## NCERT10th MathematicsExercise 1.4Topic: 1 Real Numbers

## Click here for ⇨ NCERT-10-1-Real Numbers - Ex-1.3

### EXERCISE 1.4

**Q1. Without actually performing the long division, state whether the following rational**

**numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:**

**(i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 (v) 29/343**

**(vi) 23/(2**

^{3}5^{2}) (vii) 129/**(2**

^{2}**5**

^{7}**7**

^{5})**(viii) 6/15 (ix) 35/50 (x) 77/210**

### Explanation:

1) Let x = p/q be a rational number, such that the prime factorisation of q is of form 2

^{n}5^{m}, where n and m are non-negative integers. Then x has a decimal expansion which terminates.### Solution:

**(i) 13/3125**

1) Here the denominator is 3125.

2) Find the factors of 3125, we get,

3125 = 5 x 625

3125 = 5 x 5 x 1253125 = 5 x 5 x 5 x 253125 = 5 x 5 x 5 x 5 x 53125 = 5^{5}

3) Here our expression is 13/5

^{5}, and the denominator is 2^{n}x 5^{m }where n=0 and m = 5, so the decimal expansion of 13/3125 is terminating.**(ii) 17/8**

1) Here the denominator is 2

^{3}, which is of the form 2^{n}x 5^{m }where n=3 and m = 0.2) So the decimal expansion of 17/8 is terminating.

**(iii) 64/455**

1) Here the denominator is 455.

2) Find the factors of 455, we get,

455 = 5 x 91

455 = 5 x 7 x 13

3) Here our expression is 13/(5 x 7 x 13), and the denominator is (5 x 7 x 13), which is not of the form 2

^{n}5^{m}, so the decimal expansion of 64/455 is a non-terminating repeating decimal expansion.**(iv) 15/1600**

1) Here the denominator is 1600.

2) Find the factors of 1600, we get,

1600 = 16 x 100

1600 = 16 x 4 x 25

1600 = 2^{6}x 5^{2}

3) Here our expression is 15/(2

^{6}x 5^{2}) and the denominator is 2^{n}x 5^{m }where n=6 and m = 2, so the decimal expansion of 15/1600 is terminating.**(v) 29/343**

1) Here the denominator can be written as 7

^{3}.2) The decimal expansion of 29/7

^{3}, the denominator is (7^{3}), which is not of form 2^{n}5^{m}, so the decimal expansion of 29/343 is a non-terminating repeating decimal expansion.**(vi) 23/(2**

^{3}5^{2})1) Here the denominator is of the form 2

^{n}x 5^{m }where n=3 and m = 2.2) The decimal expansion of 23/(2

^{3}5^{2}) is terminating.**(vii) 129/**

**(2**

^{2}**5**

^{7}**7**

^{5})1) Here the denominator is not of the form 2

^{n}x 5^{m }and has 7^{5 }as a factor.2) So the decimal expansion of 129/(2

^{2}5^{7}7^{5}) is a non-terminating repeating decimal expansion.**(viii) 6/15**

1) Here the denominator is 15.

15 = 3 x 5

2) Here the denominator is (3 x 5), which is not of form 2

^{n}5^{m}, so the decimal expansion of 6/15 is a non-terminating repeating decimal expansion.**(ix) 35/50**

1) Here the denominator is 50.

2) Find the factors of 1600, we get,

50 = 2 x 25

50 = 2^{1}x 5^{2}

3) Here our expression is 35/(2

^{1}x 5^{2}) and the denominator is of the form 2^{n}x 5^{m }where n=1 and m = 2, so the decimal expansion of 35/50 is terminating.**(x) 77/210**

1) Here the denominator is 210.

210 = 21 x 10

210 = 3 x 7 x 2 x 5

2) Here the denominator is (3 x 7 x 2 x 5), which is not of the form 2

^{n}5^{m}, so the decimal expansion of 77/210 is a non-terminating repeating decimal expansion.

Q2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

### Solution:

**(i) 13/3125**

13/3125 = (13 x 2)/(3125 x 2)

= (26)/(6250)= (2.6)/(625)= (2.6 x 2)/(625 x 2)= (5.2)/(1250)= (0.52)/(125)= (0.52 x 2)/(125 x 2)= (1.04)/(250)= (0.104)/(25)= (0.104 x 4)/(25 x 4)= (0.416)/(100)= (0.00416)

15/1600 = (15)/(16 x 100)

= (0.15)/(16)= (0.15 x 5)/(16 x 5)= (0.75)/(80)= (0.075)/(8)= (0.0375)/(4)= (0.01875)/(2)= (0.009375)

So 15/1600 = 0.009375

**(v) 29/343.**

**It**

**is non-terminating.**

**(vi) 23/(2**

^{3}5^{2})23/(2^{3}5^{2}) = (23)/(2^{1}2^{2}5^{2})

= (23)/(2 x 10^{2})= (23)/(2 x 100)

= (11.5)/(100)= (0.115)

So 23/(2

^{3}5^{2}) = 0.115**(vii) 129/**

**(2**

^{2}**5**

^{7}**7**

^{5}).

**It**

**is non-terminating.**

**(viii) 6/15**

6/15 = (6)/(15)

= (2)/(5)= (2 x 2)/(5 x 2)= (4)/(10)= (0.4)

So 6/15 = 0.4

**(ix) 35/50**

35/50 = (35 x 2)/(50 x 2)

= (70)/(100)

= (0.7)

So 35/50 = 0.7

**(x) 77/210.**

**It**

**is non-terminating.**

Q3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p/q what can you say about the prime factors of q?

### Explanation:

1) Let x = p/q be a rational number, such that the prime factorisation of q is of form 2

^{n}5^{m}, where n and m are non-negative integers. Then x has a decimal expansion which terminates.### Solution:

(i) 43.123456789

1) 43.123456789 = (43123456789)/(10

^{9})= (43123456789)/(2^{9}x 5^{9})

2) Here the denominator is of the form 2

^{n}5^{m}, so the number is rational.(ii) 0.120120012000120000

1) Here the decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

---------------

(iii) 43.123456789

1) Here the decimal expansion is non-terminating and recurring, so the given number is a rational number of the form p/q where q is not of form 2

^{n}5^{m}. The prime factors of q will also have a factor other than 2 or 5.