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

If $\sqrt[4]{\dfrac{x+1}{2}} = \dfrac{1}{2}$, then find $x $.

  1. $-0.969$
  2. $-0.875$
  3. $0$
  4. $0.875$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given is $\sqrt [ 4 ]{ \dfrac { x+1 }{ 2 }  } = \dfrac { 1 }{ 2 } $
Now Raising the power to $4$ on both sides, we get
$\dfrac { x+1 }{ 2 } = { \left (\dfrac {1}{2}\right) }^{ 4 }\\ \Rightarrow \dfrac { x+1 }{ 2 } =\dfrac { 1 }{ 16 } \\ \Rightarrow 16x+16=2\\ \Rightarrow x=-0.875$
Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $\dfrac{5}{x+3} = \dfrac{1}{x}+\dfrac{1}{2x}$, calculate the value of $x$.

  1. $\dfrac{3}{14}$
  2. $\dfrac{1}{3}$
  3. $\dfrac{6}{13}$
  4. $\dfrac{3}{4}$
  5. $\dfrac{9}{7}$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Given, $\dfrac { 5 }{ x+3 } =\dfrac { 1 }{ x } +\dfrac { 1 }{ 2x } $

Taking RHS:
$\dfrac { 1 }{ x } +\dfrac { 1 }{ 2x } $
LCM of these is $2x$
$\Rightarrow \dfrac { 2 }{ 2x } +\dfrac { 1 }{ 2x } \ \Rightarrow \dfrac { 3 }{ 2x } $
Now taking LHS:
$\dfrac { 5 }{ x+3 } $
It is given, LHS $=$ RHS
$\dfrac { 5 }{ x+3 } =\dfrac { 3 }{ 2x } $
$\Rightarrow 5\times 2x=3\times (x+3)\ \Rightarrow 10x=3x+9\ \Rightarrow x=\dfrac {9}{7}$

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

Compute the approximate value of $x$: $\sqrt [3]{\dfrac {2x + 3}{5}} = \dfrac {2}{3}$

  1. $-0.76$
  2. $-0.69$
  3. $-0.67$
  4. $0.69$
  5. $0.76$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given is $\sqrt [ 3 ]{ \dfrac { 2x+3 }{ 5 }  } = \dfrac { 2 }{ 3 } $
Now raising the power to $3$ on both sides, we get
$\dfrac { 2x+3 }{ 5 } = { (2/3) }^{ 3 }\\ \Rightarrow \dfrac { 2x+3 }{ 5 } =\dfrac { 8 }{ 27 } \\ \Rightarrow 54x+81=40\\ \Rightarrow x=-0.76$
Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $x=\displaystyle\frac{1}{\displaystyle 2-\frac{1}{\displaystyle 2-\frac{1}{2-x}}}, (x\neq 2)$, then the value of x is ________?

  1. $1$
  2. $3$
  3. $2$
  4. $5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x=\dfrac{1}{2-\dfrac{1}{2-\dfrac{1}{2-x}}}$


$\Rightarrow x=\dfrac{1}{2-\dfrac{2-x}{3-2x}}$


$\Rightarrow x=\dfrac{3-2x}{4-3x}$

On solving, we get $x=1$
So, Option (A)

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

Find the value of $x$.
$\displaystyle \left ( -\frac {1}{4} \right )^{-3} \times \left ( \frac {1}{4} \right )^{4} \div 4^{-2}=-(4^{10x+1})$

  1. $\displaystyle -\frac {2}{5}$
  2. $4$
  3. $\displaystyle \frac {-3}{5}$
  4. $0$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$
-{ \frac { 1 }{ 4 }  }^{ -3 }{ \times \frac { 1 }{ 4 }  }^{ 4 }\div { 4 }^{ -2 }{ \quad =\quad -4 }^{ (10x+1) }\ \ -{ 4 }^{ 3 }{ \times 4 }^{ -4 }\times { 4 }^{ 2 }{ \quad =\quad -4 }^{ (10x+1) }\ { -4 }^{ 3-4+2 }{ \quad =\quad -4 }^{ (10x+1) }\ { -4 }^{ 1 }{ \quad =\quad -4 }^{ (10x+1) }\ 1\quad =\quad 10x\quad +1\ 10x\quad =\quad 0\ x\quad =\quad 0
$

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

If $\displaystyle \left ( 1^{2}+2^{2}+3^{2}+...+10^{2} \right )=385$ then the value of $\displaystyle \left ( 2^{2}+4^{2}+6^{2}+...+20^{2} \right )$

  1. 770

  2. 1155

  3. 1540

  4. (385 x 385)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\left(2^2 + 4^2 + 6^2 + ... + 20^2\right) = \left(1 \times 2\right)^2 + \left(2\times 2\right)^2 + \left(2 \times 3\right)^2 + ... + \left(2 \times 10\right)^2$



$= \left(2^2 \times 1^2\right) + \left(2^2 \times 2^2\right) + \left(2^2 \times 3^2\right) + ... + \left(2^2 \times 10^2\right)$

$= 2^2 \times \left [1^2 + 2^2 + 3^2 + ... + 10^2\right]$

$\left(1^2+2^2+3^2+....+n^2\right) = \frac{1}{6}n\left(n+1\right)\left(2n+1\right)$

$=\left(4\times\frac{1}{6}\times 10\times 11\times 21\right)$

$=\left(4\times 5\times 77\right)$

$= 1540$