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

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

If the fractions $\cfrac{3}{5}$,$\cfrac{2}{11}$, $\cfrac{4}{7}$, $\cfrac{1}{3}$, $\cfrac{5}{6},$ and $\cfrac{3}{8}$are arranged in the ascending order which fraction will be at the 3rd place ?

  1. $\cfrac{1}{3}$
  2. $\cfrac{3}{5}$
  3. $\cfrac{2}{11}$
  4. $\cfrac{3}{8}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

LCM of the denominators of given rational numbers i.e. 5, 11, 7, 3, 6, and 8 = 9240

Now by equating the denominators we get, 
$\dfrac { 3 }{ 5 } =\dfrac { 5652 }{ 9240 }$

$\dfrac { 2 }{ 11 } =\dfrac { 1680 }{ 9240 }$

$\dfrac { 4 }{ 7 } =\dfrac { 5280 }{ 9240 }$

$\dfrac { 1 }{ 3 } =\dfrac { 3080 }{ 9240 }$

$\dfrac { 5 }{ 6 } =\dfrac { 7700 }{ 9240 }$

$\dfrac { 3 }{ 8 } =\dfrac { 3465 }{ 9240 }$
 On seeing the rational numbers it is clear that the rational number at third position in ascending order is $\dfrac { 3465 }{ 9240 } \  i.e.\dfrac { 3 }{ 8 }$
So, correct answer is option D. 

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

The fraction equivalent to $\displaystyle \frac{1}{2}$ is

  1. $\displaystyle \frac{2}{4}$
  2. $\displaystyle \frac{3}{6}$
  3. $\displaystyle \frac{8}{16}$

  4. all the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle \frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$
$\displaystyle \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$
$\displaystyle \frac{1}{2} = \frac{1 \times 8}{2 \times 8} = \frac{8}{16}$
So, $\displaystyle \frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{8}{16}$

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

Which of the following fractions is less than $\displaystyle \frac{7}{8}$ and greater than $\displaystyle \frac{1}{3}$?

  1. $\displaystyle \frac{1}{4}$
  2. $\displaystyle \frac{23}{24}$
  3. $\displaystyle \frac{11}{12}$
  4. $\displaystyle \frac{17}{24}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle \frac{1}{3} = 0.333000,$ $\displaystyle \frac{7}{8} = 0.875$
$\displaystyle \frac{1}{4} = 0.25,$ $\displaystyle \frac{23}{24} = 0.9583000,$ $\displaystyle \frac{11}{12} = 0.9166000$
$\displaystyle \frac{17}{24} = 0.7083000$
Since $0.7083000 \displaystyle \left ( =\frac{17}{24} \right )$ is greater than 
$0.333000 \displaystyle \left ( =\frac{1}{3} \right )$ and less than $0.875 \displaystyle \left ( =\frac{17}{24} \right )$$\displaystyle \left ( =\frac{7}{8} \right )$
Therefore $\displaystyle \frac{17}{24}$ lies between $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{7}{8}$

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

Which of the following fractions is the largest ?

  1. $\displaystyle \frac{13}{16}$
  2. $\displaystyle \frac{7}{8}$
  3. $\displaystyle \frac{31}{40}$
  4. $\displaystyle \frac{63}{80}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\displaystyle \frac{13}{16}=\frac{13\times 5}{16\times 5}=\frac{65}{80}, \frac{7\times 10}{8\times 10}=\frac{70}{80},\frac{31}{40}=\frac{31\times 2}{40\times 2}$
$\displaystyle =\frac{62}{80}$ and last fraction is $\displaystyle =\frac{63}{80}$
Out of these the largest fraction is $\displaystyle \frac{70}{80}$ $\displaystyle =\frac{7}{8}$

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

$\displaystyle \frac{4}{15}$of $\displaystyle \frac{5}{7}$ of a number is greater than $\displaystyle \frac{4}{9}$ of $\displaystyle \frac{2}{5}$ of the same number by $8$. What is half of that number?

  1. $630$
  2. $315$
  3. $210$
  4. $105$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the number be $x$

So from the question, we have
$\dfrac{4}{15}.\dfrac{5}{7}.x-\dfrac{4}{9}.\dfrac{2}{5}.x=8$
$\Rightarrow \dfrac {4x}{21}-\dfrac {8x}{45}=8$
$\Rightarrow x=\dfrac {24\times 7\times 15}{4}$
$\Rightarrow x=6\times 7\times 15$
$\Rightarrow x=630$
Half of that number is equal to $315$.

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

Compare $\displaystyle \frac {9}{16}$ .......... $\displaystyle \frac {13}{5}$

  1. $=$
  2. $>$
  3. $<$

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

Given fractions are

$\displaystyle \frac{9}{16} = 0.5625$

$\displaystyle \frac{13}{5}=2.6$

Hence$ \displaystyle \frac{9}{16}<\frac{13}{5}$

OR
$9\times 5<13\times16$
 $ \displaystyle \frac{9}{16}<\frac{13}{5}$

Multiple choice maths equivalent fractions comparing and ordering fractions comparing fractions fractions and its related operations

By how much is $\displaystyle \frac {19}{20}$ greater than $\displaystyle \frac {2}{20}$ ?

  1. $\displaystyle \frac {21}{10}$
  2. $\displaystyle \frac {21}{40}$
  3. $\displaystyle \frac {17}{20}$
  4. $\displaystyle \frac {17}{40}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle \frac {19}{20}\, -\, \displaystyle \frac {2}{20}\, =\, \displaystyle \frac {19-2}{20}\, =\, \displaystyle \frac {17}{20}$