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

Given that $\dfrac {1}{7} = 0.\overline {142857}$, which is a repeating decimal having six different digits. If $x$ is the sum of such first three positive integers $n$ such that $\dfrac {1}{n} = 0.\overline {abcdef}$, where $a, b, c, d, e$ and $f$ are different digits, then the value of $x$ is

  1. $20$
  2. $21$
  3. $41$
  4. $42$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$1^{st}$ number
$x _{1} = 7$
$\Rightarrow \dfrac {1}{x _{1}} = 0.\overline {142857}$
such that,
$2^{nd}$ number
$x _{2} = 13$
$\Rightarrow \dfrac {1}{x _{2}} = 0.\overline {076923}$
$x _{3} = 21$
$\Rightarrow \dfrac {1}{x _{3}} = \dfrac {1}{21} = 0.\overline {047619}$
$x = x _{1} + x _{2} + x _{3}$
$\Rightarrow 7 + 13 + 21 = 41$.

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

If $x - \dfrac {1}{x} = \sqrt {6}$, then $x^{2} + \dfrac {1}{x^{2}}$ is ________.

  1. $2$
  2. $4$
  3. $6$
  4. $8$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, $\dfrac {x - 1}{ x} = 6$

Multiplying and divide the above equation with $x - \dfrac {1}{x}$
Thus $ \dfrac{x-\dfrac{1}{x}\times x-\dfrac{1}{x}}{x-\dfrac{1}{x}} = \sqrt{6} $
Using $(a-b)^{2} = a^{2} + b^{2} - 2ab $
and substituting $x-\dfrac{1}{x} = \sqrt{6}$  in denominator, we get
$\dfrac{x^{2} + \dfrac{1}{x^{2}} - 2x\dfrac{1}{x}}{\sqrt{6}} = \sqrt{6}$
$\Rightarrow x^{2} + \dfrac{1}{x^{2}} - 2  =\sqrt{6}\times \sqrt{6}$
$\Rightarrow x^{2} + \dfrac{1}{x^{2}} = 6+2 = 8$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

If $x^{2}+\dfrac{1}{^{x^2}}=18$, then the value of $\left(x+\dfrac{1}{x}\right)$ is ?

  1. $1$
  2. $3$
  3. $\sqrt {20}$
  4. $6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$x^2+\dfrac{1}{x^2}=18$


$\Rightarrow$  $x^2+\dfrac{1}{x^2}=20-2$                           [ Since, $20-2=18$ ]

$\Rightarrow$  $x^2+\dfrac{1}{x^2}+2=20$

$\Rightarrow$  $x^2+\dfrac{1}{x^2}+2\times x\times\dfrac{1}{x}=20$

$\Rightarrow$  $\left(x+\dfrac{1}{x}\right)^2=20$                      [ Since, $a^2+b^2+2ab=(a+b)^2$ ]

$\Rightarrow$  $\left(x+\dfrac{1}{x}\right)=\sqrt{20}$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

If $5a\sqrt{b}-\dfrac{3}{2b\sqrt{a}}=12$ and $a=8b$, then the value of $25a^{2}b+\dfrac{9}{4ab^{2}}$ is ?

  1. $144-15\sqrt{8}$
  2. $144+15\sqrt{8}$
  3. $15\sqrt{8}-144$
  4. $-15\sqrt{8}-144$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given,


$5a\sqrt{b}-\dfrac{3}{2b\sqrt{a}}=12$


Formula,


$(a-b)^2=a^2+b^2-2ab$


squaring the given on both sides,


$25a^2b+\dfrac{9}{4b^2a}-2\left ( 5a\sqrt{b} \right )\left ( \dfrac{3}{2b\sqrt{a}} \right )=144$


$25a^2b+\dfrac{9}{4b^2a}=144+2\left ( 5a\sqrt{b} \right )\left ( \dfrac{3}{2b\sqrt{a}} \right )$


$=144+15\sqrt{\dfrac{a}{b}}$


$=144+15\sqrt{\dfrac{8b}{b}}$


$=144+15\sqrt{8}$


Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

Evaluate $\displaystyle \left ( \frac{2x}{7} - \frac{7y}{4} \right )^{2}$

  1. $\displaystyle \frac{x^{2}}{49} + \frac{17y^{2}}{16} - xy$
  2. $\displaystyle \frac{4x^{2}}{49} + \frac{49y^{2}}{16} - xy$
  3. $\displaystyle \frac{4x^{2}}{9} + \frac{49y^{2}}{4} - xy$
  4. $\displaystyle \frac{x^{2}}{13} + \frac{49y^{2}}{13} - xy$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\left ( \dfrac{2x}{7} - \dfrac{7y}{4} \right ) ^{2}$


Using,

$(a-b)^2=a^2-2ab+b^2$

$=\left( \dfrac{2x}{7}\right)^2-2\left( \dfrac{2x}{7}\right)\left(\dfrac{7y}{4} \right)+\left(\dfrac{7y}{4} \right)^2$

$=\dfrac{4x^2}{49}-xy+\dfrac{49y^2}{16}$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

Evaluate $\displaystyle \left ( \frac{7}{8}x + \frac{4}{5}y \right ) ^{2}$

  1. $\displaystyle \frac{49}{64}x^{2} + \frac{16}{25}y^{3} + \frac{7}{5}xy$
  2. $\displaystyle \frac{78}{32}x^{2} + \frac{16}{25}y^{2} + \frac{1}{5}xy$
  3. $\displaystyle \frac{49}{64}x^{2} + \frac{16}{25}y^{2} + \frac{7}{5}xy$
  4. $\displaystyle \frac{78}{32}x^{2} + \frac{16}{25}y^{2} + \frac{1}{6}xy$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\left ( \frac{7}{8}x + \frac{4}{5}y \right ) ^{2}$
Using,
$(a+b)^2=a^2+2ab+b^2$
$=( \frac{7}{8}x)^2+2( \frac{7}{8}x)(\frac{4}{5}y )+(\frac{4}{5}y )^2$
$=\frac{49}{64}x^2+\frac{7}{5}xy+\frac{16}{25}y^2$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

Find square of the following expression

$\displaystyle 3a - 4b$

  1. $\displaystyle a^{2} - 24ab + 48b^{2}$
  2. $\displaystyle 9a^{2} - 4ab + 48b^{2}$
  3. $\displaystyle 9a^{2} - 24ab + 16b^{2}$
  4. $\displaystyle a^{2} - 24ab + 16b^{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Squaring,
$(3a - 4b)^2$
Using, $(a-b)^2=a^2-2ab+b^2$
$=(3a)^2-2(3a)(4b)+(4b)^2$
$=9a^2-24ab+16b^2$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

Find square of the following expression

$\displaystyle \frac{3a}{2b} - \dfrac{2b}{3a}$

  1. $\displaystyle \dfrac{9a^{2}}{4b^{2}} - 2 + \dfrac{b^{2}}{a^{2}}$
  2. $\displaystyle \dfrac{9a^{2}}{4b^{2}} + 2 + \dfrac{4b^{2}}{9a^{2}}$
  3. $\displaystyle \dfrac{a^{2}}{4b^{2}} - 2 + \dfrac{4b^{2}}{9a^{2}}$
  4. $\displaystyle \dfrac{9a^{2}}{4b^{2}} - 2 + \dfrac{4b^{2}}{9a^{2}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Squaring,
$(\dfrac{3a}{2b} - \dfrac{2b}{3a})^2$
Using, $(a-b)^2=a^2-2ab+b^2$
$=(\dfrac{3a}{2b})^2-2(\dfrac{3a}{2b})(\dfrac{2b}{3a})+(\dfrac{2b}{3a})^2$
$=\dfrac{9a^2}{4b^2}-2+\dfrac{4b^2}{9a^2}$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

Find the square of  $\displaystyle a + 2b + c$

  1. $\displaystyle a^{2} + b^{2} + c^{2} + ab + bc + ac$
  2. $\displaystyle a^{2} + 4b^{2} + c^{2} + 4ab + 4bc + 2ac$
  3. $\displaystyle a^{3} + 4b^{3} + c^{3} + 8ab + 8bc + 8ac$
  4. $\displaystyle a^{3} + b^{3} + c^{3} + 4ab + 4bc + 2ac$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Squaring,
$(a + 2b + c)^2$
Using $(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ac$
$=a^2+(2b)^2+c^2+2(a)(2b)+2(2b)(c)+2(a)(c)$
$=a^2+4b^2+c^2+4ab+4bc+2ac$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

Find the square of $\displaystyle 2a - b - 3c$

  1. $\displaystyle 4a^{2} + b^{2} + 9c^{2} - 4ab + 6bc - 12ca$
  2. $\displaystyle a^{3} + b^{3} + c^{2} - 4ab + 6bc - ca$
  3. $\displaystyle a^{2} + b^{2} - c^{2} - 4ab + 6bc - 2ca$
  4. $\displaystyle 4a^{3} + b^{3} + 9c^{2} - 4ab - 6bc - 12ca$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

   $2a-b-3c$
Squaring, we get
   $(2a-b-3c)^2$
Using $(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ac$
$=(2a)^2+(-b)^2+(-3c)^2+2(2a)(-b)+2(-b)(-3c)+2(2a)(-3c)$
$=4a^{2} + b^{2} + 9c^{2} - 4ab + 6bc - 12ca$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

Evaluate :

$\displaystyle \left ( \frac{2x}{5} + \frac{3y}{4} - \frac{4z}{7} \right )^{2}$

  1. $\displaystyle \frac{x^{2}}{25} + \frac{9y^{2}}{16} + \frac{z^{2}}{49} + \frac{3}{5} xy - \frac{6}{7}yz - \frac{16}{35}zx$
  2. $\displaystyle \frac{4x^{2}}{25} + \frac{9y^{2}}{16} + \frac{16z^{2}}{49} + \frac{1}{5} xy - \frac{6}{3}yz - \frac{36}{35}zx$
  3. $\displaystyle \frac{x^{2}}{25} + \frac{9y^{2}}{16} + \frac{5z^{2}}{49} + \frac{3}{5} xy - \frac{6}{7}yz - \frac{16}{35}zx$
  4. $\displaystyle \frac{4x^{2}}{25} + \frac{9y^{2}}{16} + \frac{16z^{2}}{49} + \frac{3}{5} xy - \frac{6}{7}yz - \frac{16}{35}zx$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

    $ \left ( \dfrac{2x}{5} + \dfrac{3y}{4} - \dfrac{4z}{7} \right )^{2}$
Squaring, we get
Using $(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ac$
$=( \dfrac{2x}{5})^2+(\dfrac{3y}{4})^2+(- \dfrac{4z}{7})^2+2( \dfrac{2x}{5})(\dfrac{3y}{4})+2(\dfrac{3y}{4})(- \dfrac{4z}{7})+2( \dfrac{2x}{5})(- \dfrac{4z}{7})$
$=\dfrac{4x^{2}}{25} + \dfrac{9y^{2}}{16} + \dfrac{16z^{2}}{49} + \dfrac{3}{5} xy - \dfrac{6}{7}yz - \dfrac{16}{35}zx$