CBSE Class 10 Mathematics Real Numbers Worksheet Set 08

Question. 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) \( \frac{12}{75} \)
(ii) \( \frac{7}{500} \)
(iii) \( \frac{11}{5^3 \cdot 2^2 \times 7^4} \)
(iv) \( \frac{13}{5^3 \times 2^{11}} \)
Answer: Solution. (i) Simplest form of \( \frac{12}{75} = \frac{4}{25} \), since prime factors of denominator 25 is of the form \( 2^0 \times 5^2 \). Hence, \( \frac{12}{75} \) or \( \frac{4}{25} \), is a terminating decimal.
(ii) Since the factors of denominator is 500 is in the form \( 2^2 \times 5^3 \). Hence \( \frac{7}{500} \) is a terminating decimal.
(iii) Since the factors of denominator is \( 5^3 \times 2^2 \times 7^4 \) so, it is not in the form \( 2^m \times 5^n \) hence \( \frac{11}{5^3 \times 2^2 \times 7^4} \) is a non-terminating decimal.
(iv) Clearly prime factors of denominator are \( 5^3 \times 2^{11} \) is in the from \( 2^m \times 5^n \) so \( \frac{13}{5^3 \times 2^{11}} \) is a terminating decimal.

Question. 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) \( \frac{13}{3125} \)
(ii) \( \frac{17}{8} \)
(iii) \( \frac{64}{455} \)
(iv) \( \frac{15}{1600} \)
(v) \( \frac{29}{343} \)
(vi) \( \frac{23}{2^3 5^2} \)
(vii) \( \frac{129}{2^2 5^7 7^5} \)
(viii) \( \frac{6}{15} \)
(ix) \( \frac{35}{50} \)
(x) \( \frac{77}{210} \)
Answer: Solution. (i) Since the factors of the denominator 3125 are \( 2^0 \times 5^5 \). Therefore \( \frac{13}{3125} \) is a terminating decimal.
(ii) Since the factors of the denominator 8 are \( 2^3 \times 5^0 \). So, \( \frac{17}{8} \) is a terminating decimal.
(iii) Since the factors of the denominator 455 is not in the form \( 2^n \times 5^m \). So \( \frac{64}{455} \) is non-terminating repeating decimal.
(iv) Since the factors of the denominator 1600 are \( 2^6 \times 5^2 \), So, \( \frac{15}{1600} \) is a terminating decimal.
(v) Since the factors of the denominator 343 is not of the form \( 2^n \times 5^m \). So, it is non-terminating repeating decimal.
(vi) Since the denominator is of the form \( 2^3 \times 5^2 \). So, \( \frac{23}{2^3 \times 5^2} \) is a terminating decimal.
(vii) Since the factors of the denominator \( 2^2 5^7 7^5 \) is not of the form \( 2^n \times 5^m \). So, \( \frac{129}{2^2 5^7 7^5} \) is non-terminating repeating decimal.
(viii) \( \frac{6}{15} = \frac{2}{5} \) here factors of the denominator 5 is of the form \( 2^0 \times 5^1 \). So, \( \frac{6}{15} \) is a terminating decimal.
(ix) Since the factors of the denominator 50 is of the form \( 2^1 \times 5^2 \). So, \( \frac{35}{50} \) is terminating decimal.
(x) Since the factors of the denominator 210 is not of the form \( 2^n \times 5^m \). So, \( \frac{77}{210} \) is non-terminating repeating decimal.

Question. 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 \( \frac{p}{q} \) what can you say about the prime factors of \( q \)?
(i) 2.060060006.......
(ii) 23.232323.......
Answer: Solution. (i) Given number is non terminating and non repeating hence, cannot be expressed in the form of \( \frac{p}{q} \) so no comment on prime factors of '\( q \)' because \( q \) does not exist.
(ii) Given number is non terminating and repeating rational number so, it can be expressed in the form of \( \frac{p}{q} \) where prime factors of \( q \) will also have prime numbers other than 2 or 5.

Question. Write down the decimal expansion of those rational numbers in Example (2) which have terminating decimal expansions. 
Answer: Solution. (i) \( \frac{13}{3125} = \frac{13}{5 \times 5 \times 5 \times 5 \times 5} = \frac{13 \times 2 \times 2 \times 2 \times 2 \times 2}{5 \times 2 \times 5 \times 2 \times 5 \times 2 \times 5 \times 2 \times 5 \times 2} = \frac{13 \times 32}{10 \times 10 \times 10 \times 10 \times 10} = \frac{416}{100000} = 0.0046 \)
(ii) \( \frac{17}{8} = \frac{17 \times 5^3}{2^3 \times 5^3} = \frac{17 \times 125}{10^3} = \frac{2125}{1000} = 2.125 \)
(iii) Non-terminating repeating.
(iv) \( \frac{15}{1600} = \frac{15}{2^6 \times 5^2} = \frac{15}{2^4 \times 2^2 \times 5^2} = \frac{15}{2^4 \times 10^2} = \frac{15 \times 5^4}{2^4 \times 5^4 \times 10^2} = \frac{15 \times 625}{10^4 \times 10^2} = \frac{9375}{1000000} = 0.009375 \)
(v) Non-terminating repeating.
(vi) \( \frac{23}{2^3 \cdot 5^2} = \frac{23}{2 \cdot 2^2 \cdot 5^2} = \frac{23}{2 \cdot 10^2} = \frac{23 \times 5}{2 \times 5 \times 10^2} = \frac{115}{1000} = 0.115 \)
(vii) Non-terminating repeating.
(viii) \( \frac{6}{15} = \frac{2}{5} = \frac{4}{10} = 0.4 \)
(ix) \( \frac{35}{50} = \frac{35 \times 2}{50 \times 2} = \frac{70}{100} = 0.70 \)
(x) Non-terminating repeating.

Question. 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 \( \frac{p}{q} \), what can you say about the prime factors of \( q \)? 
(i) 43.123456789
(ii) 0.120120012000120000....
(iii) 43.123456789 (bar on decimal part)
Answer: Solution. (i) 43.123456789 is terminating. So, it represents a rational number. Thus, \( 43.123456789 = \frac{43123456789}{1000000000} = \frac{p}{q} \). Thus, \( q = 10^9 \).
(ii) 0.12012001200012000... is non-terminating and non-repeating. So, it is an irrational number.
(iii) 43.12345789 (bar) is non-terminating but repeating. So, it is a rational. Thus, \( 43.123456789 = \frac{4312345646}{999999999} = \frac{p}{q} \). Thus, \( q = 999999999 \).

Question. 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) \( \frac{19}{250} \)
(ii) \( \frac{23}{2048} \)
(iii) \( \frac{127}{910} \)
(iv) \( \frac{35}{800} \)
(v) \( \frac{54}{343} \)
(vi) \( \frac{52}{2^3 \times 5^4} \)
(vii) \( \frac{1111}{7^4 \times 13^2} \)
(viii) \( \frac{328}{2^2 \times 5^3 \times 7^4} \)
(ix) \( \frac{1001}{2^0 \times 5^3 \times 11^5} \)
(x) \( \frac{922}{2^1 \times 5^2 \times 7^3 \times 11^4} \)
Answer: 1. (i) Terminating (ii) Terminating (iii) Non-terminating recurring (iv) Terminating (v) Non-terminating recurring (vi) Terminating (vii) Non-terminating recurring (viii) Non-terminating recurring (ix) Non-terminating recurring (x) Non-terminating recurring

Question. Write down the decimal expansion of the rational number of above which have terminating decimal expansion.
Answer: 2. (i) 0.076 (ii) 0.011230468 (iv) 0.04375 (vi) 0.0104

Question. Which of the following decimal expansions are rational. If they are rational \( \frac{p}{q} \), then write down the prime factors of \( q \).
(i) 125.085
(ii) 23.243456789
(iii) 0.240240024000240000....
(iv) 11.21478
(v) 12.123456789 (bar)
Answer: 3. (i) Rational Number; \( q = 2^3 \times 5^3 \) (ii) Rational Number; \( q = 2^9 \times 5^9 \) (iii) Irrational Number (iv) Rational Number; \( q = 2^3 \times 3^2 \times 5^3 \times 11 \) (v) Rational Number; \( q = 3^2 \times 111111111 \)

Question. Without actually performing the long division, find if \( \frac{987}{10500} \) will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.
Answer: 4. Terminating because \( 987/10500 = 47/500 \). Now, prime factors of denominator i.e. \( 500 = 2^2 \times 5^3 \).

Question. A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of \( q \), when this number is expressed in the form \( p/q \)? Give reasons.
Answer: 5. As 327.7081 is a terminating decimal number so \( q \) must be of form \( (2)^m \times (5)^n \) where \( (m, n) \in N \).

REVISION EXERCISE

SECTION – A

Multiple Choice Questions (MCQs) 

Question. The [HCF \times LCM] for the numbers 50 and 20 is
(a) 10
(b) 100
(c) 1000
(d) 50
Answer: (c) 1000

Question. Which of the following numbers has terminating decimal expansion ?
(a) \( \frac{37}{45} \)
(b) \( \frac{21}{2^3 5^6} \)
(c) \( \frac{17}{49} \)
(d) \( \frac{89}{2^2 3^2} \)
Answer: (b) \( \frac{21}{2^3 5^6} \)

Question. If least prime factor of 'a' is 3 and least prime factor of 'b' is 7, the least prime factor of \( (a + b) \) is:
(a) 2
(b) 3
(c) 5
(d) 11
Answer: (a) 2

Question. If \( a, b \) are coprime, then \( a^2, b^2 \) are :
(a) Coprime
(b) Not coprime
(c) Odd numbers
(d) Even numbers
Answer: (a) Coprime

Question. If the HCF of 65 and 117 is expressible in the form \( 65m - 117 \), then the value of \( m \) is :
(a) 4
(b) 2
(c) 1
(d) 3
Answer: (b) 2

Question. \( 119^2 - 111^2 \) is:
(a) Prime number
(b) Composite number
(c) An odd prime number
(d) An odd composite number
Answer: (b) Composite number

Question. The decimal expansion of \( \frac{33}{2^2 \times 5} \) will terminate after :
(a) One decimal place
(b) Two decimal places
(c) Three decimal places
(d) More than three decimal places
Answer: (b) Two decimal places

Question. A rational number can be expressed as a terminating decimal if the denominator has factors.
(a) 2, 3 or 5
(b) 2 or 3
(c) 3 or 5
(d) 2 or 5
Answer: (d) 2 or 5

Question. Euclid’s division lemma states that if \( a \) and \( b \) are any two positive integers, then there exists unique integers \( q \) and \( r \) such that
(a) \( a = bq + r, 0 < r < b \)
(b) \( a = bq + r, 0 \le r \le b \)
(c) \( a = bq + r, 0 \le r < b \)
(d) \( a = bq + r, 0 < b < r \)
Answer: (c) \( a = bq + r, 0 \le r < b \)

Question. Which of the following is not an irrational number ?
(a) \( 5 - \sqrt{3} \)
(b) \( \sqrt{5} + \sqrt{3} \)
(c) \( 4 + \sqrt{2} \)
(d) \( 5 + \sqrt{9} \)
Answer: (d) \( 5 + \sqrt{9} \)

Question. Which is not an Irrational number ?
(a) \( 5 - \sqrt{3} \)
(b) \( \sqrt{2} + \sqrt{5} \)
(c) \( 4 + \sqrt{2} \)
(d) \( 6 + \sqrt{9} \)
Answer: (d) \( 6 + \sqrt{9} \)

Question. If two positive integers \( a \) and \( b \) are written as \( a = x^2 y^2 \) and \( b = xy^2 \); \( x, y \) are prime numbers then HCF \( (a, b) \) is :
(a) \( xy \)
(b) \( xy^2 \)
(c) \( x^2 y^3 \)
(d) \( x^2 y^2 \)
Answer: (b) \( xy^2 \)

Question. For the decimal number \( 0.\bar{6} \), the rational number is :
(a) \( \frac{33}{50} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{111}{167} \)
(d) \( \frac{1}{3} \)
Answer: (b) \( \frac{2}{3} \)

Question. Which of the following is a non-terminating repeating decimal ?
(a) \( \frac{35}{14} \)
(b) \( \frac{14}{35} \)
(c) \( \frac{1}{7} \)
(d) \( \frac{7}{8} \)
Answer: (c) \( \frac{1}{7} \)

Question. If \( x = 2^3 \times 3 \times 5^2, y = 2^2 \times 3^3 \), then HCF \( (x, y) \) is:
(a) 12
(b) 108
(c) 6
(d) 36
Answer: (a) 12

Question. Given that HCF (2520, 6600) = 120, LCM (2520, 6600) = \( 252 \times k \), then the value of \( k \) is :
(a) 550
(b) 1600
(c) 165
(d) 1625
Answer: (a) 550

Question. If \( p, q \) are two co-prime numbers, then HCF \( (p, q) \) is :
(a) \( p \)
(b) \( q \)
(c) \( pq \)
(d) 1
Answer: (d) 1

Question. The decimal expansion of the rational number \( \frac{43}{2^4 \times 5^3} \) will terminate after :
(a) 3 places
(b) 4 places
(c) 5 places
(d) 1 place
Answer: (b) 4 places

Question. The product of the HCF and LCM of the smallest prime number and smallest composite number is :
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (d) 8

Question. The decimal expansion of the rational number \( \frac{23}{2^2 \cdot 5} \) will terminate after.
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) more than three decimal places
Answer: (b) two decimal places

Question. \( n^2 - 1 \) is divisible by 8, if \( n \) is
(a) an integer
(b) a natural number
(c) an odd integer
(d) an even integer
Answer: (c) an odd integer

Question. If \( p, q \) are two prime numbers, then LCM \( (p, q) \) is :
(a) 1
(b) \( P \)
(c) \( q \)
(d) \( pq \)
Answer: (d) \( pq \)

Question. If \( d = \text{HCF } (48, 72) \), the value of \( d \) is :
(a) 24
(b) 48
(c) 12
(d) 72
Answer: (a) 24

Question. Given that LCM (91, 26) = 182, then HCF (91, 26) is:
(a) 13
(b) 26
(c) 7
(d) 9
Answer: (a) 13

Question. The decimal expansion of the rational number \( \frac{11}{2^3 \cdot 5^2} \) will terminate after :
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) more than 3 decimal places
Answer: (c) three decimal places

Question. If \( d = \text{LCM } (36, 198) \), then the value of \( d \) is :
(a) 396
(b) 198
(c) 36
(d) 1
Answer: (a) 396

Question. Given that HCF (26, 91) = 13, then LCM of (26, 91) is :
(a) 2366
(b) 182
(c) 91
(d) 364
Answer: (b) 182

Question. \( (2 + \sqrt{5})(2 - \sqrt{5}) \) expression is:
(a) A rational number
(b) A whole number
(c) An integer
(d) All of the options
Answer: (d) All of the options

Question. \( (\sqrt{2} - \sqrt{3})(\sqrt{3} + \sqrt{2}) \) is
(a) A rational number
(b) A whole number
(c) An irrational number
(d) A natural number
Answer: (a) A rational number

Question. The decimal expansion of \( \frac{141}{120} \) will terminate after how many places of decimals ?
(a) 3
(b) 4
(c) 1
(d) 2
Answer: (a) 3

Question. Given that HCF (253, 440) = 11 and LCM (253, 440) = \( 253 \times R \). The value of \( R \) is :
(a) 400
(b) 40
(c) 440
(d) 253
Answer: (b) 40

Question. The decimal expansion of \( \frac{131}{120} \) will terminate after how many places of decimal?
(a) 1
(b) 2
(c) 3
(d) will not terminate
Answer: (d) will not terminate

Question. If \( n \) is any natural number, then which of the following expressions ends with 0 :
(a) \( (3 \times 2)^n \)
(b) \( (4 \times 3)^n \)
(c) \( (2 \times 5)^n \)
(d) \( (6 \times 2)^n \)
Answer: (c) \( (2 \times 5)^n \)

Question. How many prime factors are there in prime factorization of 5005.
(a) 2
(b) 4
(c) 6
(d) 7
Answer: (b) 4