# NCERT Solutions for Class 9 Math | Exercise 1.3

###### CBSE Chapter 1 | Number Systems | Decimal Expansion of Rational Numbers

Chapter 1 of CBSE NCERT Class 9 Math covers number system. Concepts covered in exercise 1.3 include rational numbers, decimal expansion of rational numbers, terminating decimals - also known as finite decimals, non-terminating decimals - also known as infinite decimals, recurring decimals also known as repeating decimals, fractional equivalents of non-terminating recurring decimals.

Solutions to Question 1, Question 7, Question 8, and Question 9 are given in this page. For the other questions, click on the solution links given adjacent to the questions.

### NCERT Solutions for Class 9 Math | Exercise 1.3

1. Question 1

Write the following in decimal form and say what kind of decimal expansion each has:
i) $$frac{36}{100}$ ii) $\frac{1}{11}$ iii) 4$\frac{1}{8}$ iv) $\frac{3}{13}$ v) $\frac{2}{11}$ vi) $\frac{329}{400}$ #### Answer i) $\frac{36}{100}$ = 0.36. It is a terminating decimal. ii) $\frac{1}{11}$= 0.$\overline{09}$. It is a non-terminating repeating decimal expansion. iii) 4$\frac{1}{8}$ = 4.125. It is a terminating decimal. iv) $\frac{3}{13}$ = 0.$\overline{230769}$. It is a non-terminating repeating decimal expansion. v) $\frac{2}{11}$ = 0.$\overline{18}$. It is a non-terminating repeating decimal expansion. vi) $\frac{329}{400}$ = 0.8225. It is a terminating decimal. 2. Question 2 You know that the value of $\frac{1}{7}$ = 0.$\overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7}$, $\frac{3}{7}$, $\frac{4}{7}$, $\frac{5}{7}$, and $\frac{6}{7}$ are, without doing the long division? If so, how? 3. Question 3 Express the following in the form $\frac{p}{q}$, where p and q are integers and q ≠ 0: i) 0.$\overline{6}$ ii) 0.4$\overline{7}$ iii) 0.$\overline{001}$ 4. Question 4 Express 0.$\overline{9}$ in the form $\frac{p}{q}$: 5. Question 5 What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$? Perform the division to check. 6. Question 6 Look at several examples of rational numbers in the form $\frac{p}{q}$$q ≠ 0), where p and q are integers with no common factors other than 1 and have terminating decimal expansions. What property must q satisfy?

7. Question 7

Write three numbers whose decimal expansions are non-terminating non-recurring.

1. 0.1401400140001400001400000...
2. 0.080080008000080000080000008...
3. 0.40400400040000400000...

8. Question 8

Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.

First, let us find the decimal expansions of $\frac{5}{7}$ and $\frac{9}{11}$.
$\frac{5}{7} = 0.\overline{714285}$
$\frac{9}{11} = 0.\overline{81}$
We have to find irrational numbers between 0.$\overline{714285}$ and 0.$\overline{81}$

If we write three numbers that are non-terminating and non-recurring, we will have written three irrational numbers between the two given fractions.
0.73073007300073..., 0.7890789007890007890000..., and 0.80800800080000... are irrational numbers between the two given fractions.

9. Question 9

Classify the following numbers as rational or irrational:
i) √23
ii) √225
iii) 0.3769
iv) 7.478478....
v) 1.101001000100001...

i) √23 = 4.753... It is a non-terminating and non-recurring decimal. It is an irrational number. It cannot be expressed in the form $\frac{p}{q}$ where p and q are integers and q ≠ 0.

ii) √225 = 15. It is a terminating decimal. It is a rational number. It can be expressed in the form $\frac{p}{q}$ as $\frac{15}{1}$.

iii) 0.3769. It is a terminating decimal. It is a rational number. It can be expressed in the form $\frac{p}{q}$ as $\frac{3769}{10000}$.

iv) 7.478478.... It is a non-terminating but recurring decimal. It is a rational number. It can be expressed in the form $\frac{p}{q}$ as 7$\frac{478}{999}$ or $\frac{7471}{999}$.

v) 1.101001000100001... It is a non-terminating and non-recurring decimal. It is an irrational number. It cannot be expressed in the form $\frac{p}{q}$ where p and q are integers and q ≠ 0.

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