Tag: maths

Questions Related to maths

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

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. $x>z>y$

Show answer
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.