Mathematics · Quantitative Aptitude

Algebraic Equations and Expressions

142 Questions

Algebraic equations and expressions test mathematical skills in solving for unknown variables. Questions involve linear equations, ratios, and evaluating complex expressions. This topic forms a core component of quantitative aptitude sections in various competitive exams.

Linear equationsEvaluating expressionsRatio proportionsExponential equationsHypergeometric functions

Algebraic Equations and Expressions Questions

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $2^{x} + 2^{x + 2} = 40$, then the value of $x$ is

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

Given, ${ 2 }^{ x }+{ 2 }^{ x+2 }=40\ \Rightarrow { 2 }^{ x }+{ 2 }^{ 2 }{ 2 }^{ x }=40\ \Rightarrow { 2 }^{ x }+(4){ 2 }^{ x }=40\ \Rightarrow (5){ 2 }^{ x }=40\ \Rightarrow { 2 }^{ x }=8\ \Rightarrow { 2 }^{ x }={ 2 }^{ 3 }$

So, $ x=3$

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

Find the value of $x: \dfrac {1}{x} + \dfrac {4}{5x} = \dfrac {2}{x + 5}$

  1. $0.71$
  2. $3.57$
  3. $5.8$
  4. $45$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given  $\dfrac { 2 }{ x+5 } =\dfrac { 1 }{ x } +\dfrac { 4 }{ 5x } $
Taking RHS:

$\dfrac { 1 }{ x } +\dfrac { 4 }{ 5x } $

LCM is $5x$
$\Rightarrow \dfrac { 5 }{ 5x } +\dfrac { 4 }{ 5x } \\ \Rightarrow \dfrac { 9 }{ 5x } $
Now taking LHS:
$\dfrac { 2 }{ x+5 } $
LHS $=$ RHS
$\dfrac { 2 }{ x+5 } =\dfrac { 9 }{ 5x } $
$\Rightarrow 5x\times 2=9(x+5)\\ \Rightarrow 10x=9x+45\\ \Rightarrow x=45$
Multiple choice maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

If  $2x  + 2(4 + 3x) < 2 + 3x > 2x + \dfrac{x}{2}$ then $x$ can take which of the following values. 

  1. $-3$
  2. $1$
  3. $0$
  4. $-1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given,
$2x+2(4+3x) < 2+3x > 2x+\dfrac{x}{2}$

Solving $1^{st}$ inequality

$2x+8+6x < 2+3x$

$5x < -6$

$x < \dfrac{-6}{5}$

Solving $2^{nd}$ inequality

$2+3x > 2x+\dfrac{x}{2}$

$\dfrac{x}{2} > -2$

$x > -4$

$x\in \left (-4, \dfrac{-6}5\right )$

$x$ can take values $-3$

$A$ is correct.
Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

The value of $x$ is ____ if $(3x + 1) : (5x - 4)$ is the duplicate ratio of $5 : 6$

  1. $2$
  2. $4$
  3. $8$
  4. $6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(3x + 1) : (5x - 4)$ is the duplicate ratio of $5 : 6$.
Also, the duplicate ratio of $5 : 6$ is $5^{2} : 6^{2} = 25 : 36$
$\therefore \dfrac {3x + 1}{5x - 4} = \dfrac {25}{36}$
$\therefore 108 x + 36 = 125x - 100$
$\therefore 125x - 108x = 36 + 100$
$\therefore 17x = 136$
$\therefore x = \dfrac {136}{17} = 8$

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

If $(4x + 3) : (9x + 10)$ is the triplicate ratio of $3 : 4$, then the value of x is ___

  1. $6$
  2. $12$
  3. $5$
  4. $4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The triplicate ratio of $3 : 4$ is $3^{3} : 4^{3} = 27 : 64$
$\therefore \dfrac {4x + 3}{9x + 10} = \dfrac {27}{64}$
$\Rightarrow 256x + 192 = 243x + 270$
$\Rightarrow 256x - 243x = 270 - 192$
$\Rightarrow 13x = 78$
$\Rightarrow x = \dfrac {78}{13} = 6$
$\Rightarrow x = 6$

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

The value of $x$ is ____ if $(x - 4) : (x + 2)$ is the triplicate ratio of $1 : 2$

  1. $\dfrac {34}{7}$
  2. $\dfrac {7}{34}$
  3. $\dfrac {30}{7}$
  4. $\dfrac {7}{30}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The triplicate ratio of $1 : 2$ is $1^{3} : 2^{3} = 1 : 8$
$\therefore \dfrac {x - 4}{x + 2} = \dfrac {1}{8}\Rightarrow 8x - 32 = x + 2$
$\Rightarrow 8x - x = 2 + 32$
$\Rightarrow 7x = 34$
$\Rightarrow x = \dfrac {34}{7}$

Multiple choice

22x-1 = (16/√8)6

  1. 7

  2. 0

  3. 8

  4. 0.25
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Simplify the right side: (16 / sqrt(8))^6 = (16 / (2*sqrt(2)))^6 = (8 / sqrt(2))^6 = (4*sqrt(2))^6. This simplifies to (2^2 * 2^0.5)^6 = (2^2.5)^6 = 2^15. Setting 2^(2x-1) = 2^15, we get 2x-1 = 15, so 2x = 16, x = 8.