Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.4 Solutions: The World of Numbers
Rational numbers are not limited to whole-number positions on the number line; they can also lie between integers. Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.4 Solutions focus on representing rational numbers, finding numbers between two given values, simplifying fractions and using the idea that rational numbers are dense.
In Ganita Manjari Class 9 Chapter 3 Exercise 3.4, students move from calculating with rational numbers to placing them correctly on a number line. The exercise includes values like 2/3, −5/4 and 11/2, along with questions on finding rational numbers between two given numbers. These Class 9 Maths Chapter 3 Exercise 3.4 Solutions also cover close decimal values such as 3.1415 and 3.1416, helping students understand that rational numbers are densely spread across the number line.
Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.4 Solutions
1. Represent the rational numbers 2/3, −5/4 and 11/2 on a single number line.
Solution:
We need to locate:
2/3, −5/4, 11/2
First understand their positions.
2/3 lies between 0 and 1.
To represent 2/3, divide the distance between 0 and 1 into 3 equal parts. Mark the second part from 0 on the right side.
−5/4 = −1 1/4
So, −5/4 lies between −2 and −1.
To represent −5/4, divide the distance between −1 and −2 into 4 equal parts. Mark the first part to the left of −1.
11/2 = 5 1/2
So, 11/2 lies between 5 and 6.
To represent 11/2, divide the distance between 5 and 6 into 2 equal parts. Mark the midpoint between 5 and 6.
Answer: On the number line, −5/4 lies between −2 and −1, 2/3 lies between 0 and 1, and 11/2 lies between 5 and 6.
2. Find three distinct rational numbers that lie strictly between −1/2 and 1/4.
Solution:
We need rational numbers strictly between:
−1/2 and 1/4
Convert them to a common denominator.
−1/2 = −4/8
1/4 = 2/8
Now choose rational numbers between −4/8 and 2/8.
Examples:
−3/8, −1/8, 1/8
Check:
−4/8 < −3/8 < −1/8 < 1/8 < 2/8
So, all three lie strictly between −1/2 and 1/4.
Answer: Three rational numbers between −1/2 and 1/4 are −3/8, −1/8 and 1/8.
3. Simplify the expression: −1/4 + 5/12.
Solution:
Given expression:
−1/4 + 5/12
Find the LCM of 4 and 12.
LCM = 12
Convert −1/4 to denominator 12.
−1/4 = −3/12
Now add:
−1/4 + 5/12 = −3/12 + 5/12
= 2/12
= 1/6
Answer: 1/6
4. A tailor has 15 3/4 metres of fine silk. If making one kurta requires 2 1/4 metres of silk, exactly how many kurtas can he make?
Solution:
Total silk = 15 3/4 metres
Silk required for one kurta = 2 1/4 metres
Convert mixed numbers into improper fractions.
15 3/4 = 63/4
2 1/4 = 9/4
Number of kurtas:
63/4 ÷ 9/4
To divide by a fraction, multiply by its reciprocal.
63/4 × 4/9
Cancel 4 from numerator and denominator.
= 63/9
= 7
Answer: The tailor can make exactly 7 kurtas.
5. Find three rational numbers between 3.1415 and 3.1416.
Solution:
We need three rational numbers between:
3.1415 and 3.1416
Terminating decimals are rational numbers. So, we can choose any three terminating decimals between these two values.
Examples:
3.14151, 3.14155, 3.14159
Check:
3.1415 < 3.14151 < 3.14155 < 3.14159 < 3.1416
Answer: Three rational numbers between 3.1415 and 3.1416 are 3.14151, 3.14155 and 3.14159.
6. Can you think of other way(s) to find a rational number between any two rational numbers?
Solution:
Yes. There are many ways to find a rational number between any two rational numbers.
Method 1: Taking the average
If a and b are two rational numbers, then:
(a + b) / 2
lies between a and b.
For example, between 1/2 and 3/4:
Average = (1/2 + 3/4) / 2
= (2/4 + 3/4) / 2
= (5/4) / 2
= 5/8
So, 5/8 lies between 1/2 and 3/4.
Method 2: Using a common denominator
Convert both rational numbers to the same denominator and choose a numerator between them.
For example:
1/2 = 4/8
3/4 = 6/8
A rational number between them is:
5/8
Method 3: Using decimal form
Write the numbers as decimals and choose a decimal between them.
For example, between 0.5 and 0.75, we can choose 0.6, which is rational.
Answer: We can find a rational number between two rational numbers by taking their average, converting them to a common denominator, or using decimal form.
Related Links – Class 9 Maths Chapter 3