Tag: reducing simple equations to simpler form

Questions Related to reducing simple equations to simpler form

On a car trip Sam drove  $m$  miles, Kara drove twice as many miles as Sam, and Darin drove  $20$  fewer miles than Kara. In terms of  $m$ , how many miles did Darin drive?

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

  1. $2m+20$

  2. $2m-20$

  3. $\frac{m}{2}+20$

  4. $\frac{m+20}{2}$

Show answer
Correct Option: B
Explanation:

Given that

Number of miles driven by $Sam$ $=$ $m$
As $Kara$ drove twice as many miles as $Sam$,
Number of miles drove by $Kara$ $=$ $2m$
As $Darin$ drove 20 fewer miles than $Kara$,

Hence, Number of miles drove by $Darin$ $=$ Number of miles drove by $Kara$ $-$ $20$
$=$ $2m$ $-$ $20$ 
Therfore, $Darin$ drove $'2m$ $-$ $20'$ miles.

If $\dfrac{19}{5x+17} = \dfrac{19}{31}$, then find $x $.

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

  1. $0.4$

  2. $1.4$

  3. $2.8$

  4. $3.4$

Show answer
Correct Option: C
Explanation:

Given $\dfrac{19}{5x+17} = \dfrac{19}{31}$

$\Rightarrow 19\times 31=19(5x+17)\ \Rightarrow 31=5x+17\ \Rightarrow 5x=14\ \Rightarrow x=\dfrac {14}{5}=2.8$

If $\cfrac{37}{4\sqrt{j}-19} = \cfrac{37}{17}$, then find the value of $j$.

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

  1. $64$

  2. $72.25$

  3. $81$

  4. $90.25$

Show answer
Correct Option: C
Explanation:

Given, $\dfrac {{ 37 }}{{ (4\sqrt { j }  }-19)}={ 37 }/{ 17 }$

Since the numerators are equal, we can equate the denominators.
$\Rightarrow { (4\sqrt { j }  }-19)={ 17 }$
$\Rightarrow 4\sqrt { j } =36$
$\Rightarrow \sqrt { j } =9$
So, $j=81$
Hence, option C is correct.

If $\displaystyle \frac{a-b}{b}=\frac{3}{7}$, which of the following must also be true?

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

  1. $\displaystyle \frac{a}{b}=-\frac{4}{7}$

  2. $\displaystyle \frac{a}{b}=\frac{10}{7}$

  3. $\displaystyle \frac{a+b}{b}=\frac{10}{7}$

  4. $\displaystyle \frac{a-2b}{b}=-\frac{11}{7}$

Show answer
Correct Option: B
Explanation:

Given: $\displaystyle \frac {a-b}{b}=\frac 37$

Separating denominators,

$\Rightarrow \displaystyle \frac ab -1=\frac 37$
$\Rightarrow \displaystyle \frac ab=\frac 37+1=\frac {10}{7}$
$\Rightarrow \dfrac {a}{b}=\dfrac {10}{7}$
Therefore, option B is correct.

If $( 2m) k=6$, then $mk = $

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

  1. $3$

  2. $4$

  3. $5$

  4. $6$

Show answer
Correct Option: A
Explanation:

Given, $(2m)k$ $=$ $6$

$\Rightarrow 2$ $\times$ $m$ $\times$ $k$ $=$ $6$
$\Rightarrow m$ $\times$ $k$ $=$ $\dfrac {6}{2}$
$\Rightarrow mk$ $=$ $3$
Therefore, $mk$ $=$ $3$

If $a\neq 0$ and $\dfrac{5}{x}=\dfrac{5+a}{x+a}$, what is the value of $x$?

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

  1. $-5$

  2. $-1$

  3. $5$

  4. $2$

Show answer
Correct Option: C
Explanation:

Given:

$a$ $\neq$ $0$ and $\dfrac {5}{x}$ $=$ $\dfrac {5 \space + \space a}{x \space + \space a}$
To find the value of $x$,
$\Rightarrow \dfrac {5}{x}$ $=$ $\dfrac {5 \space + \space a}{x \space + \space a}$
$\Rightarrow 5$ $\times$ $(x$ $+$ $a)$ $=$ $x$ $\times$ $(5$ $+$ $a)$
$\Rightarrow 5x$ $+$ $5a$ $=$ $5x$ $+$ $xa$
Get the co-efficients of $'x'$ on one-side,
$\Rightarrow 5x$ $+$ $xa$ $-$ $5x$ $=$ $5a$
$\Rightarrow xa$ $=$ $5a$
As $(a$ $\neq$ $0)$,    $x$ $=$ $5$
Therefore, the value of $'x'$ is $'5'$.

If one-third of a two digit number exceeds its one-fourth by $8$, then what is the sum of the digits of the number?

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

  1. $6$

  2. $13$

  3. $15$

  4. $17$

Show answer
Correct Option: C
Explanation:

$(10x + y)\left (\dfrac {1}{3} - \dfrac {1}{4}\right ) = 8$
$\Rightarrow 10x + y = 96$
$\Rightarrow x = 9, y = 6$
$\therefore x + y = 15$

The total cost of three prizes is Rs. $2550$. If the value of second prize is $\left(\displaystyle\frac{3}{4}\right)^{th}$ of the first prize and the value of $3rd$ prize is $\displaystyle\frac{1}{2}$ of the second prize, then the value of the first prize is ___________.

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

  1. Rs. $1308$

  2. Rs. $1028$

  3. Rs. $1200$

  4. Rs. $1450$

Show answer
Correct Option: C
Explanation:

let the first,second and third prizes be $x,y$ and $z$ respectively

$x+y+z=2550$.....(1)
$y=\dfrac{3}{4}x$
$z=\dfrac{1}{2}y= \dfrac{3}{8}x$
Putting values of $y$ and $z$ in equation (1)
$x+\dfrac{3}{4}x+\dfrac{3}{8}x=2550$

$\dfrac{8x+6x+3x}{8}=2550$
$17x=2550\times8$
$x=1200$

If $\sqrt {x-1}- \sqrt {x+1}+1 =0$, then $4x$ is equal to ____. 

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

  1. $4 \sqrt {-1}$

  2. $0$

  3. $5$

  4. $1 \dfrac {1}{4}$

Show answer
Correct Option: C
Explanation:

We can write it as $\sqrt{x-1} + 1 = \sqrt{x+1}$

Squaring both sides we get,
$x-1 + 1 +2\sqrt{x-1} = x+1$
$\Rightarrow 2\sqrt{x-1} = 1$

Squaring both sides, we get
$4(x-1) = 1$ 
$\therefore 4x = 5$

Solve the following linear equations. If $\cfrac{x-5}{3} = \cfrac{x-3}{5}$, then $x  $is equal to

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

  1. $8$

  2. $6$

  3. $2$

  4. $3$

Show answer
Correct Option: A
Explanation:
Given, $\cfrac { x - 5 }{ 3 }  =\cfrac { x-3 }{ 5 }$
Taking L.C.M., we get
$ \cfrac { 5x-25 }{ 15 }  = \cfrac { 3x - 9 }{ 15 } $
$5x - 25 = 3x -9$
Take $x$ terms on one sides and constants on another side, we get
$2x = 16$
$x = 8$