Class 8 Maths Chapter 1 Practice Question Answers - Rational Numbers

Q1: Represent −𝟐/𝟏𝟏, −𝟓/𝟏𝟏, and −𝟗/ 𝟏𝟏 on the number line.
Sol: To represent these numbers, divide the number line into 11 parts. Now, the given rational numbers will be 2, 5 and 9 points away from 0.

Q2: If the product of any two rational numbers is 2 and one of them is 1/7, find the other?
Sol: Consider 2 rational numbers as “a” and “b”.
Given, a = 1/7 and a × b = 2
Now, 1/7 × b = 2
⇒ b = 7 × 2 = 14
So, the other rational number will be 14.

Q3: Mention the commutativity, associative and distributive properties of rational numbers. Also, check a × b = b × a and a + b = b + a for a = ½ and b = ¾
Sol:
Commutative property:
For any two rational numbers a and b, a + b = b + a.
For any two rational numbers a and b, a × b = b × a.
Associative Property:
For any three rational numbers a, b and c,
(a + b) + c = a + (b + c)
Distributive property states that for any three numbers x, y and z,
x × ( y + z ) = (x × y) + ( x × z)
a x b = b x a
a x b = ½ * ¾ = 3/8
b x a = ¾ * ½ = 3/8
a + b = ¾ + ½ = 5/4
b + a = ½ + ¾ = 5/4

Q4: Mention a rational number which has no reciprocal.
Sol: A rational number “0” has no reciprocal or multiplicative inverse.

Q5: Write the additive inverse of 19/-6 and -⅔
Sol: 19/-6 = 19/6 and -⅔ = 2/3

Q6: What are the multiplicative and additive identities of rational numbers?
Sol: 0 and 1 are the additive and multiplicative identity of rational numbers respectively.

Q7: Write the multiplicative inverse of -13/19 and -7
Sol: -13/19 = -19/13 and -7 = -1/7

Q8: Mention any 4 rational numbers which are less than 5.
Sol: -1, 1, 2 and 3.

Q9: Write any 5 rational numbers between −2/5 and ½.
Sol:
−2/5 can be written as −8/20.
1/2 can be written as 10/20.
So, rational numbers between these two numbers can be,
−7/20,−6/20,−5/20,−4/20,−3/20,−2/20,−1/20,0,1/20,2/20,3/20,4/20.

Q10: Mr X went shopping with a certain amount of money. He spent Rs. 10(¼) on buying a pen and Rs. 25(¾) in food. He then gave the remaining Rs. 16(½) to his friend. Calculate how much money he initially had.
Sol: To get the amount of money Mr X had initially, his purchases have to be added.
So,
Initial Money = Rs. [10(¼) + 25(¾) + 16(½)]
= Rs. (41/4 + 103/4 + 33/2)
By taking LCM, we get;
= Rs. (41  + 103 + 66)/4
= Rs. 210/4
Initial Money = Rs. 105/2 (or) Rs. 52(½)

Q11: Give three rational numbers between 3/6 and 3/4.
Sol: To find three rational numbers between 3/6 and 3/4, we can first simplify the given fractions:
3/6 = 1/2
3/4 = 3/4
Now, we need to find three rational numbers between 1/2 and 3/4. To do this, we can take the average of the two fractions and then adjust the numerators while keeping the denominators the same. Here's how:
Average of 1/2 and 3/4 = (1/2 + 3/4) / 2 = 5/8
Now, let's find the numbers between 1/2 and 5/8:
The first rational number: (1/2 + 5/8) / 2 = 9/16
The second rational number: (5/8 + 3/4) / 2 = 11/16
The third rational number: (1/2 + 11/16) / 2 = 13/32
So, the three rational numbers between 3/6 and 3/4 are 9/16, 11/16, and 13/32.

Q12: What is the property used for the expression given  -2/3 × 3/5 + 5/2 -3/5 × 1/6 ?
Sol: The property used for the given expression -2/3 × 3/5 + 5/2 - 3/5 × 1/6 is the Distributive Property of Multiplication over Addition and Subtraction.
The Distributive Property states that for any real numbers a, b, and c:
a × (b + c) = a × b + a × c
a × (b - c) = a × b - a × c
In the given expression, you have multiplication and addition/subtraction combined. You can apply the Distributive Property to simplify the expression step by step:
-2/3 × 3/5 = (-2/3) × (3/5) = (-2 × 3) / (3 × 5) = -6/15 = -2/5
-3/5 × 1/6 = (-3/5) × (1/6) = -3/30 = -1/10
Substitute these values back into the original expression:
-2/5 + 5/2 - 1/10
Now, combine the fractions with common denominators:
-4/10 + 25/10 - 1/10 = (25 - 4 - 1)/10 = 20/10 = 2
So, the simplified value of the expression is 2. The Distributive Property was used to simplify the multiplication and addition/subtraction operations within the expression.

Q13: Give 2 rational numbers whose multiplicative inverse is same as they are.
Sol: Two rational numbers whose multiplicative inverses are the same as they are would be 1 and -1.
The multiplicative inverse (reciprocal) of a number x is 1/x. For both 1 and -1, their multiplicative inverses are also 1 and -1 respectively:
1 x (1/1) = 1
-1 x (-1/1) = 1
So, the numbers 1 and -1 fulfill the condition of having the same multiplicative inverse as themselves.

Q14: What is the additive inverse of 3/4?
(a) -3/4
(b) 4/3
(c) -4/3
(d) -2/3
Ans: (a)
Sol: The additive inverse of a number x is the number that, when added to x, gives a sum of 0. For a rational number, the additive inverse is the negative of that number.
The additive inverse of 3/4 is -3/4.
So, the correct answer is (a) -3/4.

Q15: Give a rational number that is equivalent to 4/7 with \
(a) numerator 20
(b) denominator 28
Sol: To find a rational number that is equivalent to 4/7 with a specific numerator or denominator, you need to scale the fraction while maintaining its ratio.
(a) To have a numerator of 20, you can calculate the equivalent fraction by multiplying both the numerator and the denominator of 4/7 by 5:
4/7 x 5/5 = 20/35
So, the equivalent fraction to 4/7 with a numerator of 20 is 20/35.
(b) To have a denominator of 28, you can calculate the equivalent fraction by multiplying both the numerator and the denominator of 4/7 by 4:
4/7 x 4/4 = 16/28
So, the equivalent fraction to 4/7 with a denominator of 28 is 16/28.