Tag: decimal expansions of real numbers

Questions Related to decimal expansions of real numbers

$3.24636363....$ is _____________.

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

  1. An integer

  2. An irrational number

  3. A rational number

  4. Not a real number

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Correct Option: C
Explanation:

$3.24636363.......$ is a non-terminating, but repeating number as after one-hundredth place, digits are recurring.

$\therefore$ It is a rational number.

$\dfrac{p}{q}$ form of the number $0.\overline{3}$ is :

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

  1. $\dfrac{3}{10}$

  2. $\dfrac{3}{100}$

  3. $\dfrac{1}{3}$

  4. $\dfrac{1}{2}$

Show answer
Correct Option: C
Explanation:

Let $x$ = .33333......

Multiplying by 10 on both sides we get
$10x=3.3333....\ on\quad subtracting\quad both\quad equations\quad we\quad get,\ 9x=3\ x=\dfrac { 1 }{ 3 } $
So, correct answer is option C.

$\dfrac{35}{50}$ has a non-terminating decimal expansion.

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

  1. True

  2. False

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Correct Option: B
Explanation:

$35/50=7/10=0.7$
This is a terminating decimal number.

The decimal representation of $\dfrac { 93 }{ 1500 }$  will be

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

  1. Terminating

  2. Non-terminating

  3. Non-terminating, repeating

  4. Non-terminating, non-repeating

Show answer
Correct Option: A
Explanation:

Checking the termination of $\cfrac{93}{1500}$ is same as checking the termination of $\cfrac{31}{500}$ which is equal to $\cfrac{62}{1000}$


As the value is $0.062$, we can say the fraction is terminating.

$\therefore \cfrac{93}{1500}$ is terminating.

The fraction, $\dfrac{1}{3}$

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

  1. equals $0.33333333$

  2. is less than $0.33333333\ by\ \dfrac{1}{3.10^{8}}$

  3. is less than $0.33333333\ by\ \dfrac{1}{3.10^{9}}$

  4. is greater than $0.33333333\ by\ \dfrac{1}{3.10^{8}}$

  5. is greater than $0.33333333\ by\ \dfrac{1}{3.10^{9}}$

Show answer
Correct Option: D
Explanation:

$\cfrac { 1 }{ 3 } -0.33333333=\cfrac { 1 }{ 3 } -\cfrac { 33333333 }{ { 10 }^{ 8 } } \ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\cfrac { { 10 }^{ 8 }-99999999 }{ 3\cdot { 10 }^{ 8 } } \ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\cfrac { 1 }{ 3\cdot { 10 }^{ 8 } } $

$\therefore \cfrac { 1 }{ 3 } $ is greater than 0.33333333 by $\cfrac { 1 }{ 3\cdot { 10 }^{ 8 } } $.

Let $x=\dfrac { p }{ q } $ be a rational number, such that the prime factorization of $q$ is of the form $2^n 5^m$, where $n, m$ are non-negative integers. Then $x$ has a decimal expansion which terminates.

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

  1. True

  2. False

  3. Neither

  4. Either

Show answer
Correct Option: A
Explanation:

The form of q is $2^n*5^m$
q can be $1,2,5,10,20,40....$
Any integer divided by these numbers will always give a terminating decimal number.

Without actually performing the long division, state whether the following rational number will have terminating decimal expansion or a non-terminating repeating decimal expansion. Also, find the numbers of places of decimals after which the decimal expansion terminates.
$\dfrac { 13 }{ 3125 } $

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

  1. $3$

  2. $4$

  3. $5$

  4. $6$

Show answer
Correct Option: C
Explanation:

The given value is $\dfrac{13}{3125}$ the denominator is 3125 which can be written as:


$3125=2^0 \times 5^5$ it is in the form of $2^m \times 5^n$

$max(m,n)=5$

$\therefore$ the expansion is terminating decimal it terminates after 

$max(m,n)=5$ places from the decimal [since  $ m=0,n=5$]

State whether the following statement is true/false.

$\dfrac{2375}{375}$ is not a terminating decimal

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

For $\cfrac{2375}{375}$


$375=5^3\times 3$ and $2375=5^3\times 19$


Since, denominator contains $3$ as a factor other than only $2$ or $5$,

So, $\cfrac{2375}{375}$ is is non terminating.

$9.1 \overline { 7 }$ is

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

  1. Terminating decimal

  2. Mixed repeating decimal

  3. Pure repeating decimal

  4. None of these

Show answer
Correct Option: C
Explanation:

Given 


$9.1\bar 7$

Here the bar representation implies that the decimal is purely repeating one 

$\implies 9.1\bar 7=9.17777777777777777777777.....$

$\dfrac { 317 } { 3125 }$  represents ______.

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

  1. A terminating decimal

  2. A non-recurring decimal

  3. A recurring decimal

  4. An Integer

Show answer
Correct Option: A