Maharashtra Board Class 8 Maths Chapter 1 Rational and Irrational Numbers Set 1.1 Solutions

Question 1. Show the following numbers on a number line. Draw a separate number line for each example.
(i) \(3/2\), \(5/2\), \(-3/2\)
(ii) \(7/5\), \(-2/5\), \(-4/5\)
(iii) \(-5/8\), \(11/8\)
(iv) \(13/10\), \(-17/10\)
Solution:
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक संख्या रेखा दर्शाता है जिस पर प्रत्येक इकाई को दो बराबर भागों में बांटा गया है। इस पर \(3/2\), \(5/2\) और \(-3/2\) जैसी संख्याएँ बिंदु लगाकर प्रदर्शित की गई हैं।
(i) Here, the denominator of each fraction is 2.
∴ Each unit will be divided into 2 equal parts.
Answer: The numbers \(3/2\), \(5/2\), and \(-3/2\) are shown on the number line by dividing each unit into 2 equal parts and marking the respective points.
In simple words: To show fractions with a denominator of 2, divide each whole unit on the number line into two equal segments, then count those segments to find the exact position of each given fraction.

🎯 Exam Tip: When representing fractions on a number line, ensure that the divisions within each unit accurately reflect the denominator of the given fractions for precise marking.

ℹ️ चित्र व्याख्या (Diagram Explanation): इस संख्या रेखा पर प्रत्येक इकाई को पांच बराबर भागों में विभाजित किया गया है। यहाँ \(7/5\), \(-2/5\) और \(-4/5\) जैसी परिमेय संख्याओं को उनके संबंधित स्थानों पर चिह्नित किया गया है।
(ii) Here, the denominator of each fraction is 5.
∴ Each unit will be divided into 5 equal parts.
Answer: The numbers \(7/5\), \(-2/5\), and \(-4/5\) are shown on the number line by dividing each unit into 5 equal parts and marking the respective points.
In simple words: For fractions with a denominator of 5, divide each unit into five equal sections and then locate each fraction by counting the appropriate number of sections from zero.

🎯 Exam Tip: Clearly label the origin (0) and the unit points (1, -1, etc.) on your number line. This helps in accurately placing the fractional values.

ℹ️ चित्र व्याख्या (Diagram Explanation): यह संख्या रेखा प्रत्येक इकाई को आठ बराबर भागों में विभाजित करती है। इस पर \(-5/8\) और \(11/8\) जैसी संख्याओं को उनके उचित स्थानों पर दर्शाया गया है ताकि उनके सापेक्ष मान को समझा जा सके।
(iii) Here, the denominator of each fraction is 8.
∴ Each unit will be divided into 8 equal parts.
Answer: The numbers \(-5/8\) and \(11/8\) are shown on the number line by dividing each unit into 8 equal parts and marking the respective points.
In simple words: When dealing with fractions having a denominator of 8, divide each whole unit into eight equal segments and then mark the given fractions according to their numerator.

🎯 Exam Tip: Always make sure your divisions within each unit are consistent and clearly visible to avoid errors in plotting the numbers.

ℹ️ चित्र व्याख्या (Diagram Explanation): यह संख्या रेखा प्रत्येक इकाई को दस बराबर उप-भागों में बांटती है। इस पर \(13/10\) और \(-17/10\) जैसी परिमेय संख्याओं को स्पष्ट रूप से चिह्नित किया गया है, जो उनके मान को दर्शाता है।
(iv) Here, the denominator of each fraction is 10.
∴ Each unit will be divided into 10 equal parts.
Answer: The numbers \(13/10\) and \(-17/10\) are shown on the number line by dividing each unit into 10 equal parts and marking the respective points.
In simple words: For fractions with a denominator of 10, divide each unit into ten equal mini-segments and then pinpoint the location of each fraction based on its numerator.

🎯 Exam Tip: Practice drawing number lines neatly and precisely. A clear diagram often fetches full marks, even if calculations are slightly off.

Question 2. Observe the number line and answer the questions.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र एक संख्या रेखा को दर्शाता है जिस पर \(–3\) से \(3\) तक की संख्याएँ अंकित हैं, और प्रत्येक इकाई को चार बराबर भागों में विभाजित किया गया है। इस पर विभिन्न बिंदुओं जैसे A, B, E, O, F, C, D को चिह्नित किया गया है, जो विशिष्ट परिमेय संख्याओं को दर्शाते हैं।
Answer:
(i) Which number is indicated by point B?
Here, each unit is divided into 4 equal parts.
Point B is marked on the 10th equal part on the left side of O.
∴ The number indicated by point B is \(-10/4\).
In simple words: Since each unit is divided into 4 parts, and B is 10 divisions to the left of zero, point B represents the fraction \(-10/4\).

🎯 Exam Tip: When observing a number line, first determine the value of each small division by counting the segments between two whole numbers.

(ii) Which point indicates the number \(1 3/4\)?
Answer:
\(1 3/4 = (1 \times 4+3)/4 = (4+3)/4 = 7/4\)
Point C is marked on the 7th equal part on the right side of O.
∴ The number \(1 3/4\) is indicated by point C.
In simple words: The mixed fraction \(1 3/4\) converts to \(7/4\). On the number line, point C is exactly seven divisions to the right of zero, meaning C indicates \(1 3/4\).

🎯 Exam Tip: Convert mixed fractions to improper fractions to easily locate them on a number line, as improper fractions directly show how many divisions from zero to count.

(iii) State whether the statement, 'the point D denotes the number \(5/2\)' is true or false.
Answer: True
Point D is marked on the 10th equal part on the right side of O.
∴ D denotes the number \(10/4 = (5 \times 2)/(2 \times 2) = 5/2\)
In simple words: Point D is at the 10th mark from zero, which is \(10/4\). Simplifying this fraction gives \(5/2\), so the statement is true.

🎯 Exam Tip: Always simplify fractions to their lowest terms to confirm if a point represents the given number accurately. This helps avoid confusion with equivalent fractions.