CBSE Class 7 Mathematics Rational Numbers Worksheet Set 03

Question 1. Write a rational number for each of the following
(a) Numerator is a negative integer and denominator is a positive integer________________________
(b) Numerator is a positive integer and denominator is a negative integer________________________
(c) Numerator and denominator both are positive integers________________________
(d) Numerator and denominator both are negative integers________________________
Answer:
(a) Example: \(\frac{-3}{5}\)
(b) Example: \(\frac{3}{-5}\)
(c) Example: \(\frac{3}{5}\)
(d) Example: \(\frac{-3}{-5}\)
In simple words: A rational number consists of a numerator and a denominator where the signs follow basic integer rules.
Exam Tip: Remember that a rational number is expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).

Question 2. Fill in the boxes
(a) \(\frac{5}{4} = \frac{\square}{16} = \frac{25}{\square} = \frac{-15}{\square}\)
(b) \(\frac{-3}{7} = \frac{\square}{14} = \frac{9}{\square} = \frac{-6}{\square}\)
Answer:
(a) For \(\frac{5}{4} = \frac{\square}{16}\), since \(4 \times 4 = 16\), multiply numerator by 4: \(5 \times 4 = 20\). Box = 20.
For \(\frac{5}{4} = \frac{25}{\square}\), since \(5 \times 5 = 25\), multiply denominator by 5: \(4 \times 5 = 20\). Box = 20.
For \(\frac{5}{4} = \frac{-15}{\square}\), since \(5 \times (-3) = -15\), multiply denominator by -3: \(4 \times (-3) = -12\). Box = -12.
Completed series: \(\frac{5}{4} = \frac{20}{16} = \frac{25}{20} = \frac{-15}{-12}\)

(b) For \(\frac{-3}{7} = \frac{\square}{14}\), since \(7 \times 2 = 14\), multiply numerator by 2: \(-3 \times 2 = -6\). Box = -6.
For \(\frac{-3}{7} = \frac{9}{\square}\), since \(-3 \times (-3) = 9\), multiply denominator by -3: \(7 \times (-3) = -21\). Box = -21.
For \(\frac{-3}{7} = \frac{-6}{\square}\), since \(-3 \times 2 = -6\), multiply denominator by 2: \(7 \times 2 = 14\). Box = 14.
Completed series: \(\frac{-3}{7} = \frac{-6}{14} = \frac{9}{-21} = \frac{-6}{14}\)
In simple words: Equivalent fractions are found by multiplying or dividing both the numerator and the denominator by the same non-zero number.

Exam Tip: Always check the given equivalent term (either numerator or denominator) to find the scaling factor.

Question 3. Which of these are negative rational numbers
(a) \(\frac{-2}{5}\)     (b) \(\frac{5}{7}\)     (c) \(\frac{3}{-5}\)     (d) \(0\)     (e) \(\frac{6}{11}\)     (f) \(\frac{-2}{-9}\)
Answer: (a) and (c)
In simple words: A rational number is negative if either its numerator or its denominator has a negative sign, but not both.

Exam Tip: If both numerator and denominator are negative (like in \(\frac{-2}{-9}\)), the negative signs cancel out, making it a positive rational number.

Question 4. Find the standard form using the H.C.F method
(a) \(\frac{-18}{45} = \dots\)
(b) \(\frac{-12}{18} = \dots\)
(c) \(\frac{36}{-24} = \dots\)
(d) \(\frac{-3}{-15} = \dots\)
Answer:
(a) H.C.F of 18 and 45 is 9.
Divide numerator and denominator by 9:
\(\frac{-18 \div 9}{45 \div 9} = \frac{-2}{5}\)

(b) H.C.F of 12 and 18 is 6.
Divide numerator and denominator by 6:
\(\frac{-12 \div 6}{18 \div 6} = \frac{-2}{3}\)

(c) H.C.F of 36 and 24 is 12.
Divide numerator and denominator by 12:
\(\frac{36 \div 12}{-24 \div 12} = \frac{3}{-2} = \frac{-3}{2}\)

(d) H.C.F of 3 and 15 is 3.
Divide numerator and denominator by 3:
\(\frac{-3 \div 3}{-15 \div 3} = \frac{-1}{-5} = \frac{1}{5}\)
In simple words: To write a rational number in standard form, divide both the numerator and denominator by their H.C.F so that the denominator is positive and they have no common factors other than 1.

Exam Tip: Always ensure the final answer has a positive denominator.

Question 5.
(a) Six whole numbers between 3 and 10 are ________________________________________
(b) Five integers between -3 and 3 are _____________________________________________
(c) Between two successive integers the number of integers is _______
(d) The number of integers between two integers are ____________(finite/infinite)
(e) There are ___________(limited/unlimited) number of rational numbers between two rational numbers.
(f) The additive inverse of \(\frac{-2}{5}\) is _______
(g) Reciprocal of \(\frac{-6}{11}\) is ____________
Answer:
(a) 4, 5, 6, 7, 8, 9
(b) -2, -1, 0, 1, 2
(c) 0
(d) finite
(e) unlimited
(f) \(\frac{2}{5}\)
(g) \(\frac{11}{-6}\) or \(\frac{-11}{6}\)
In simple words: These are fundamental properties of numbers, integers, and rational numbers.

Exam Tip: The additive inverse of a number \(a\) is \(-a\), and the multiplicative inverse (reciprocal) of \(\frac{p}{q}\) is \(\frac{q}{p}\).

Question 6. Find the value of
(a) \(\frac{7}{9} - \frac{2}{5} = \dots\)
(b) \(2\frac{1}{5} - \frac{(-1)}{3} = \dots\)
(c) \(\frac{2}{3} \div \frac{8}{-7} = \dots\)
Answer:
(a) L.C.F of 9 and 5 is 45.
\(\frac{7 \times 5}{9 \times 5} - \frac{2 \times 9}{5 \times 9}\)
\(= \frac{35}{45} - \frac{18}{45}\)
\(= \frac{35 - 18}{45} = \frac{17}{45}\)

(b) Convert mixed fraction to improper fraction: \(2\frac{1}{5} = \frac{11}{5}\)
\(\frac{11}{5} - \left(\frac{-1}{3}\right) = \frac{11}{5} + \frac{1}{3}\)
L.C.M of 5 and 3 is 15.
\(= \frac{11 \times 3}{5 \times 3} + \frac{1 \times 5}{3 \times 5}\)
\(= \frac{33}{15} + \frac{5}{15} = \frac{38}{15}\)

(c) \(\frac{2}{3} \div \frac{8}{-7}\)
\(= \frac{2}{3} \times \frac{-7}{8}\)
\(= \frac{2 \times (-7)}{3 \times 8} = \frac{-14}{24}\)
Simplify by dividing numerator and denominator by 2:
\(= \frac{-7}{12}\)
In simple words: For addition/subtraction find a common denominator, and for division multiply by the reciprocal of the divisor.

Exam Tip: Always reduce the final fractional answer to its simplest form.

Question. Give the additive inverse of 4/5 + 1/5
Answer : –1

Question. A child walks 7/24 km towards north and then 17/36 km towards south. How far is the child now from the starting point?
Answer : 13/72

Question. The product of two rational numbers is -7/9 If one of the numbers is -4/9 find the other number.
Answer : 7/4

Question. Give the additive inverse of 6/7 + 2/7
Answer : -4/7

Question. Simplify and express the result as a rational number in standard form : (-14/9) x (-27)
Answer : 42

Question. Subtract 4/5 from 5/4
Answer : 9/20

Question. Multiply (9/-11) by (22/-27)
Answer : 2/3

Question. What number should be subtracted from -7/8 to get 5/9 ?
Answer : -103/72

Question. Find the reciprocal of -7/26 + (-11/39)
Answer : -78/43

Question. Express the number 1/2 + 1/5 as a decimal.
Answer : 0.7

Question. Multiply (-3/7) by 7/5
Answer : -3/5

Question. Find the multiplicate inverse of 3/11 + 5/9
Answer : 99/82

Question. Simplify : 56/25 ÷ (-7/50)
Answer : –16

Question. Multiply (-2/9) by (33/54)
Answer : -11/81

Question. A man travels a distance of 13/11 km by car and 9/11 km by bus to reach his destination. How much distance did he travell in all?
Answer : 2 km