Tag: rational numbers as recurring/terminating decimals

Questions Related to rational numbers as recurring/terminating decimals

Given that $\dfrac {1}{7} = 0.\overline {142857}$, which is a repeating decimal having six different digits. If $x$ is the sum of such first three positive integers $n$ such that $\dfrac {1}{n} = 0.\overline {abcdef}$, where $a, b, c, d, e$ and $f$ are different digits, then the value of $x$ is

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. $20$

  2. $21$

  3. $41$

  4. $42$

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Correct Option: C
Explanation:

$1^{st}$ number
$x _{1} = 7$
$\Rightarrow \dfrac {1}{x _{1}} = 0.\overline {142857}$
such that,
$2^{nd}$ number
$x _{2} = 13$
$\Rightarrow \dfrac {1}{x _{2}} = 0.\overline {076923}$
$x _{3} = 21$
$\Rightarrow \dfrac {1}{x _{3}} = \dfrac {1}{21} = 0.\overline {047619}$
$x = x _{1} + x _{2} + x _{3}$
$\Rightarrow 7 + 13 + 21 = 41$.

If $x =\dfrac{p}{q}$  be a rational number such that the prime factorization of $q$ is not of the form $2^n 5^m$, where $n, m$ are non-negative integers. Then $x$ has a decimal expansion which is terminating.

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. True

  2. False

  3. Neither

  4. Either

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Correct Option: B
Explanation:

If q is not of the form $2^n*5^m$ then definitely q can take any of the values 3,6,9,12,15,...etc.
As we know 4/3,20/6 all are non-terminating, thus required decimal expansion is non terminating.

The numbers 7.478478.... and 1.101001000100001.....are

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. Rational and irrational respectively

  2. Both rationals

  3. Both irrationals

  4. None of these

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Correct Option: A
Explanation:

In 7.478478.......  we can see that after decimal the digits 478 are repeating itself again and again.

Hence it is a non-terminating and repeating decimal. Therefore it is a rational number.

In 1.101001000100001........  it is a non-terminating and non-repeating decimal.
Therefore it is an irrational number.
Hence option A is correct.