9th Maths 1.3

NCERT Class 9th solution of Exercise 1.1

NCERT Class 9th solution of Exercise 1.2

NCERT Class 9th solution of Exercise 1.4

NCERT Class 9th solution of Exercise 1.5

NCERT Class 9th solution of Exercise 1.6

NCERT Class 9th Maths Projects

Exercise 1.3

Q1. 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}{200}`

Answer :

(i) `\0.36`, terminating. (ii) `0.\overline{09}`, non-terminating repating. (iii) `4.\125`, terminating. (iv)`0.\overline{230769}`, non-terminating repeating. (v) `0.\overline{18}`, non-terminating repeating. (vi)`0.\8225`, terminating.

Q2. You Know that  `\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}`,`\frac{6}{7}` are, without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of `\frac{1}{7}` carefully. ]

Answer :

`\frac{2}{7}` = `2×\frac{1}{7}` = `0.\overline{285714}`, 

`\frac{3}{7}` = `3×\frac{1}{7}` = `0.\overline{428571}`,

`\frac{4}{7}` = `4×\frac{1}{7}` = `0.\overline{571428}`,

`\frac{5}{7}` = `5×\frac{1}{7}` = `0.\overline{714285}`,

`\frac{6}{7}` = `6×\frac{1}{7}` = `0.\overline{857142}`.

Q3. 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}`

 Sol. : 

(i) Let `\x` = `0.\overline{6}` = `0.\66....`__________(1)

Multiply by `\10` both side

`\10x` = `\10×0.\66`

`\10x` = `\6.\66`____________________________(2)

Equation (2) - (1)

`\10x - x` = `\6.\66 - 0.\66....`

`\9x`  = `\6`

`\x` = `\frac{6}{9}` = `\frac{2}{3}`

Answer : 

Thus the required `\frac{p}{q}` form of  `0.\overline{6}` = `\frac{2}{3}`.

(ii) (i) Let `\x`= `0.4\overline{7}` = `0.4\77....`__________(1)

Multiply by `\10` both side

`\10x` = `\10×0.4\77`

`\10x` = `4.\7\7`____________________________(2)

Equation (2) - (1)

`\10x - x` = `4.\7\7 - 0.4\77....`

`\9x`  = `4.\3`

`\x` = `\frac{4.3}{9}` = `\frac{43}{90}`  

Answer :

Thus the required `\frac{p}{q}` form of  `0.4\overline{7}` = `\frac{43}{90}`.

 iii)Let `x` = `0.overline{001}` = `0.\001001....`__________(1) 

Multiply by `\1000` both side 

`\1000x` = `\1000×0.\001` 

`\1000x` = `\1.\001`____________________________(2) 

Equation (2) - (1) 

`\1000x - x` = `\1.\001 - 0.\001....` 

`\999x`  = `\1` 

`\x` = `\frac{1}{999}`

Answer : 

Thus the required `\frac{p}{q}` form of  `0.\overline{001}` = `\frac{1}{999}`.

Q4. Express `0.9999......` in the form `p/q`. Are you surprised by your answer? With your teacher and classmate discuss why the answer makes sense?

Sol. :

Let `\x` = `0.\overline{9}` = `0.\99....`__________(1)

Multiply by `\10` both side

`\10x` = `\10×0.\99`

`\10x` = `\9.\99`____________________________(2)

Equation (2) - (1)

`\10x - x` = `\9.\99 - 0.\99....`

`\9x`  = `\9`

`\x` = `\frac{9}{9}` = `\1`

Answer : 

Thus the required `\frac{p}{q}` form of  `0.\overline{9}` = `\1`.

 Q5. 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 your answer.

Answer :

Sixteen`(16)` digitsi.e. `\frac{1}{17}` =   `0.\overline{0588235294117647}` 

Q6. 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 having terminating decimal representations (expressions) can you guess what property q must satisfy?

Answer :

The prime factorisation of q has only powers of 2 or powers of 5 or both.

for example   `\frac{3}{2}` = 1.5,   `\frac{5}{4}` =  `\frac{5}{2×2}` = 1.25,  `\frac{4}{5}` =  0.8,  `\frac{3}{25}` = 0.12,  `\frac{7}{10}` =  `\frac{7}{2×5}` = 0.7,  `\frac{11}{100}` =  `\frac{11}{2×2×5×5}` = 0.11

Q7. Write three numbers whose decimal expansions are non-terminating non-recurring.

Answer :

The required three examples are :

`(i) 0.01001000100001....`

`(ii) 0.202002000200002.....`

`(iii) 0.003000300003.......`

Q8. Find three different irrational  numbers between rational numbers `\frac{5}{7}` and `\frac{9}{11}`

Sol. : 

`\frac{5}{7}` After division we get `0.\overline{714285}` and   `\frac{9}{11} ` After division we get  `0.\overline{81}` 

Different irrational numbers between these two given numbers.

Answer :

Those are `0.75075007500075......`,  `0.767076700767000767.....`,  `0.8080080008.....`

Q9. Classify the following numbers as rational or irrational :

(i) `\sqrt{23}` (ii) `\sqrt{225}` 

(iii) `0.3796`     (iv) `7.478478` 

(v) `1.101001000100001....`

Answer :

(i), (iv) and (v) are irrational and (ii), (iii) are rational.