Tag: decimal expansions of real numbers

Questions Related to decimal expansions of real numbers

$0.\overline{585}$ is equal to

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

  1. $\displaystyle\frac{585}{99}$

  2. $\displaystyle\frac{585}{999}$

  3. $\displaystyle\frac{999}{585}$

  4. none of these

Show answer
Correct Option: B
Explanation:

Given that, $0.\overline{585}$.

Let,

$ x=0.\overline{585} $

$ x=0.585585585........ $

 

Multiply by 1000 on both sides,

$ 1000x=1000\times 0.585585585........ $

$ =585.585585585.... $

$ =585+0.585585585........ $

$ 1000x=585+x $

$ 999x=585 $

$ x=\dfrac{585}{999} $

 

Hence, this is the answer.

A terminating decimal has a ............ number of terms after the decimal point.

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

  1. zero

  2. infinite

  3. finite

  4. none of the above

Show answer
Correct Option: C
Explanation:

A terminating decimal is a decimal that ends. It's a decimal with a finite number of digits.
Therefore, $C$ is the correct answer.

As the decimal of $\dfrac {1}{3}$ repeats$,$ $\dfrac {1}{3}$ is a $.........$ decimal.

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

  1. exact

  2. negative

  3. terminating

  4. non-terminating

Show answer
Correct Option: D
Explanation:

$\dfrac {1}{3} = 0.33333333333333333333333333....$ 

It is non-terminating recurring decimal.
Therefore, $D$ is the correct answer.

If the quotient is terminating decimal, the division is complete only when ...............

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

  1. we get the remainder $1$

  2. we get the remainder zero

  3. we get the remainder as the repeated numbers

  4. All of the above

Show answer
Correct Option: B
Explanation:
Any division is complete only when we get the remainder zero.
Therefore, $B$ is the correct answer.

A ............. decimal representation can be repeating or non-repeating decimal

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

  1. real

  2. non-terminating

  3. terminating

  4. none of these

Show answer
Correct Option: B
Explanation:

A non- terminating representation can be repeating $\dfrac{7}{22} = 0.31818181....$
or non-repeating decimal $\pi = 3.1415....$
Therefore, $B$ is the correct answer.

Which option will have a terminating decimal expansion?

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

  1. $\dfrac {77}{210}$

  2. $\dfrac {23}{30}$

  3. $\dfrac {125}{441}$

  4. $\dfrac {23}{8}$

Show answer
Correct Option: D
Explanation:

The decimal expansion of a number may terminate in which case the number is called a regular number or finite decimal. In the given options:

a. $\dfrac{77}{210} = 0.366666666....... \; or  \; 0.3\bar6$

b. $\dfrac{23}{30} = 0.766666666....... \; or  \; 0.7\bar6$

c. $\dfrac{125}{441} = 0.283446712......$ i.e. it goes on infintely

d. $\dfrac{23}{8} = 2.875$

A number having non-terminating and recurring decimal expansion is.

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

  1. An integer

  2. A rational number

  3. An irrational number

  4. A whole number

Show answer
Correct Option: B
Explanation:

A number having non-terminating and recurring decimal expansion is  a Rational Number


for example 

$\dfrac{2}{7}$  is a rational number 

$\dfrac{2}{7} =0.285714285714285714.......... $

or   $\dfrac{2}{7} =\overline{0.285714}.......... $

the number has non-terminating decimal expansion but recurring after every 6 digits after decimal

So option $B $ is correct

$1.23 \bar{48}$ is:

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

  1. An integer

  2. A rational number

  3. An irrational number

  4. A natural number

Show answer
Correct Option: B
Explanation:

$1.23\bar{48}$ can be written as $1.23484848484848....$ and so on 

Hence this is a recurring decimal as decimal part is repeating after one-thousand's place.
Thus, it is a rational number.

Which of the following numbers has the terminating decimal representation?

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

  1. $\displaystyle \frac{1}{7}$

  2. $\displaystyle \frac{1}{3}$

  3. $\displaystyle \frac{3}{5}$

  4. $\displaystyle \frac{17}{3}$

Show answer
Correct Option: C
Explanation:

$\dfrac{3}{5} = 0.6$ which is terminating decimal. 


$\dfrac{1}{7} = 0.142857.....$

$\dfrac{1}{3} = 0.333......$

$\dfrac{17}{3} = 5.6666......$

Show the correct sequence of the given four option in ascending order
(1) Prime minister  (2) Chief Minister  (3) Mayor  (4) President  (5) Sarpanch

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

  1. $5,3,2,4,1$

  2. $5,2,3,4,1$

  3. $5,3,2,1,4$

  4. $5,4,2,1,3$

Show answer
Correct Option: C
Explanation:
In ascending order 
Sarpanch < Mayore < Chief Minister < Prime minister < President.
$\therefore$ Option $ C$  is the correct answer.