UP Board Solutions Class 9 Maths Chapter 1 Real Numbers Ex 1.1

Balaji Class 9 Maths Solutions Chapter 1 Real Numbers Ex 1.1

Exercise 1.1 Real Numbers अतिलघु उत्तरीय प्रश्न (Very Short Answer Type Questions)

Question 1. निम्नलिखित प्रत्येक परिमेय संख्या को दशमलव रूप में व्यक्त कीजिए ।
(i) 8/45
(ii) 1/7
Answer:
(i) \( \frac{8}{45} = 0.177777... = 0.1\overline{7} \)
(ii) \( \frac{1}{7} = 0.142857142857... = 0.\overline{142857} \)
In simple words: To convert a rational number to a decimal, perform the division. If the decimal representation is repeating, use a bar or a dot over the repeating digit(s).

🎯 Exam Tip: For repeating decimals, ensure the bar or dot is placed accurately over the repeating block of digits to convey precision.

Exercise 1.1 Real Numbers लघु उत्तरीय प्रश्न - I (Short Answer Type Questions - I)

Question 2. निम्न को m के रूप में अर्थात् परिमेय संख्या में व्यक्त कीजिए।
(i) \(0.5\dot { 3 } \)
(ii) \(0.2 \overline{104}\)
(iii) \(2.24 \overline{689}\)
(iv) \(4.0\dot {5} \)
Answer:
(i) माना \(x = 0.5\dot { 3 }\)
10 से गुणा \( \implies 10x = 5.3\dot { 3 }\)
\(10x = 5 + 0.3\dot { 3 }\)
\(10x = 5 + \frac{3}{9}\)
\(10x = 5 + \frac{1}{3}\)
\(10x = \frac{15+1}{3}\)
\(10x = \frac{16}{3}\)
\(x = \frac{16}{3 \times 10}\)
\(x = \frac{16}{30}\)
\(x = \frac{8}{15}\)

(ii) माना \(x = 0.2 \overline{104}\)
10 से गुणा \( \implies 10x = 2.\overline{104}\)
\(10x = 2 + 0.\overline{104}\)
\(10x = 2 + \frac{104}{999}\)
\(10x = \frac{2 \times 999 + 104}{999}\)
\(10x = \frac{1998 + 104}{999}\)
\(10x = \frac{2102}{999}\)
\(x = \frac{2102}{10 \times 999}\)
\(x = \frac{2102}{9990}\)
\(x = \frac{1051}{4995}\)

(iii) माना \(x = 2.24 \overline{689}\)
100 से गुणा \( \implies 100x = 224.\overline{689}\)
\(100x = 224 + 0.\overline{689}\)
\(100x = 224 + \frac{689}{999}\)
\(100x = \frac{224 \times 999 + 689}{999}\)
\(100x = \frac{223776 + 689}{999}\)
\(100x = \frac{224465}{999}\)
\(x = \frac{224465}{999 \times 100}\)
\(x = \frac{224465}{99900}\)
\(x = \frac{44893}{19980}\)

(iv) माना \(x = 4.0\dot { 5 }\)
10 से गुणा \( \implies 10x = 40.5\dot { 5 }\)
\(10x = 40 + 0.5\dot { 5 }\)
\(10x = 40 + \frac{5}{9}\)
\(10x = \frac{40 \times 9 + 5}{9}\)
\(10x = \frac{360 + 5}{9}\)
\(10x = \frac{365}{9}\)
\(x = \frac{365}{9 \times 10}\)
\(x = \frac{365}{90}\)
\(x = \frac{73}{18}\)
\(x = 4\frac{1}{18}\)
In simple words: To convert a non-terminating repeating decimal to a fraction, set it equal to 'x', multiply by powers of 10 to shift the decimal and repeating block, then subtract the original equation to eliminate the repeating part and solve for 'x'.

🎯 Exam Tip: Pay close attention to the position of the dot or bar to correctly identify the repeating digits when setting up the equations.

Exercise 1.1 Real Numbers लघु उत्तरीय प्रश्न - II (Short Answer Type Questions - II)

Question 3. निम्न को परिमेय संख्या, \( \frac{m}{n} \) के रूप में व्यक्त कीजिए ।
(i) 0.4704
(ii) \(0 . \overline{572}\)
(iii) \(1 . \overline{63}\)
(iv) \(0.22\dot { 4 } \)
Answer:
(i) माना \(x = 0.4704\)
\(x = \frac{4704}{10000}\)
\(x = \frac{294}{625}\)

(ii) माना \(x = 0.\overline{572}\)
\(x = \frac{572}{999}\)

(iii) माना \(x = 1.\overline{63}\)
\(x = 1 + 0.\overline{63}\)
\(x = 1 + \frac{63}{99}\)
\(x = \frac{99+63}{99}\)
\(x = \frac{162}{99}\)
\(x = \frac{18}{11}\)

(iv) माना \(x = 0.22\dot { 4 }\)
100 से गुणा \( \implies 100x = 22.4\dot { 4 }\)
\(100x = 22 + 0.4\dot { 4 }\)
\(100x = 22 + \frac{4}{9}\)
\(100x = \frac{22 \times 9 + 4}{9}\)
\(100x = \frac{198+4}{9}\)
\(100x = \frac{202}{9}\)
\(x = \frac{202}{9 \times 100}\)
\(x = \frac{202}{900}\)
\(x = \frac{101}{450}\)
In simple words: A terminating decimal can be written as a fraction by dividing it by a power of 10 equal to the number of decimal places. For repeating decimals, use algebraic methods (as in Question 2) to convert them into a fraction of the form m/n.

🎯 Exam Tip: Always simplify the resulting fraction to its lowest terms. Understanding the number of 9s and 0s in the denominator for repeating decimals is crucial for efficiency.

Balaji Publications Mathematics Class 9 Solutions