Class 9 Maths Chapter 3 Exercise 3.4 Solutions Ganita Manjari NCERT
Welcome to Sid Classes, your premier online resource for accurate and reliable academic solutions! If you are studying for your school assessments and looking for well-structured, easy-to-follow answers, you have arrived at the perfect destination. In this comprehensive study guide, we provide complete, step-by-step solutions for NCERT Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.4.
Our expert-crafted solutions are structured precisely the way students write in their notebooks, utilizing clean mathematical notation and direct, logical problem-solving steps without unnecessary fluff. Whether you want to verify your homework answers, understand how to convert mixed fractions for division word problems, or learn how to represent negative and positive numbers on a single number line, this guide ensures you can approach your school tests and board examinations with total confidence.
🔑 Key Concepts to Remember:
- Number Line Scaling: Divide each unit segment into parts equal to the denominator, then mark the required position along the numerator.
- Finding Rational Numbers: Convert fractions to equivalent forms with a common denominator, or expand decimal intervals to find intermediate values.
- Mixed to Improper Fractions: Always convert mixed numbers into improper form before performing multiplication or division operations.
NCERT Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.4 Solutions
Question 1
Represent the rational numbers \(\frac{2}{3}\), \(-\frac{5}{4}\) and \(1 \frac{1}{2}\) on a single number line.
Solution:
Given numbers: \(\frac{2}{3}\), \(-\frac{5}{4} = -1 \frac{1}{4}\), and \(1 \frac{1}{2} = \frac{3}{2}\).
Number Line Representation:
<---|---------|---------|---------|---------|---------|---------|---------|--->
-2 -1 -5/4 0 1 2/3 1 1 1/2 2
Marked Points on the Line:
- \(-\frac{5}{4}\) lies between \(-1\) and \(-2\) (divided into 4 equal parts, 1 part left of \(-1\)).
- \(\frac{2}{3}\) lies between \(0\) and \(1\) (divided into 3 equal parts, 2 parts right of \(0\)).
- \(1 \frac{1}{2}\) lies exactly midway between \(1\) and \(2\).
Question 2
Find three distinct rational numbers that lie strictly between \(-\frac{1}{2}\) and \(\frac{1}{4}\).
Solution:
Given rational numbers: \(-\frac{1}{2}\) and \(\frac{1}{4}\)
Making denominators equal (LCM = 4):
$$-\frac{1}{2} = -\frac{2}{4}$$
$$\frac{1}{4} = \frac{1}{4}$$
Multiplying numerator and denominator by 2 to get more space between them:
$$-\frac{2}{4} = -\frac{4}{8}$$
$$\frac{1}{4} = \frac{2}{8}$$
Three rational numbers between \(-\frac{4}{8}\) and \(\frac{2}{8}\) are:
$$\mathbf{-\frac{3}{8}, -\frac{2}{8}, -\frac{1}{8}}$$
Question 3
Simplify the expression: \(\left(-\frac{1}{4}\right) + \left(\frac{5}{12}\right)\).
Solution:
LCM of 4 and 12 is 12.
$$= \frac{-1 \times 3}{4 \times 3} + \frac{5}{12}$$
$$= \frac{-3}{12} + \frac{5}{12}$$
$$= \frac{-3 + 5}{12}$$
$$= \frac{2}{12} = \mathbf{\frac{1}{6}}$$
Question 4
A tailor has \(15 \frac{3}{4}\) metres of fine silk. If making one kurta requires \(2 \frac{1}{4}\) metres of silk, exactly how many kurtas can he make?
Solution:
Total silk = \(15 \frac{3}{4} = \frac{63}{4}\text{ m}\)
Silk required for 1 kurta = \(2 \frac{1}{4} = \frac{9}{4}\text{ m}\)
Number of kurtas = \(\frac{63}{4} \div \frac{9}{4}\)
$$= \frac{63}{4} \times \frac{4}{9}$$
$$= \frac{63}{9} = \mathbf{7}$$
Answer: He can make 7 kurtas.
Question 5
Find three rational numbers between 3.1415 and 3.1416.
Solution:
Given decimals can be written as:
\(3.1415 = 3.14150\)
\(3.1416 = 3.14160\)
Three rational numbers between them are:
$$\mathbf{3.14151, \ 3.14155, \ 3.14158}$$
Question 6
Can you think of other way(s) to find a rational number between any two rational numbers?
Solution:
Yes, by using the Mean Method (Average Method). For any two rational numbers \(a\) and \(b\), a rational number between them is given by:
$$\frac{a + b}{2}$$
Frequently Asked Questions (FAQs)
Q1: How do you represent positive and negative fractions on a single number line?
A: Zero is placed in the center, positive fractions are plotted to the right, and negative fractions are plotted to the left by partitioning each unit interval according to the denominator value.
Q2: How can we find multiple rational numbers between two given fractions?
A: By equalizing their denominators using the LCM or by scaling up both the numerator and denominator by factors like 10 or 100 to create more numerical space.
Q3: What textbook do these solutions belong to?
A: These answers follow the latest Ganita Manjari Part I textbook issued for NCERT Class 9 Mathematics.
Q4: How do you handle word problems with mixed fractions like Question 4?
A: First convert all mixed numbers into improper fractions, and then perform the required division or multiplication as dictated by the word problem.
Q5: Can terminating decimals like 3.1415 and 3.1416 be treated as rational numbers?
A: Yes, terminating decimals can be expressed in fraction form, and we can find infinite rational or decimal numbers between them by adding trailing placeholder zeros.
Q6: What is the mean method for finding rational numbers?
A: The mean method involves adding two rational numbers together and dividing their sum by 2 to determine the exact midpoint value.
Quick Navigation:
Last updated according to the latest NCERT Ganita Manjari syllabus.