please if somebody know the solution of Questions number-9,11
please tell correct answer
thanks
Answers
Step-by-step explanation:
Solutions :-
9)
i)
Given decimal number = 0.130130013000...
This is non terminating and non recurring decimal.
It is an irrational number.
It is not a rational number.
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ii) Given decimal number = 10.01456
This is a terminating decimal.
It is a rational number.
=> 10.01456
=> 1001456/10000
=> 250364/2500
=> 62591/625
It is in the form of p/q
q = 625
625 = 5×5×5×5
Prime factorization of q = 5×5×5×5
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iii) Given decimal number = 122.058 bar on 058
=>122.058058058...
It is a non terminating recurring decimal.
It is a rational number.
Let x = 122.058058058...-------(1)
Since the periodicity is 3 then multiplying (1) with 1000 then
=> 1000×x = 122.058058...×1000
=> 1000x = 122058.058058...-----(2)
On Substituting (1) from (2)
1000x = 122058.058058...
x = 122.058058...
(-)
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999x = 121936.000000...
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=> 999x = 121936
=> x = 121936/999
122.058 bar on 058 = 121936/999
This is in the form of p/q
q = 999
q = 3×3×3×37
Prime factorization of q = 3×3×3×37
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10)
Given number = 13/120
It can be written as 13/(2×2×2×3×5)
=> 13/(2³×3¹×5¹)
The denominator is in the form of 2³×3¹×5¹
We know that
X = p/q is a rational number if q is in the form of 2^m×5^n then it is a terminating decimal.
It is not in the form of 2^m×5^n
So , It is not a terminating decimal
It is non terminating recurring decimal.
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11)
Given numbers are 798 , 168 and 56
We know that
Euclid's Division Lemma:-
For two Positive integers a and b there exist two positive integers q and r satisfying a = bq+r, where 0≤r<b.
On taking 798 and 168
Let a = 798 and b = 168
on writing it a=bq+r
=> 798 = 168×4 + 126
and now a = 168 and b = 126
=> 168 = 126 × 1 + 42
and a = 126 and b = 42
126 = 42×3 + 0
HCF of 798 and 168 is 42
now
On taking 42 and 56
Let a = 56 and b = 42
On writing it a= bq +r
=> 56 = 42×1+14
and
Now a = 42 and b = 14
42 = 14×3 + 0
HCF of 56 and 42 is 14
HCF of 798 , 168 and 56 is 14
Used formulae :-
- The decimal expansion of a rational number is either a terminating decimal or non terminating recurring decimal.
- The decimal expansion of an irrational number is non terminating and non recurring decimal.
- The number of digits in the recurring part of a decimal number after the decimal point is called periodicity.
- X = p/q is a rational number if q is in the form of 2^m×5^n then it is a terminating decimal.
Euclid's Division Lemma:-
For two Positive integers a and b there exist two positive integers q and r satisfying a = bq+r, where 0≤r<b.