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

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

  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 $0.33333333\ by\ \dfrac{1}{3.10^{9}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
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 } } $.

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

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.

  1. True

  2. False

  3. Neither

  4. Either
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

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

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 } $

  1. $3$
  2. $4$
  3. $5$
  4. $6$
Reveal answer Fill a bubble to check yourself
C Correct answer
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$]

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

State whether the following statement is true/false.

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

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice maths real number first forms rational numbers as recurring/terminating decimals 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?

  1. $x>y>z$
  2. $y$
  3. $z>x$
  4. $x>z>y$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Expanding the decimals: x = 0.123444..., y = 0.123434..., and z = 0.1234234... Comparing the digits from left to right shows that x is the largest, followed by y, and then z.

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

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) $\displaystyle \dfrac{123}{250}$

  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)
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

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

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 $\cfrac {p}{q}$?

  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
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

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

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 ?

  1. $1$ only
  2. $2$ only
  3. $1$ and $3$
  4. $2$ and $3$
Reveal answer Fill a bubble to check yourself
C Correct answer
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.