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 ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

If $\displaystyle a=(2^{-2}-2^{-3}),b=(2^{-3}-2^{-4})and: c=(2^{-4}-2^{-2})$ then find the value of 3 abc

  1. $\displaystyle \frac{-63}{1024}$
  2. $\displaystyle \frac{-63}{2048}$
  3. $\displaystyle \frac{-9}{2048}$
  4. $\displaystyle \frac{9}{1024}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$a=(2^{-2}-2^{-3}),b=(2^{-3}-2^{-4})and: c=(2^{-4}-2^{-2})$
So,  $3abc=3\times (2^{ -2 }-2^{ -3 })\times (2^{ -3 }-2^{ -4 })\times (2^{ -4 }-2^{ -2 })$
$3abc=3\times (1/4-1/8)\times (1/8-1/16)\times (1/16-1/4)$
$3abc=3\times (1/8)\times (1/16)\times (-3/16)$
$3abc=\frac { -9 }{ 2048 } $
Answer (C) $\displaystyle \frac{-9}{2048}$

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

If $\displaystyle 15\frac {2}{3}\times 3\frac {1}{6}+6\frac {1}{3}=11\frac {7}{18}+x$, then the value of $x$ is ______ .

  1. $\displaystyle 39\frac {5}{9}$
  2. $\displaystyle 137\frac {4}{9}$
  3. $\displaystyle 29\frac {7}{9}$
  4. $\displaystyle 44\frac {5}{9}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$15\frac{2}{3}\times 3\frac{1}{6}+6\frac{1}{3}= 11\frac{7}{18}+x$
Or $\frac{47}{3}\times \frac{19}{6}+\frac{19}{3}= \frac{205}{18}+x$
Or $\frac{893}{18}+\frac{19}{3}= \frac{205}{18}+x$
Or 893+114=205+18x
Or 18x=893+114-205=$\frac{802}{18}=44\frac{5}{9}$

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

The value of $\displaystyle\frac{2^{n+4}-2\times2^{n}}{2\times2^{(n+3)}}+2^{-3}$ is equal to ........... 

  1. $2^{n+1}$
  2. $\begin{pmatrix}\displaystyle\frac{9}{8}-2^{n}\end{pmatrix}$
  3. $\begin{pmatrix}-2^{n+1}+\displaystyle\frac{1}{8}\end{pmatrix}$
  4. $1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle\frac{2^{n+4}-2\times2^{n}}{2\times2^{(n+3)}}+2^{-3}$
$\frac { 2^{ n+4 }-2^{ n+1 } }{ 2^{ (n+4) } } +2^{ -3 }$
$\frac { 2^{ n }(2^{ 4 }-2^{ 1 }) }{ 2^{ n }\times { 2 }^{ 4 } } +2^{ -3 }$
$\frac { (2^{ 4 }-2^{ 1 }) }{ { 2 }^{ 4 } } +2^{ -3 }$
$\frac { (16-2) }{ 16 } +\frac { 1 }{ 8 } $
$\frac { 14 }{ 16 } +\frac { 1 }{ 8 } $
$\frac { 14+2 }{ 16 } =1$
Answer (D) 1


Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

Tony needs to solve the equation given below.
$\displaystyle  \frac {3}{4}t=\frac {6}{20}$
What operation should Tony perform to solve the equation for $t$ ?

  1. Multiply both sides by $\displaystyle \frac {1}{4}$
  2. Divide both sides by $\displaystyle \frac {3}{4}$
  3. Add $\displaystyle \frac {3}{4}$ to both sides
  4. Subtract $\displaystyle \frac {3}{4}$ from both sides
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

As t term is 3t/4,So to find t we need to divide is by 3/4.
Answer (B) Divide both sides by 3/4

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

Which of the following statements do not accurately describe the multiplicative inverse ?

  1. If a given fraction with numerator $1$ , the multiplicative inverse will be its denominator.
  2. If a given whole number the multiplicative inverse will a fraction containing that number as numerator and $1$ as denominator.
  3. In a statement $\sqrt3\times x=1$ , $x$ is the multiplicative inverse.
  4. One pair of number when multiplied together gives the number $1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Option $B$ is correct.
The multiplicative inverse will include $1$ as numerator.

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

What is the multiplicative inverse of $3-\sqrt8$ ?

  1. $\dfrac { 1 }{ 3+\sqrt { 8 } } $
  2. $3-\sqrt8$
  3. $8+\sqrt3$
  4. $3+\sqrt8$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let $x$ be the multiplicative inverse


$\Rightarrow 3-\sqrt { 8 } \times x=1$

$ \Rightarrow x=\dfrac { 1 }{ 3-\sqrt { 8 }  } $

Rationalising the number

$\Rightarrow x=\dfrac { 1 }{ 3-\sqrt { 8 }  } \times \dfrac { 3+\sqrt { 8 }  }{ 3+\sqrt { 8 }  } $

$ \Rightarrow x=\dfrac { 3+\sqrt { 8 }  }{ { (3 })^{ 2 }-{ ({ \sqrt { 8 }  })^{ 2 } } } =\dfrac { 3+\sqrt { 8 }  }{ 9-8 } =\dfrac { 3+\sqrt { 8 }  }{ 1 } $

$ \Rightarrow x=3+\sqrt { 8 } $

So option $D$ is correct.

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

The value of $\dfrac{(469 +174)^2 - (469-174)^2}{469 \times 174}$ is

  1. $2$
  2. $4$
  3. $689$
  4. $1023$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$(a+b)^2-(a-b)^2=(a^2+2ab+b^2)-(a^2-2ab+b^2)=4ab$.


Using the above identity,

$\dfrac{(469+174)^2-(469-174)^2}{469\times{174}}=\dfrac{4\times{469}\times{174}}{469\times174}=4$.

$\therefore$  The answer is $4$.  $[B]$

Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

Find the value of $\displaystyle\left(\frac{P+Q}{R}\right)\times S$.
(i) $100$ lakhs $=$ _________[Q] millions
(ii) ____[R] crores $=100$ millions
(iii) $100$ thousands $=$ ___[P] lakhs
(iv) $10$ crores $=$ _______[S] millions.

  1. $10$
  2. $100$
  3. $110$
  4. $1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$1$ Million $=$ $10$ lakh
$100$lakh $=$ Q million
Q $=$ ($100$/$10$) $=$$10$
R crore $=$ $100$ Million
R $=$ ($100$X$1000000$) / ($10000000$)
R $=$ $10$
$100$ X $1000$ $=$ $1$lakh $=$ P lakhs
P $=$ $1$
$10$X$10000000$ $=$ S Million
S $=$ $100000000$/$1000000$
S $=$  $100$
So, P $=$ $1$,   Q $=$ $10$,  R $=$ $10$,  S $=$ $100$
P $+$ Q  $=$ $11$
$\dfrac{(P+Q)}{R}=$ $11/10$
$\left(\dfrac{(P+Q)}{R}\right)\times S=\dfrac{11}{10}\times 100$ $=$ $110$
Hence, option C is correct.
Multiple choice maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

If $b \neq d$, the fractions $\dfrac{ax + b}{cx + d}$ and $\dfrac{b}{d}$ are unequal if:

  1. $a = c = 1 \ and \ x \neq 0$
  2. $a = b = 0$
  3. $a = c = 0$
  4. $x _3 = 0$
  5. $ad = bc$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\cfrac { ax+b }{ cx+d } \neq \cfrac { b }{ d } \Longrightarrow adx+bd\neq bcx+bd\Longrightarrow (ad-bc)x\neq 0\ \therefore x\neq 0\quad and\quad ad\neq bc$

We know, $b \neq d$
$\therefore a=c=1$
$\therefore a=c=1 $ and $x \neq 0$

Multiple choice maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

Given that $\displaystyle\frac{4p + 9q}{p} = \frac{5q}{p - q}$ and $p$ and $q$ are both positive. calculate $\displaystyle\frac{p}{q}$

  1. $\displaystyle\frac{1}{3}$
  2. $\displaystyle\frac{2}{3}$
  3. $\displaystyle\frac{3}{2}$
  4. $\displaystyle\frac{3}{5}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given 

$\dfrac{4p+9q}{p}=\dfrac{5q}{p-q}$
or, $4p^2-4pq+9pq-9q^2=5pq$
or, $4p^2=9q^2$
or, $\dfrac{p}{q}=\dfrac{3}{2}$ [Since $p$ and $q$ are both positive]

Multiple choice maths ratio, proportion and unitary method converting to ratios finding ratios other quantities

If $33\displaystyle\frac{1}{3}\%$ of $A=1.5$ of $B=\displaystyle\frac{1}{8}$ of $C$, then what is $A\,\colon\,B\,\colon\,C$?

  1. $5 : 3 : 2$
  2. $9 : 2 : 24$
  3. $6 : 2 : 17$

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

$33\displaystyle\frac{1}{3}\%$ of $A=1.5$ of $B=\displaystyle\frac{1}{8}$ of $C$ $\Rightarrow\displaystyle\frac{100}{3\times100}A=\displaystyle\frac{3}{2}B=\displaystyle\frac{1}{8}C\;\;\Rightarrow:\displaystyle\frac{1}{3}A=\displaystyle\frac{3}{2}B=\displaystyle\frac{1}{8}C=K:(say)$  $\Rightarrow:A=3K,\,B=\displaystyle\frac{2}{3}K,\,C=8K$  $\Rightarrow:A\,\colon\,B\,\colon\,C=3\,\colon\,\displaystyle\frac{2}{3}\,\colon\,8=3\times3\,\colon\,\displaystyle\frac{2}{3}\times3\,\colon\,8\times3=9\,\colon\,2\,\colon\,24$