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 fraction lowest form of a fraction simplest ratio lowest form of fractions

The value of $\left[\left(-2\displaystyle\frac{3}{4}\right)-\left(\displaystyle -1\frac{3}{4}\right)\right]+\left[\left(\displaystyle -2\frac{3}{4}\right)-\left(\displaystyle -1\frac{3}{4}\right)\right]+......$ upto $30$ times is:

  1. $-1$
  2. $1$
  3. $30$
  4. $-30$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Consider the given expression.

$\left [ \left ( -2\dfrac{3}{4} \right )- \left ( -1\dfrac{3}{4} \right )\right ]+\left [ \left ( -2\dfrac{3}{4} \right )- \left ( -1\dfrac{3}{4} \right )\right ]+.......$ upto $30$ times

Sum $=\left[\left(\dfrac{-11}{4}\right)-\left(\dfrac{-7}{4}\right)\right]+\left[\left(\dfrac{-11}{4}\right)-\left(\dfrac{-7}{4}\right)\right]+......$ upto $30$ times

        $=[-1]+[-1]+......upto\ 30\ times$

        $=-30$

Hence, the value of the expression is $-30$. 

Multiple choice maths fraction lowest form of a fraction simplest ratio lowest form of fractions

The fraction $\dfrac {a^{2} + b^{2} - c^{2} + 2ab}{a^{2} + c^{2} - b^{2} + 2ac}$ is (with suitable restrictions on the values of $a, b,$ and $c$).

  1. Irreducible
  2. Reducible to $-1$

  3. Reducible to a polynomial of three terms
  4. Reducible to $\dfrac {a - b + c}{a + b - c}$
  5. Reducible to $\dfrac {a + b - c}{a - b + c}$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

$\dfrac {a^{2} + b^{2} - c^{2} + 2ab}{a^{2} + c^{2} - b^{2} + 2ac} = \dfrac {(a + b)^{2} - c^{2}}{(a + c)^{2} - b^{2}} = \dfrac {(a + b + c)(a + b - c)}{(a + c + b)(a + c - b)}$
$= \dfrac {a + b - c}{a + c - b}$ with $(a + c)^{2} \neq b^{2}$.

Multiple choice maths multiplication and division of integers division of integers and its properties multiplying and dividing integers multiplication of integers

The value of $\displaystyle\ \frac{15\times (-24)\times-18}{-(27)\times (20)}$ is

  1. -16

  2. -15

  3. -12

  4. 12
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

As there are $ 2 $ negative numbers in the numerator and $ 1 $ in the denominator, the answer will be negative. 

Cancelling out the common factors and simplifying we get 


$ \dfrac { 15\times (-24)\times -18 }{ -(27)\times (20) } =\dfrac {3 \times 8 \times 18}{-9\times 4}=-3\times 2\times2 = -12 $


Multiple choice maths multiplication and division of integers division of integers and its properties multiplying and dividing integers multiplication of integers

Evaluate $\displaystyle\ \frac{(-8)\times8\times(-8)\times(-8)}{(-4)\times(-4)\times(-4)\times(-4)}$

  1. -1

  2. -6

  3. -16

  4. 16
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given, $

\dfrac { (-8) \times 8 \times (-8) \times (-8) }{ (-4)\times (-4) \times (-4) \times (-4) } $

As there are $ 3 $ negative numbers in the numerator and $ 4 $ in the denominator, the answer will be negative.

Canceling out the common factors and simplifying we get

$ \dfrac { (-8) \times 8 \times (-8) \times (-8) }{ (-4)\times (-4) \times (-4) \times (-4) }   =2\times -2\times2\times2 = - 16 $

Multiple choice maths multiplication and division of integers division of integers and its properties multiplying and dividing integers multiplication of integers

Evaluate  $\displaystyle\ \frac {(-16)\times (-8)\times (-81)}{(-18)\times 32}$

  1. $-18$
  2. $18$
  3. $16$
  4. $-16$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given, $

\dfrac { (-16)\times (-8) \times (-81) }{ (-18)\times 32 } $

As there are $ 3 $ negative numbers in the numerator and $ 1 $ in the denominator, the answer will be positive. 

 
Canceling out the common factors and simplifying we get

$ \dfrac { (-16)\times (-8) \times (-81) }{ (-18)\times 32 }=\dfrac{4 \times 81}{9 \times2}   = 18 $

Multiple choice maths multiplication and division of integers division of integers and its properties multiplying and dividing integers multiplication of integers

The value of $\displaystyle \frac{1\div\frac{2}{3}\times \frac{3}{4} }{1\div \frac{2}{3}\times \frac{3}{4}}$ on simplification is

  1. $\displaystyle \frac{3}{16}$
  2. $\displaystyle \frac{9}{16}$
  3. $\displaystyle \frac{1}{16}$
  4. $\displaystyle \frac{7}{16}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\frac{1\div \frac{2}{3}\times \frac{3}{4}}{1+\div \frac{2}{3}of\frac{4}{3}}$

$\frac{1\div \frac{2}{3}\times \frac{3}{4}}{1\div \frac{2}{3}\times \frac{4}{3}}$
$\frac{1\times \frac{1}{2}}{1\times \frac{8}{9}}$
$\Rightarrow \frac{1\times 2}{1\times \frac{8}{9}}$
$\Rightarrow \frac{\frac{1}{2}}{\frac{8}{9}} =\frac{9}{16}$

Multiple choice maths multiplication and division of integers division of integers and its properties multiplying and dividing integers multiplication of integers

The value of $\displaystyle \frac{0.9\times 0.9\times 0.9+0.1\times 0.1\times 0.1}{0.9\times 0.9-0.9\times 0.1+0.1\times 0.1}$ on simplification is

  1. 0

  2. -1

  3. 2

  4. 1
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$\frac{0.9\times 0.9\times 0.9+0.1\times 0.1\times 0.1}{0.9\times 0.9-0.9\times 0.1+0.1\times 0.1}$
=$\frac{(0.9)^{3}+(0.1)^{3}}{(0.9)^{2}-0.9\times 0.1+(0.1)^{2}}$
We know that$ a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})$
Then $\frac{(0.9+0.1)\left [ (0.9)^{2}-0.9\times 0.1+(0.1)^{2} \right ]}{(0.9)^{2}-0.9\times 0.1+(0.1)^{2}}=(0.9+0.1)=1$
Multiple choice maths multiplication and division of integers division of integers and its properties multiplying and dividing integers multiplication of integers

The simplified value of $\displaystyle y^{4}\div y$ of $\displaystyle y^{3}\div y^{2}\times y^{5}\div y^{3}$ is

  1. 2

  2. 0

  3. -1

  4. 1
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$y^{4}\div y ofy^{3}\div y^{2}\times y^{5}\div y^{3}$
=$y^{4}\div y\times y^{3}\div y^{2}\times y^{2}$
=$\frac{y^{4}}{y^{4}}\times \frac{y^{4}}{y^{4}}=1\times 1=1$
Multiple choice maths multiplication and division of integers division of integers and its properties multiplying and dividing integers multiplication of integers

Value of $\displaystyle { 2 }^{ 2 }{ \times (-3) }^{ 2 }{ \times 2 }^{ 2 }{ \times (-4) }^{ 2 } $ is

  1. $2034$
  2. $2403$
  3. $2304$

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

$\displaystyle { 2 }^{ 2 }{ \times (-3) }^{ 2 }{ \times 2 }^{ 2 }{ \times (-4) }^{ 2 }\ =4\times +9\times 4\times 16\ =2304$ 

Multiple choice maths multiplication and division of integers division of integers and its properties multiplying and dividing integers multiplication of integers

Evaluate $\displaystyle\ \frac {5\times (-144)\times (-27)}{(-15)\times(18)\times(-16)}$

  1. $\dfrac {9}{4}$
  2. $\dfrac {9}{8}$
  3. $\dfrac {9}{2}$
  4. $\dfrac {-9}{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

As there are $ 2 $ negative numbers in the numerator and $ 2 $ in the denominator, the answer will be positive.

Canceling out the common factors and simplifying we get

$ \dfrac { 5\times (-144)\times (-27) }{ -(15)\times 18 \times (-16) } =\dfrac{9 \times 27}{3 \times 18} =\dfrac{9}{2}$