Tag: decimal expansions of real numbers

Questions Related to decimal expansions of real numbers

If $x=0.123\bar{4}, y=0.12\bar{34}$ and $z=0.1\bar{234}$, then which of the following is correct?

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

  1. $x>y>z$

  2. $y$

  3. $z>x$

  4. $x>z>y$

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Correct Option: D

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :
(i) $\displaystyle \dfrac{7}{16}$ (ii) $\displaystyle \dfrac{23}{125}$
(iii) $\displaystyle \dfrac{9}{14}$ (iv) $\displaystyle \dfrac{32}{45}$
(v) $\displaystyle \dfrac{43}{50}$ (vi) $\displaystyle \dfrac{17}{40}$
(vii) $\displaystyle \dfrac{61}{75}$ (viii) $\displaystyle \dfrac{123}{250}$

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

  1. (i), (iii), (v), (vi) and (vii)

  2. (i), (ii), (v), (vi) and (viii)

  3. (i), (iii), (v), (vi) and (viii)

  4. (i), (ii), (v), (vi) and (vii)

Show answer
Correct Option: B
Explanation:

 The rational no having denominator $3, 7, 9, 11, 13, 17, 23, 27$.............. and multiple of these number will have non terminating decimal .
(1) $\dfrac{7}{16}$ the denominator of this rational number is not having these above number and multiple of these number, so this will have  terminating decimal.
(2) $\dfrac{23}{125}$ -- the denominator of this rational number is not having these above number and multiple of these number, so this will have  terminating decimal.
(3) $\dfrac{9}{14}$ --he denominator of this rational number is having these above number multiple of $7$, so this will have non terminating decimal.
(4)$\dfrac{32}{45}$--he denominator of this rational number is having these above number multiple of 9, so this will have non terminating decimal.
(5) $\dfrac{43}{50}$-- the denominator of this rational number is not having these above number and multiple of these number, so this will have  terminating decimal.
(6)$\dfrac{17}{40}$ -- the denominator of this rational number is not having these above number and multiple of these number, so this will have  terminating decimal.
(7)$\dfrac{61}{75}$-- he denominator of this rational number is having these above number multiple of 3, so this will have non terminating decimal.
(8)$\dfrac{123}{250}$--the denominator of this rational number is not having these above number and multiple of these number, so this will have  terminating decimal.
(i), (ii), (v), (vi) and (viii) will have  terminating decimal.

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 $\cfrac {p}{q}$?

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

  1. $q$ has prime factors $2$ or $5$ or both.

  2. $q$ has prime factors except $2$ and $5.$

  3. $q$ has no prime factors

  4. None of these

Show answer
Correct Option: A
Explanation:

We know that The rational no having denominator 3, 7, 9, 11, 13, 17, 23, 27.............. and multiple of these number will have non terminating decimal .
As  decimal expansion is 327.7081 which is terminating.
prime factors of q, when this number is expressed in the form p/q will not be above number, it will be 2 or 5 or both.

Consider the following statements :
1. $\displaystyle \frac{1}{22}$ can not be written as terminating decimal 


2. $\displaystyle \frac{2}{15}$ can be written as a terminating decimal 

3. $\displaystyle \frac{1}{16}$ can be written as a terminating decimal 

Which of the statements given above is/are correct ?

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

  1. $1$ only

  2. $2$ only

  3. $1$ and $3$

  4. $2$ and $3$

Show answer
Correct Option: C
Explanation:

$\displaystyle \frac{1}{22} = 0.04545454545$ is not a terminating decimal.

$\displaystyle \frac{2}{15}  = 0.133333333$ is not a terminating decimal.

$\displaystyle \frac{1}{16} = 0.0625$ is a terminating decimal.

Hence, statement $1$ and $3$ are correct.

Which one of the following is not a correct statement ?

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

  1. $\displaystyle 0.\overline{01}=\frac{1}{90}$

  2. $\displaystyle 0.\overline{1}=\frac{1}{9}$

  3. $\displaystyle 0.\overline{2}=\frac{2}{9}$

  4. $\displaystyle 0.\overline{3}=\frac{1}{3}$

Show answer
Correct Option: A
Explanation:

$\dfrac{1}{90} = 0.0111111111 = 0.0\bar{1}$


$\dfrac{1}{9} = 0.11111111 = 0.\bar{1}$

$\dfrac{2}{9} = 0.222222222 = 0.\bar{2}$

$\dfrac{1}{3} = 0.3333333333= 0.\bar{3}$

Hence, option $A$ is not correct.

The decimal form of $5\dfrac{3}{8}$ is

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

  1. $5.375$

  2. $5.000$

  3. $5.255$

  4. $2.325$

Show answer
Correct Option: A
Explanation:

First change the mixed fraction into proper fraction.
$5\dfrac{3}{8}=\dfrac{43}{8}$

$\dfrac{43}{8}= 5 + \dfrac38 = 5 + 0.375 = 5.375$

Arrange the following decimal numbers in ascending order.
$5.5, 0.55, 0.055, 0.005$

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

  1. $5.5, 0.055, 0.005, 0.55$

  2. $0.55, 0.005, 0.055, 5.5$

  3. $5.5, 0.55, 0.055, 0.005$

  4. $0.005, 0.055, 0.55, 5.5$

Show answer
Correct Option: D
Explanation:

We need to arrange the numbers from smallest to largest.
So, $0.005, 0.055, 0.55, 5.5$ is in ascending order

............... numbers have terminating and non- terminating repeating decimals.

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

  1. Integers

  2. Whole

  3. Rational

  4. Irrational

Show answer
Correct Option: C
Explanation:

$\dfrac {1}{4} = 0.25$ is a terminating decimal.


$\dfrac {8}{3} = 2.666666666......$ is a non-terminating repeating decimal.

Both are rational numbers but it was non repeating then they are irrational numbers.
Therefore, $C$ is the correct answer.

If the denominator of a fraction has factors other then $2$ and $5$, the decimal expression ..............

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

  1. repeats

  2. is that of a whole number

  3. has equal numerator and denominator

  4. terminates

Show answer
Correct Option: A
Explanation:

If there are prime factors in the denominator other than $2$ or $5$, then the decimals repeat.
$\dfrac {1}{24} = \dfrac {1}{3\times 2\times 2\times 2}$ (there is a factor of $3$, the decimal will repeat.)
Therefore, $A$ is the correct answer.

If the denominator of a fraction has only factors of $2$ and factors of $5$, the decimal expression ............. 

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

  1. has equal numerator and denominator

  2. becomes a whole number

  3. does not terminate

  4. terminates

Show answer
Correct Option: D
Explanation:

When the prime factorization of the denominator of a fraction has only factors of $2$ and factors of $5$, we can always express the decimal as terminating decimal. 
For examples $\dfrac {1}{25} = \dfrac {1}{5\times 5}$ repeats (just powers of $5$, the decimal terminates.)
Therefore, $D$ is the correct answer.