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 applications of quadratic equations solving (simple) problems word problems based on quadratic equations quadratic equation maths

If $\displaystyle a^{2}+b^{2}+c^{2} = 1,$ then which of the following cannot be the value of $( ab + bc + ca)$?

  1. $0$
  2. $\displaystyle \frac{1}{2}$
  3. $\displaystyle \frac{-1}{2}$
  4. $-1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We know that $(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2(ab+bc+ca)$


$\Rightarrow (a+b+c) ^{2}=1+2(ab+bc+ca)$

$(a+b+c)^{2}$ should always be positive 

By checking options one by one, we get the result

1)ab+bc+ca=0

$\Rightarrow (a+b+c)^{2}=1$ which can be true 

Same way b, c options are true 

4th option where ab+bc+ca=-1/2

We get $(a+b+c)^{2}=-1$ which is false 

Multiple choice maths powers and exponents scientific notation use of exponents power of 10

If $3^x=5^y=7^z=105$, then $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}$ is equal to?

  1. $1$
  2. $0$
  3. $-1$

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

$x = \cfrac{\log 105}{\log 3}, y = \cfrac{\log 105}{\log 5} , z = \cfrac{\log 105}{\log 7}$

$\cfrac{1}{x} + \cfrac{1}{y} + \cfrac{1}{z} = \cfrac{\log 3 + \log 5 + \log 7}{\log 105} = \cfrac{\log 105}{\log 105} = 1$

Multiple choice maths three dimensional geometry - ii point of intersection of a line and a plane line and a plane three dimensional geometry

 The value of $k$ such that $\displaystyle \dfrac{{x}-4}{1}=\dfrac{{y}-2}{1}=\dfrac{{z}-{k}}{2}$ lies in the plane $2x-4y+{z}=7$ is 

  1. $7$
  2. $-7$

  3. no real value
  4. $4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given equation of straight line $\displaystyle \dfrac { x-4 }{ 1 } =\dfrac { y-2 }{ 1 } =\dfrac { z-k }{ 2 } $


Since, the line lies in the plane $2x-4y+z=7$

$\therefore $ Point $\left( 4,2,k \right) $ must satisfy the plane.

$\Rightarrow 8-8+k=7 $

$\Rightarrow k=7$

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

$\frac{1}{3}(-2p+6q-9r)-\frac{1}{6}(-4p -18q +24r) = $

  1. $-\frac{4}{3}p$

  2. 5q

  3. -7r

  4. 5q-7r
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\frac{1}{3}(-2p+6q-9r)-\frac{1}{6}(-4p -18q +24r) = \frac{1}{3}(-2p+6q-9r)-\frac{1}{3}(-2p -9q +12r) $
$=\frac{1}{3} [\left (-2p+6q-9r)-(-2p -9q +12r) \right]$
$=\frac{1}{3} (-2p + 6q - 9r + 2p + 9q -12r)$
$=\frac{1}{3} (15q - 21r)$
$=(5q - 7r)$

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

$ \frac{3}{4}(a+y) \left [ y + a - \frac{1}{3} \left ( y + a -\frac{1}{4}(a+y) \right )\right ]$

  1. $(a+y)^{2}$
  2. $\frac{3a}{16}$
  3. $\frac{9}{16}(a+y)^{2}$

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

$ \frac{3}{4}(a+y) \left [ y + a - \frac{1}{3} \left ( y + a -\frac{1}{4}(a+y) \right )\right ]$
= $ \frac{3}{4}(a+y) \left [ y + a - \frac{1}{3} \left (\frac{3}{4}(a+y) \right) \right ]$
= $ \frac{3}{4}(a+y) \left [ y + a - \frac{1}{4}(a+y)\right ]$
= $ \frac{3}{4}(a+y) \left [ \frac{3}{4}(a+y)\right ]$
= $ \frac{9}{16}(a+y)^2$

Multiple choice maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

Which of the following is equal to  $1?$

  1. $\dfrac { 0.304 \times 20 } { 304 \times 2 }$
  2. $\dfrac { 0.304 \times 20 } { 3 \cdot 04 \times 2 }$
  3. $\dfrac { 0.304 \times 2 } { 30 \cdot 4 \times 2 }$
  4. $\dfrac { 0.304 \times 2 } { 304 \times 0 \cdot 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\begin{array}{l} \dfrac { { 0.304\times 20 } }{ { 3.04\times 2 } }  \ =\dfrac { { 304\times 20\times 100 } }{ { 304\times 2\times 1000 } }  \ =1 \ Hence, \ B\, is\, the\, correct\, answer. \end{array}$

Multiple choice maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

$\displaystyle \frac{\left ( 0.35 \right )^{2}-\left ( 0.03 \right )^{2}}{0.19}=? $

  1. 0.32

  2. 0.48

  3. 0.76

  4. 0.64
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\dfrac{(0.35)^{2}-(0.03)^{2}}{0.19}$=$\dfrac{(0.35+0.03)(0.35-0.03)}{0.19}$    $[a^2-b^2=(a+b)(a-b)]$


$=\dfrac{(0.38)(0.32)}{0.19}$


$=2\times{0.32}$

$=0.64$

So the correct answer will option D

Multiple choice maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

What is $\displaystyle 0.\overline{09}\times7.\overline{3}$ equal to ?

  1. $\displaystyle 0.\overline{6}$
  2. $\displaystyle 0.\overline{7}$

  3. 0.6

  4. 0.7
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\displaystyle 0.\overline{09}\times7.\overline{3}=\frac{9}{99}\times7\frac{3}{9}=\frac{9}{99}\times\frac{66}{9}=\frac{6}{9}=0.\overline{6}$

Multiple choice maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

What is the value of $\displaystyle \left ( 4.7\times13.26+4.7\times9.43+4.7\times77.31 \right )?$

  1. $470$
  2. $235$
  3. $705$
  4. $940$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

given that 

we have find the the value of the expression .

$4.7\times 13.26 +4.7 \times 9.43 +4.7 \times 77.31$

Taking $4.7$ common 

$= 4.7 \times [13.26 + 9.43 +77.31]$

$= 4.7 \times [100]$

$ = 470$

So option $A $ is correct


Multiple choice maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

$\displaystyle \frac{\left ( 2.3 \right )^{3}-0.027}{\left ( 2.3 \right )^{2}+0.69+0.09}=? $

  1. $2.6$
  2. $2$
  3. $2.33$
  4. $2.8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$\dfrac { { \left( 2.3 \right)  }^{ 3 }-0.027 }{ { \left( 2.3 \right)  }^{ 2 }+0.69+0.09 } \\ =\dfrac { 12.167-0.027 }{ 5.29+0.69+0.09 } \\ =\dfrac { 12.14 }{ 6.07 } \\ =2$
So, correct answer is option B.
Multiple choice maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

Which of the following is equal to $1$?

  1. $\displaystyle \frac{(0.11)^{2}}{(1.1)^{2}\times 0.1}$
  2. $\displaystyle \frac{(1.1)^{2}}{11^{2}\times (0.01)^{2}}$
  3. $\displaystyle \frac{(0.011)^{2}}{1.1^{2}\times 0.01^2}$
  4. $\displaystyle \frac{(0.11)^{2}}{1.1^{2}\times 0.01}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$\Rightarrow \cfrac{(0.11)^{2}}{(1.1)^{2}\times 0.1}=\cfrac{0.0121}{1.21\times 0.1}=\cfrac{0.0121}{0.121}=0.1$

$\Rightarrow \cfrac{(1.1)^{2}}{11^{2}\times (0.01)^{2}}=\cfrac{1.21}{121\times 0.0001}=\cfrac{0.01}{0.0001}=100$

$\Rightarrow \cfrac{(0.011)^{2}}{(1.1)^{2}\times (0.01)^{2}}=\cfrac{0.000121}{1.21\times 0.0001}=1$

$\Rightarrow \cfrac{(0.11)^{2}}{1.1^{2}\times 0.01}=\cfrac{0.0121}{1.21\times 0.01}=\cfrac{0.0121}{1.21}=1.21$
Multiple choice maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

$\displaystyle \frac{24.23\times 1.423\times 34.21}{521.3\times 413.32\times 2.53}$ is same is 

  1. $\displaystyle \frac{2423\times 1423\times 3421}{5213\times 41332\times 253}$
  2. $\displaystyle \frac{2423\times 1423\times 3421}{5213\times 4133.2\times 2.53}$
  3. $\displaystyle \frac{2.423\times 14.23\times 342.1}{521.3\times 4133.2\times 2.53}$
  4. $\displaystyle \frac{24.23\times 14.23\times 3.421}{5.213\times 41332\times 0.253}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Option c is correct answer.

 
As in that expression total numbers after decimal is same as the given expression.


$\displaystyle \frac{24.23\times 1.423\times 34.21}{521.3\times 413.32\times 2.53}$  $=\displaystyle \frac{2.423\times 14.23\times 342.1}{521.3\times 4133.2\times 2.53}$

Multiple choice maths calculations and mental strategies 4 mental additions and subtractions of decimals multiplication and division of decimals mental multiplication and division

Simplify : $\displaystyle \frac{3.6\times 0.48\times 2.50}{0.12\times 0.09\times 0.5}$

  1. 80

  2. 800

  3. 8000

  4. 80,000
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\displaystyle \frac{3.6\times 0.48\times 2.50}{0.12\times 0.09\times 0.5}=\frac{36\times 48\times 250}{12\times 9\times 5}=800 $


No of decimal places in num and den being equal