Quantitative Aptitude

Number System and Simplification

548 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 unchanging relations algebra aid introduction to unknowns measures and relations

If $\displaystyle x+\frac{a}{x}=b$ then the value of $\displaystyle \frac{x^{2}+bx+a}{bx^{2}-x^{3}}$ is

  1. $\displaystyle \frac{6b}{a}$
  2. $\displaystyle \frac{5b}{a}$
  3. $\displaystyle \frac{2b}{a}$
  4. $\displaystyle \frac{4b}{a}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given $x+\frac{a}{x}=b$ Multiply by x both sides

$x^{2}+a=bx$
$\Rightarrow x^{2}-bx+a=0$
So $\frac{x^{2}+bx+a}{bx^{2}-x^{3}}=\frac{x^{2}-bx+a+2bx}{-x(x^{2}-bx)}=\frac{0+2bx}{-x(-a)}=\frac{2bx}{ax}=\frac{2b}{a}$

Multiple choice maths unchanging relations algebra aid introduction to unknowns measures and relations

If $\displaystyle a-\frac{1}{3}=\frac{1}{a}$ then the value of $\displaystyle a^{3}-\frac{1}{a^{3}}$ is

  1. $\displaystyle 1\frac{1}{27}$
  2. $\displaystyle 1\frac{2}{27}$
  3. $\displaystyle 1\frac{3}{27}$
  4. $\displaystyle 1\frac{4}{27}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given $a-\frac{1}{3}=\frac{1}{a}$

=$a-\frac{1}{a}=\frac{1}{3}$
We know that
$x^{3}-y^{3}=(x-y)^{3}+3xy(x+y)$
Then $a^{3}-\left ( \frac{1}{a} \right )^{3}=(a-\left ( \frac{1}{a} \right ))^{3}+3a\left ( \frac{1}{a} \right )(a+\left ( \frac{1}{a} \right ))$ 
=$\left ( \frac{1}{3} \right )^{3}+3\times\left (  \frac{1}{3} \right )$
=$\frac{1}{27}+1=\frac{28}{27}=1\frac{1}{27}$

Multiple choice maths unchanging relations algebra aid introduction to unknowns measures and relations

If $\dfrac{2+3}{x}=\dfrac{2+x}{3}$
What one value for $x$ can be correctly entered into the answer grid?

  1. -5

  2. 3

  3. -3

  4. 2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

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

$\Rightarrow 2x+x^2=15$
$\Rightarrow x^2+2x-15=0$
$\Rightarrow x^2+5x-3x-15=0$
$\Rightarrow x(x+5)-3(x+5)=0$
$\Rightarrow (x+5)(x-3)=0$
$\Rightarrow x=-5,3$
Value of $x$ is not negative, so $x=3$.

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

The reciprocal of $\dfrac {-5}{13}$ is _____

  1. $\dfrac {5}{13}$
  2. $\dfrac {-13}{5}$
  3. $\dfrac {13}{5}$
  4. $\dfrac {-5}{13}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
The reciprocal (also known as the multiplicative inverse) is the number we have to multiply to get an answer equal to the multiplicative number with recipocal of it is 1.
Then $\frac{-5}{13}\times \frac{-13}{5}=1$.
So recipocal of $\frac{-5}{13}$ is $\frac{-13}{5}$.
So answer is (B) $\frac{-13}{5}$.

 
Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

Find the compounded ratio of $\dfrac{3}{5}, \dfrac{7}{8}$ and $\dfrac{5}{10}$

  1. $\dfrac{21}{80}$
  2. $\dfrac{21}{10}$
  3. $\dfrac{80}{27}$
  4. $\dfrac{5}{11}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

let be $ a=\dfrac{3}{5}, b=\dfrac{7}{8}, c=\dfrac{5}{10}$
therefore the $ratio=a\times b\times c=\dfrac{3\times 7\times 5}{5\times 8\times 10}=\dfrac{21}{80}$

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

_____ is the duplicate ratio of $\dfrac {x}{2} : \dfrac {y}{3}$

  1. $\dfrac {x}{4} : \dfrac {y}{9}$
  2. $\dfrac {x^{2}}{4} : \dfrac {y^{2}}{9}$
  3. $\dfrac {2}{x} : \dfrac {3}{y}$
  4. $\sqrt {\dfrac {x}{2}} : \sqrt {\dfrac {y}{3}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The duplicate ratio of $a : b$ is $b : a$
$\therefore$ The duplicate ratio of $\dfrac {x}{2} : \dfrac {y}{3}$ is $\left (\dfrac {x}{2}\right )^{2} : \left (\dfrac {y}{3}\right )^{2} = \dfrac {x^{2}}{4} : \dfrac {y^{2}}{9}$

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

______ is the triplicate ratio of $\dfrac {x}{2} : \sqrt [3]{x}$

  1. $\dfrac {x^{3}}{8} : x^{3}$
  2. $8 : x$
  3. $x^{3} : 8$
  4. $x^{2} : 8$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $\dfrac {x}{2} : \sqrt [3]{x}$ is $\left (\dfrac {x}{2}\right )^{3} : \left (\sqrt [3]{x}\right )^{3} = \dfrac {x^{3}}{8} : x = \dfrac {x^{2}}{8}$
$\therefore x^{2} : 8$.

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

The value of $\sqrt{\dfrac{64x^{2}}{49y^{2}}}$ is

  1. $\dfrac {8x}{7y}$
  2. $\dfrac {8}{7y}$
  3. $\dfrac {8x^{4}}{7y^{2}}$
  4. $none$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\sqrt{\dfrac{64x^{2}}{49y^{2}}}$


$\Rightarrow$  $\sqrt{\dfrac{(8x)^2}{(7y)^2}}$
Taking square root on both sides,

$\Rightarrow$  $\dfrac{8x}{7y}$

$\therefore$  $\sqrt{\dfrac{64x^{2}}{49y^{2}}}=\dfrac{8x}{7y}$

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

Evaluate: $\displaystyle\sqrt{5 \left(2\frac{3}{4}\, -\, \frac{3}{10}\right)}$ is $\displaystyle 3\frac{1}{m}$

$m$ is 

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

  $\sqrt{5 \left(2\dfrac{3}{4}\, -\, \dfrac{3}{10}\right)}$
$=\sqrt { 5\left( \dfrac { 11 }{ 4 } \, -\, \dfrac { 3 }{ 10 }  \right)  } $
$=\sqrt { 5\left( \dfrac { 55-6 }{ 20 } \,  \right)  } $
$ =\sqrt { \left( \dfrac { 49 }{ 4 } \,  \right)  } $
$=\dfrac{7}{2}$
$=3\dfrac{1}{2}$

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

Evaluate: $\sqrt{\displaystyle \frac{1}{16}\, +\, \displaystyle \frac{1}{9}}$

  1. $\displaystyle \frac{7}{12}$
  2. $\displaystyle \frac{25}{144}$
  3. $\displaystyle \frac{5}{12}$

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

$\sqrt{\displaystyle \frac{1}{16}\, +\, \displaystyle \frac{1}{9}}\,=\sqrt{\displaystyle \frac{(1\times9)+(1\times16)}{9\times16}} =\, \sqrt{\displaystyle \frac{25}{144}}\, =\, \displaystyle \frac{5}{12}$

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

$\sqrt {\displaystyle \frac{0.289}{0.00121}}\, =\, ?$

  1. $\displaystyle \frac{1.7}{11}$
  2. $\displaystyle \frac{17}{11}$
  3. $\displaystyle \frac{170}{11}$
  4. $\displaystyle \frac{17}{110}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle {\sqrt {\frac{0.289}{0.00121}}\, =\, \sqrt {\frac{0.28900}{0.00121}}\, =\, \sqrt {\frac{28900}{121}}}$ $=\cfrac{170}{11}$