (-25)}{11} = \frac{-10}{11}\)
(b) LCM of 2 and 12 is 12.
\(\frac{-1 \times 6}{2 \times 6} + \frac{5}{12} = \frac{-6}{12} + \frac{5}{12} = \frac{-1}{12}\)
(c) LCM of 7 and 9 is 63.
\(\frac{-5 \times 9}{7 \times 9} + \frac{-4 \times 7}{9 \times 7} = \frac{-45}{63} + \frac{-28}{63} = \frac{-73}{63}\)
(d) LCM of 4 and 9 is 36.
\(\frac{13 \times 9}{4 \times 9} + \frac{25 \times 4}{9 \times 4} = \frac{117}{36} + \frac{100}{36} = \frac{217}{36}\)
(e) LCM of 11 and 8 is 88.
\(\frac{21 \times 8}{11 \times 8} + \frac{-13 \times 11}{8 \times 11} = \frac{168}{88} + \frac{-143}{88} = \frac{25}{88}\)
(f) LCM of 4, 6, and 8 is 24.
\(\frac{3 \times 6}{24} + \frac{5 \times 4}{24} + \frac{7 \times 3}{24} = \frac{18 + 20 + 21}{24} = \frac{59}{24}\)
(g) LCM of 3, 4, 12, and 8 is 24.
\(\frac{2 \times 8}{24} + \frac{3 \times 6}{24} - \frac{5 \times 2}{24} - \frac{1 \times 3}{24}\)
\(= \frac{16 + 18 - 10 - 3}{24} = \frac{21}{24} = \frac{7}{8}\)
(h) \(\frac{3}{8} \times \frac{16}{27} = \frac{3 \times 16}{8 \times 27} = \frac{1 \times 2}{1 \times 9} = \frac{2}{9}\)
(i) \(\frac{-13}{19} \times \frac{-4}{13} = \frac{(-13) \times (-4)}{19 \times 13} = \frac{4}{19}\)
(j) Convert mixed fractions: \(\frac{44}{13} - \frac{3}{2}\)
LCM of 13 and 2 is 26.
\(\frac{44 \times 2}{26} - \frac{3 \times 13}{26} = \frac{88 - 39}{26} = \frac{49}{26} = 1\frac{23}{26}\)
In simple words: For addition/subtraction use LCM of denominators. For multiplication, cancel common factors and multiply numerators and denominators.

Exam Tip: Always reduce the final fraction to its lowest terms.

Question 13. Subtract: a) \(\frac{3}{16}\) from \(\frac{-5}{8}\) b) \(\frac{-5}{12}\) from 2
Answer:
(a) \(\frac{-5}{8} - \frac{3}{16}\)
LCM of 8 and 16 is 16.
\(= \frac{-5 \times 2}{16} - \frac{3}{16} = \frac{-10 - 3}{16} = \frac{-13}{16}\)
(b) \(2 - \left(\frac{-5}{12}\right) = 2 + \frac{5}{12}\)
\(= \frac{24}{12} + \frac{5}{12} = \frac{29}{12}\)
In simple words: "Subtract A from B" means write B first as \(B - A\).

Exam Tip: Be careful with subtraction signs, especially when subtracting negative numbers.

Question 14. Divide: a) \(\frac{2}{5}\) by \(\frac{-7}{25}\) b) \(-5\frac{4}{9}\) by \(1\frac{22}{27}\)
Answer:
(a) \(\frac{2}{5} \div \left(\frac{-7}{25}\right) = \frac{2}{5} \times \frac{25}{-7}\)
\(= \frac{2 \times 5}{1 \times (-7)} = \frac{10}{-7} = \frac{-10}{7}\)
(b) Convert to improper fractions: \(-5\frac{4}{9} = \frac{-49}{9}\) and \(1\frac{22}{27} = \frac{49}{27}\)
\(\frac{-49}{9} \div \frac{49}{27} = \frac{-49}{9} \times \frac{27}{49}\)
\(= \frac{-1}{1} \times \frac{3}{1} = -3\)
In simple words: Division of rational numbers is done by multiplying the first number by the reciprocal of the second number.

Exam Tip: Always flip (reciprocal) the divisor fraction and change the division sign to multiplication.

Question 15. Find 4 rational numbers between \(\frac{-1}{2}\) and \(\frac{1}{2}\)
Answer:
Make denominators larger by multiplying numerator and denominator by 5:
\(\frac{-1}{2} = \frac{-5}{10}\) and \(\frac{1}{2} = \frac{5}{10}\)
Four rational numbers between them are: \(\frac{-4}{10}, \frac{-3}{10}, \frac{-2}{10}, \frac{-1}{10}\) (or any other numbers between \(-5\) and 5 divided by 10).
In simple words: Expand the fractions with a larger common denominator to easily insert rational numbers in between.

Exam Tip: There are infinitely many rational numbers between any two rational numbers.

Question 16. The product of two rational numbers is \(\frac{-11}{15}\). If one of them is 3, find the other.
Answer:
Let the other rational number be \(x\).
\(3 \times x = \frac{-11}{15}\)
\(x = \frac{-11}{15} \div 3\)
\(x = \frac{-11}{15} \times \frac{1}{3} = \frac{-11}{45}\)
In simple words: Divide the product by the known number to find the missing number.

Exam Tip: Translating word problems into simple algebraic equations makes them very easy to solve.

Question 17. The sum of two rational numbers is \(\frac{-5}{8}\). If one of them is \(\frac{-2}{3}\), find the other.
Answer:
Let the other rational number be \(x\).
\(\frac{-2}{3} + x = \frac{-5}{8}\)
\(x = \frac{-5}{8} - \left(\frac{-2}{3}\right)\)
\(x = \frac{-5}{8} + \frac{2}{3}\)
LCM of 8 and 3 is 24.
\(x = \frac{-5 \times 3}{24} + \frac{2 \times 8}{24} = \frac{-15 + 16}{24} = \frac{1}{24}\)
In simple words: Subtract the given number from the total sum to find the other number.

Exam Tip: Moving a term across the equals sign changes its sign.

Question 18. Simplify: \(2\frac{1}{4} + 5\frac{5}{8} \times 1\frac{1}{3} - 1\frac{3}{4} \div 1\frac{1}{2}\)
Answer:
Convert all mixed fractions to improper fractions:
\(= \frac{9}{4} + \frac{45}{8} \times \frac{4}{3} - \frac{7}{4} \div \frac{3}{2}\)
Perform multiplication and division first (BODMAS):
Multiplication part: \(\frac{45}{8} \times \frac{4}{3} = \frac{45 \times 4}{8 \times 3} = \frac{15}{2}\)
Division part: \(\frac{7}{4} \div \frac{3}{2} = \frac{7}{4} \times \frac{2}{3} = \frac{14}{12} = \frac{7}{6}\)
Substitute back into the expression:
\(= \frac{9}{4} + \frac{15}{2} - \frac{7}{6}\)
LCM of 4, 2, and 6 is 12.
\(= \frac{9 \times 3}{12} + \frac{15 \times 6}{12} - \frac{7 \times 2}{12}\)
\(= \frac{27 + 90 - 14}{12} = \frac{103}{12} = 8\frac{7}{12}\)
In simple words: Follow BODMAS rule: convert to improper fractions first, do multiplication and division, then addition and subtraction.

Exam Tip: Never add or subtract before completing all multiplication and division operations in the expression.

Question 19. Find a rational no. which in standard form is equal to \(\frac{4}{5}\) and the sum of its numerator and denominator is 27.
Answer:
Let the numerator be \(4x\) and denominator be \(5x\).
Sum of numerator and denominator = 27
\(4x + 5x = 27\)
\(9x = 27\)
\(x = 3\)
Numerator = \(4 \times 3 = 12\)
Denominator = \(5 \times 3 = 15\)
Required rational number = \(\frac{12}{15}\)
In simple words: Use a common variable multiplier \(x\) with the ratio terms, solve for \(x\), and find the numbers.

Exam Tip: Any fraction equivalent to \(\frac{4}{5}\) can be written as \(\frac{4x}{5x}\).

Question 20. Arrange each of the following in ascending order: a) \(\frac{2}{5}, \frac{-1}{6}, \frac{5}{-12}, \frac{-11}{30}\)
Answer:
First, standardise the negative signs in denominators:
\(\frac{5}{-12} = \frac{-5}{12}\)
Fractions are: \(\frac{2}{5}, \frac{-1}{6}, \frac{-5}{12}, \frac{-11}{30}\)
LCM of denominators 5, 6, 12, and 30 is 60.
Convert each to denominator 60:
\(\frac{2 \times 12}{5 \times 12} = \frac{24}{60}\)
\(\frac{-1 \times 10}{6 \times 10} = \frac{-10}{60}\)
\(\frac{-5 \times 5}{12 \times 5} = \frac{-25}{60}\)
\(\frac{-11 \times 2}{30 \times 2} = \frac{-22}{60}\)
Comparing numerators: \(-25 < -22 < -10 < 24\)
Ascending order: \(\frac{5}{-12}, \frac{-11}{30}, \frac{-1}{6}, \frac{2}{5}\)
In simple words: Make denominators equal with LCM and arrange from smallest numerator to largest numerator.

Exam Tip: Among negative numbers, the one with the larger absolute value is actually smaller.

Question 20 (b). If you have to arrange health, wealth and character in ascending order of importance, arrange it according to your priority.
Answer:
Character, Health, Wealth (or as per individual ethical priority: Character is paramount, followed by Health, and then Wealth).
In simple words: Character and health are invaluable foundations for a meaningful life, placing wealth last in true importance.

Exam Tip: Values-based questions allow personal reasoning as long as logical justification is provided.

Question. Find a rational number whose numerator is (35 – 15) and denominator is [3 × (–7)].
Answer : -20/21

Question. Tell the rational number whose additive inverse is –7.
Answer : 7

Question. Find the value of : 13/5 - 12/25
Answer : 53/25

Question. Express -25/-115 in the standard form.
Answer : 5/23

Question. How many rational numbers are there between any two rational numbers?
Answer : infinite.

Question. Simplify : 2/3 + 3/4 + 1/12
Answer : 3/2

Question. Express -6/-7 as a rational number whose denominator is –42.
Answer : -36/-42

Question. Express -77/22 in the lowest form.
Answer : -7/2

Question. Express -8/28 in the standard form.
Answer : -2/7

Question. Simplify : -8/19 + (-2/57)
Answer : -26/57

Question. Simplify : 5/6 + (- 5/9)
Answer : 5/18

Question. Find two equivalent numbers of 7/-15
Answer : 14/-30 , -49/105

Question. Express -5/-9 as a rational number whose denominator is 45.
Answer : 25/45

Question. Find any two equivalent numbers of -10/17
Answer : -20/34, -30/51

Question. Express 27/-45 in the standard form.
Answer : -3/5

Question. Which one is greater 3/4 or 5/4.
Answer : 5/4

Question. Find a rational number whose numerator is (20 – 15) and denominator is [2 × (–3)].
Answer : -5/6

Question. Calculate 4/9 + 9.4
Answer : 117/36

Question. The sum of two rational numbers is 11/5, If one of them is -4/15 find the other rational number.
Answer : 37/15

Question. Convert -4/5 into a rational number whose numerator is 24.
Answer : 24/-30

Question. What number should be added to -5/11 so as to get 26/33 ?
Answer : 41/33

Question. Express 64/128 in the simplest form.
Answer : 1/2

Question. The sum of two rational numbers is –8. If one of the numbers is -15/7, find the other number.
Answer : -41/7

Question. Find the value of : -6/13 - (-7/13)
Answer : 1/13

Question. What is the product of a number and its multiplicative inverse.
Answer : 1

Question. Find the value of : 5/63 - (-8/21)
Answer : 29/63

Question. Express -12/-30 in the simplest form.
Answer : 2/5

Question. Simplify : -3/4 + 5/9
Answer : -7/36

Question. Pawan gives (1/5)th part of his monthly income in charity and deposits (1/6)th part in the bank. He spends the remaining amount. What part of his salary does he spend?
Answer : (19/30) th part 

Students must free download and practice these worksheets to gain more marks in exams. CBSE Class 7 Maths Worksheet - Rational Numbers (1)