Quantitative Aptitude

Number System and Simplification

585 Questions

Number system and simplification questions assess mathematical proficiency in handling fractions, decimals, and complex algebraic expressions. Test takers must simplify surds, calculate reciprocals, and solve intricate number pattern equations. This foundational quantitative aptitude topic is critical for achieving high scores in SSC and banking exams.

Fraction simplificationDecimal operationsSurds and indicesPercentage calculationsComplex number algebra

Number System and Simplification Questions

Multiple choice maths number systems existence of irrational numbers irrational numbers properties of irrational numbers

$0.\overline{35}$ is equal to

  1. $\displaystyle\frac{35}{66}$
  2. $\displaystyle\frac{35}{77}$
  3. $\displaystyle\frac{35}{99}$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$X=0.35353535$   -- i
Multiplying equation i with 100,

$100x=35.353535353$   --ii 
Subtracting equation i from ii 

$ 100x-x = 35.3535 - 0.3535 $
$99x=35$
$ x = \dfrac{35}{99}$
Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

$3.\overline{25}$ is equal to

  1. $\displaystyle\frac{320}{99}$
  2. $\displaystyle\frac{321}{99}$
  3. $\displaystyle\frac{322}{99}$
  4. $\displaystyle\frac{323}{99}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given that,$3.\overline{25}$.


Let,

$x=3.\overline{25}$

 $x=3.252525.....$


Multiply by 100 both sides,

  $ 100x=100\times 3.252525..... $

 $ 100x=325.2525..... $

 $ 100x=322+3.2525..... $

 $ 100x=322+x $

 $ 99x=322 $

 $ x=\dfrac{322}{99} $


Hence, this is the answer.

Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

$0.\overline{05}$ is equal to

  1. $\displaystyle\frac{3}{99}$
  2. $\displaystyle\frac{4}{99}$
  3. $\displaystyle\frac{5}{99}$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given that,$0.\overline{05}$

Let,

  $ x=0.\overline{05} $

 $ x=0.05050505..... $

Multiply by $100$ both sides,

 $ 100x=100\times 0.05050505..... $

 $ 100x=5.050505..... $

 $ 100x=5+0.050505..... $

 $ 100x=5+x $

 $ 99x=5 $

 $ x=\dfrac{5}{99} $


Hence, this is the answer.

Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

Which statement is true?

  1. $ \displaystyle \frac{-8}{12} $= $ \displaystyle \frac{10}{-15} $
  2. $ \displaystyle \sqrt{3} $ is not a real number

  3. Additive identity of 5 is -5
  4. $ \displaystyle \frac{2}{5} $>$ \displaystyle \frac{4}{5} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Option A is correct because both fractions simplify to -2/3. Option B is false as sqrt(3) is a real number. Option C is false because the additive identity is 0, while -5 is the additive inverse of 5. Option D is false because 2/5 is less than 4/5.

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $\sqrt { 1+\dfrac { x }{ 289 }  } =1\dfrac { 1 }{ 17 }$ then $x=$

  1. $1$
  2. $13$
  3. $35$
  4. $15$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We have,

$\sqrt{1+\dfrac{x}{289}}=1\dfrac{1}{17}$

$\sqrt{\dfrac{289+x}{289}}=\dfrac{18}{17}$

 

On squaring both sides, we get

$ {{\left( \sqrt{\dfrac{289+x}{289}} \right)}^{2}}={{\left( \dfrac{18}{17} \right)}^{2}} $

$ \dfrac{289+x}{289}=\dfrac{324}{289} $

$ 289+x=324 $

$ x=324-289 $

$ x=35 $

 

Hence, this is the answer.

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $\displaystyle \frac{a-b}{b}=\frac{3}{7}$, which of the following must also be true?

  1. $\displaystyle \frac{a}{b}=-\frac{4}{7}$
  2. $\displaystyle \frac{a}{b}=\frac{10}{7}$
  3. $\displaystyle \frac{a+b}{b}=\frac{10}{7}$
  4. $\displaystyle \frac{a-2b}{b}=-\frac{11}{7}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given: $\displaystyle \frac {a-b}{b}=\frac 37$

Separating denominators,

$\Rightarrow \displaystyle \frac ab -1=\frac 37$
$\Rightarrow \displaystyle \frac ab=\frac 37+1=\frac {10}{7}$
$\Rightarrow \dfrac {a}{b}=\dfrac {10}{7}$
Therefore, option B is correct.

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $\displaystyle \frac{a}{3y}+\frac{3b}{x}=7$ and $\displaystyle a+1=2b+1=x=5,$ find the value of $'y'.$


  1. $\displaystyle \frac{10}{77}$
  2. $\displaystyle \frac{22}{69}$
  3. $\displaystyle \frac{20}{87}$
  4. $\displaystyle \frac{14}{93}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since, $ a+1=2b+1=x=5 $
$ => a = 5 - 1 = 4 $
$ 2b = 5 - 1 = 4 => b = 2 $

$ x = 5 $

Substituting these values in $ \frac { a }{ 3y } +\frac { 3b }{ x } =7 $, we get $ \frac { 4 }{ 3y } +\frac { 6 }{ 5 } =7 $
$ => \frac { 4 }{ 3y } = 7 - \frac { 6 }{ 5 } $
$ => \frac { 4 }{ 3y } =  \frac { 29 }{ 5 } $
$ => \frac { 3y }{ 4 } =  \frac { 5 }{ 29 } $
$ => y = \frac {20}{87} $

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

The solution of the equation $\displaystyle \frac{2x+4}{3x-1}=\frac{4}{3}$ is 

  1. 6

  2. 4
  3. $\displaystyle \frac{8}{3}$
  4. $\displaystyle \frac{3}{4}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given equation is $\displaystyle \frac{2x+4}{3x-1}=\frac{4}{3}$
Cross multiplying, we get
$3(2x+4)=4(3x-1)$
$\Rightarrow 6x+12=12x-4\Rightarrow 6x=16$
$\Rightarrow \displaystyle x=\frac{16}{6}=\frac{8}{3}$.
Hence, the solution is $x=\cfrac{8}{3}$.

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $\displaystyle \sqrt{\left ( x-1 \right )\left ( y+2 \right )}=7$, $x$ and $y$ being positive whole numbers, then the values of $x$ and $y$ are, respectively

  1. $8,5$
  2. $15,12$
  3. $22,19$
  4. $6,8$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$ \sqrt{(x-1)(y+2)}=7\Rightarrow (x-1)(y+2)=7^{2}$
$ \Rightarrow (x-1)=7:and:(y+2)=7$
$ x=8$ and $\displaystyle y=5$

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

$\displaystyle \sqrt{6+\sqrt{6+\sqrt{6+...}}}$ equals 

  1. $\displaystyle 6^{\frac{2}{3}}$

  2. 6
  3. $\displaystyle 6^{\frac{1}{3}}$

  4. 3
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let  $ x =\sqrt{6+\sqrt{6+\sqrt{6+...}}}$


sqare it on both sides


$x^2=\sqrt{6+\sqrt{6+\sqrt{6+...}}} = 6+x$


$\Rightarrow x^2=6+x $


$\Rightarrow x^2-x-6=0 $


$\Rightarrow x^2-3x+2x-6=0 $


$\Rightarrow x(x-3)+2(x-3)=0 $


$\Rightarrow (x-3)(x+2)=0 $


$either  x-3=0---> x=3 $


$ or  x+2=0 ----> x=-2 $


since result of sqrt of anything will be positive only, therefore,


Answer $x=3$





Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $\displaystyle \frac{x^2\, -\, (x\, +\, 1)(x\, +\, 2)}{5x\, +\, 1}\, =\, 6$, then $x$ is equal to

  1. $\displaystyle \frac{8}{33}$
  2. $\displaystyle \frac{8}{3}$
  3. $\displaystyle \frac{-8}{33}$
  4. $\displaystyle \frac{-6}{33}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given, $\displaystyle \frac{x^2\, -\, (x\, +\, 1)(x\, +\, 2)}{5x\, +\, 1}\, =\, 6$
$\Rightarrow x^2\, -\, (x^2\, +\, 3x\, +\, 2)\, =\, 6(5x\, +\, 1)$, .....(on cross multiplying )
$\Rightarrow x^2\, -\, x^2\, -\, 3x\, -\, 2\, =\, 30x\, +\, 6$
$\Rightarrow -3x - 2 = 30x + 6$
$\Rightarrow -3x - 30x = 6 + 2 \Rightarrow -33x = 8$
$\Rightarrow x\, =\, \displaystyle \frac{-8}{33}$
Hence, the solution is, $x=-\cfrac{8}{33}$.