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 brackets order operations and algebra using brackets in algebraic expressions order of operations

If $\displaystyle b=6-\left [ \frac{4b+3}{2a-5} \right ]$, express a in terms of b.

  1. $\displaystyle a=\frac{33-b}{2\left ( 6-b \right )}$
  2. $\displaystyle a=\frac{b-33}{2\left ( 6-b \right )}$
  3. $\displaystyle a=\frac{33-b}{2\left ( b-6 \right )}$

  4. none of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given,
$ b = 6 - \left[ \frac { 4b+3 }{ 2a-5 }  \right]  $
$ => \left[ \frac { 4b+3 }{ 2a-5 }  \right] = 6 - b $
$ => 4b + 3 = (6-b)(2a-5) $
$ => 4b + 3 = 12a - 30 - 2ab +5b $
$ => 12a - 33  -2ab + b = 0 $
$ =>  a(12 -2b) = 33 - b $
$ => 2a(6-b) = 33-b $
$ => a = \frac {33-b}{2(6-b)} $

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Given $\displaystyle b=\frac{2a}{a-2}$ and $\displaystyle c=\frac{3b-4}{4b+3}$, express c in terms of a.

  1. $\displaystyle c=\frac{2a+8}{11a+6}$
  2. $\displaystyle c=\frac{2a-8}{11a-6}$
  3. $\displaystyle c=\frac{2a+8}{11a-6}$

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

Given, $ c = \frac {3b-4}{4b + 3} $

But, $ b = \frac {2a}{a-2} $
Hence, $ c = \frac {3(\frac {2a}{a-2} )-4}{4(\frac {2a}{a-2} ) + 3} $
$ => c = \frac {6a-4a + 8}{8a +3a - 6} $
$ => c = \frac {2a+8}{11a -6} $

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Find the value of $\displaystyle \frac{2}{1+\frac{1}{1-\frac{1}{2}}}\times\frac{3}{\frac{5}{6}of\frac{3}{2}\div 1\frac{1}{4}}$.

  1. 4

  2. 3

  3. 2

  4. 1
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given exp.$\displaystyle \frac{2}{1+\frac{1}{\frac{1}{2}}}\times\frac{3}{\frac{15}{12}\div \frac{5}{4}}=\frac{2}{1+2}\times\frac{3}{\frac{15}{12}\times\frac{4}{5}}=\frac{2}{3}\times\frac{3}{1}=2$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

 Make b the subject of formula : $\displaystyle a=\frac{1+b^2}{1-b^2}$. 

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

Given, $ a = \frac {1 +{b}^{2}}{1 - {b}^{2}} $
$ => a -a{b}^{2} = 1 +{b}^{2} $
$ => a- 1 = {b}^{2} (1 + a) $
$ =>{b}^{2} = \frac {a-1}{1+a} $


Multiple choice maths decimal fractions rounding decimals non-terminating recurring decimals in rational numbers rounding of decimals estimation and rounding off estimations, bounds and rounding off

$\displaystyle 6.743\times 100$ is equal to _____ 

  1. $674.300$
  2. $.674300$
  3. $67.4300$
  4. $6.74300$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The number of decimal places from the right of the number in the question will be in the product at the same number of places.

$6.743×100=674.300$   ($3$ decimal places)
So option A is the correct answer.

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

If $x=7-4\sqrt 3$, the value of $x^2+\displaystyle\frac{1}{x^2}$ will be

  1. $146$
  2. $148$
  3. $194$
  4. $196$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given,
$x=7-4\sqrt 3$
$\therefore \displaystyle\frac{1}{x}=\displaystyle\frac{1}{(7-4\sqrt 3)}\times \displaystyle\frac{7+4\sqrt 3}{7+4\sqrt 3}$
$=\displaystyle\frac{7+4\sqrt 3}{(49-48)}$
$=7+4\sqrt 3$
Now,
$x+\displaystyle\frac{1}{x}=(7-4\sqrt 3)+(7+4\sqrt 3)$
$=14$
$=> \left( x+\displaystyle\frac{1}{x}\right)^2=(14)^2$
Using $(a+b)^2=a^2+b^2+2ab$ we get,
$=> x^{ 2 }+\frac { 1 }{ x^{ 2 } } +2(x)(\frac { 1 }{ x } )=196$
$=> x^2+\displaystyle\frac{1}{x^2}=196-2$
$\therefore x^2+\displaystyle\frac{1}{x^2}=194$

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

If $\dfrac { a }{ b } =\dfrac { b }{ c } =\dfrac { c }{ d }$ then $\dfrac { { b }^{ 3 }+{ c }^{ 3 }+{ d }^{ 3 } }{ { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 } }$ will be equal to

  1. $\dfrac{a}{b}$
  2. $\dfrac{b}{c}$
  3. $\dfrac{c}{d}$
  4. $\dfrac{d}{a}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given a/b = b/c = c/d = k, then c = dk, b = ck = dk^2, and a = bk = dk^3. Substituting these into the expression: (b^3 + c^3 + d^3) / (a^3 + b^3 + c^3) = (d^3k^6 + d^3k^3 + d^3) / (d^3k^9 + d^3k^6 + d^3k^3) = d^3(k^6 + k^3 + 1) / d^3k^3(k^6 + k^3 + 1) = 1/k^3. Since a/b = k, then b/a = 1/k, so (b/a)^3 = 1/k^3. Alternatively, d/a = d/(dk^3) = 1/k^3.

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

Mark the correct alternative of the following.
If $x : y =1 : 1$, then $\dfrac{3x+4y}{5x+6y}=?$

  1. $\dfrac{7}{11}$
  2. $\dfrac{17}{11}$
  3. $\dfrac{17}{23}$
  4. $\dfrac{4}{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, $x:y=1:1$

or, $\dfrac{x}{y}=\dfrac{1}{1}$
or,  $x=y$......(1)

Now,
$\dfrac{3x+4y}{5x+6y}$
$=\dfrac{7x}{11x}$ [ Using (1)]
$=\dfrac{7}{11}$.

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

If $\displaystyle \frac{x}{y}=\frac{6}{5}$ then $\displaystyle \frac{x^{2}+y^{2}}{x^{2}-y^{2}}$ is:

  1. $\displaystyle \frac{36}{25}$
  2. $\displaystyle \frac{25}{36}$
  3. $\displaystyle \frac{61}{11}$
  4. $\displaystyle \frac{11}{61}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle \frac{x}{y}=\frac{6}{5}$

$x = 6k, y = 5k$

$\displaystyle \frac{x^{2}+y^{2}}{x^{2}-y^{2}}=\frac{\left ( 6k \right )^{2}+\left ( 5k \right )^{2}}{\left ( 6k \right )^{2}-\left ( 5k \right )^{2}}=\frac{61k^{2}}{11k^{2}}=\frac{61}{11}$

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

If $ \displaystyle \frac {1}{x} : \frac {1}{y} : \frac {1}{z} = 2:3:5, $ then $x:y:z =?$

  1. $2:3:5$
  2. $15:10:6$
  3. $5:3:2$
  4. $6:10:15$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\dfrac { 1 }{ x } :\dfrac { 1 }{ y } :\dfrac { 1 }{ z } =\quad 2:3:5\ \dfrac { yz:xz:xy }{ xyz } =\quad 2:3:5\ yz:xz:xy\quad =\quad 2xyz:3xyz:5xyz\ 1:1:1=\quad 2x:3y:5z\ x:y:z=\quad \dfrac { 1 }{ 2 } :\dfrac { 1 }{ 3 } :\dfrac { 1 }{ 5 } =\dfrac { 15:10:6 }{ 30 } \ So,\quad x:y:z\quad is\quad 15:10:6$