Tag: solving linear equations with variable on both sides

Questions Related to solving linear equations with variable on both sides

Neglecting air resistance, the upward velocity of the water in the stream of a particular fountain is given by the formula $v = -32t + 28$, where $t$ is the number of seconds after the water leaves the fountain. While going upward, the water slows down until at the top of the stream, the water has a velocity of $0$ feet per second. How long does it take a droplet of water to reach the maximum height?

maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

  1. $0.825$ seconds

  2. $0.925$ seconds

  3. $0.875$ seconds

  4. $0.975$ seconds

Show answer
Correct Option: C
Explanation:

Given, $v = -32t + 28$
It is mentioned that at the maximum height, the velocity of water is $0$ feet per second.
Therefore, final velocity $(v) = 0$. 
$\Rightarrow 0=-32t+28$

$\Rightarrow 32t=28$      
$\Rightarrow t=\cfrac { 28 }{ 32 } =0.875$ seconds

Twelve years hence a person will be four times as he was twelve years ago, then his present age is

maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

  1. $20$ years

  2. $25$ years

  3. $28$ years

  4. $30$ years

Show answer
Correct Option: A
Explanation:

Let his present age be $x$
According to problem
$\Rightarrow\;x+12=4\;(x-12)$
$\Rightarrow\;-3x=-48-12$
$\Rightarrow\;3x=60$
$\Rightarrow\;x=20$ years.

A father is at present three as old as his son . Five years back he was four times as old as his son.  Find the age of his son

maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

  1. 12 years

  2. 15 years

  3. 18 years

  4. 20 years

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Correct Option: B
Explanation:

Present age of the son is 'x' years his father's age is 3x
Five year ago:
Son's age = x - 5 and father's age = $3x - 5$
$\displaystyle \therefore 3x-5=4(x-5)$ or x = 15

If the sum of four consecutive even integers is $212$, what is the value of the second even integer?

maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

  1. $50$

  2. $51$

  3. $52$

  4. $53$

Show answer
Correct Option: C
Explanation:

Let the four consecutive even numbers be $x, x + 2, x + 4$ and $x + 6$.

Therefore, $x + x + 2 + x + 4 + x + 6 = 212$
$4x + 12 = 212$
$4x = 212 - 12$
$4x = 200$ (Divide both sides by $4$)
$x = 50$
The second number be $x + 2$
So, the second number is $52$.

If the sum of four consecutive odd integers is $400$, what is the value of the first odd integer?

maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

  1. $95$

  2. $96$

  3. $97$

  4. $98$

Show answer
Correct Option: C
Explanation:

Let the four consecutive odd numbers be $x, x + 2, x + 4$ and $x + 6$.
Therefore, $x + x + 2 + x + 4 + x + 6 = 400$
$4x + 12 = 400$
$4x = 400 - 12$
$4x = 388$  (Divide both sides by $4$)
$x = 97$
The first number be $x$ 
So, the first number is $97$.

If the sum of four consecutive integers is $110$, what is the value of the third consecutive integer?

maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

  1. $26$

  2. $27$

  3. $28$

  4. $29$

Show answer
Correct Option: C
Explanation:

Let the four consecutive numbers be $x, x + 1, x + 2$ and $x + 3$.
Therefore, $x + x + 1 + x + 2 + x + 3 = 110$
$4x + 6 = 110$
$4x = 110 - 6$
$4x = 104$   (Divide both sides by $4$)
$x = 26$
The third number be $x + 2$ 
So, the third number is $28$.

The sum of a $2$ digit number and the number obtained by reversing its digits is $154$. If the digits differ by $4$, find the number.

maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

  1. $95$

  2. $73$

  3. $84$

  4. $62$

Show answer
Correct Option: A
Explanation:
Let two digit number $=ab=10a+b$
Sum of number and reversed number $=154$
$(10a+b)+(10b+a)=154$
$a+b=14$
difference between digits$=\left|a-b\right|=4$
$a-b=\pm 4$
$(a+b=14)+(a-b=4)=2a=18$    $a=9,b=5$
$(a+b=14)-(a-b-4)=2a=10$      $a=5,b=9$
$\therefore   \text {Number}=95 (or)59$

Two numbers are in the ratio $3 : 5$. If $9$ is subtracted from each, the new numbers are in the ratio $12 : 23$. Find the smaller number.

maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

  1. $27$

  2. $33$

  3. $49$

  4. $55$

Show answer
Correct Option: B
Explanation:

Let the numbers be $x$ and $y$

$\dfrac{x}{y}=\dfrac{3}{5}$ ...(1)

According to the question,

$\dfrac{x-9}{y-9}=\dfrac{12}{23}$ ...(2)

$x=\dfrac{3y}{5}$

$ \dfrac { \dfrac { 3y }{ 5 } -9 }{ y-9 } =\dfrac { 12 }{ 23 }  $

$  \dfrac { 3y-45 }{ 5y-45 } =\dfrac { 12 }{ 23 } $

$23(3y-45)=12(5y-45)$

$69y-1035=60y-540$

$9y=1035-540=495$

$y=55$

From (1)

$\dfrac{x}{55}=\dfrac{3}{5}$

Thus the smaller number is 33

The age of Vamsi's sister is $4\dfrac { 1 }{ 2 } $ times that of Vamsi, where as his uncle is $30$ years older than him. If the total of their ages is $56$ years, what is the age of Vamsi?

maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

  1. $12$ years

  2. $10$ years

  3. $8$ years

  4. $4$ years

Show answer
Correct Option: D
Explanation:
Let the age of Vamsi be $x$.
$\therefore$ age of Vamsi's sister  $=4 \cfrac{1}{2} \times x = \cfrac{9}{2} x$
Age of Vamsi's uncle $= 30 +$ Vamsi's age $= 30 + x$
Given that:-
age of Vamsi's sister $+$ age of Vamsi + age of Vamsi's uncle $= 56$
$\Rightarrow \; x + \cfrac{9}{2} x + 30 + x = 56$
$\Rightarrow \; \cfrac{13}{2} x = 56 - 30$
$\Rightarrow$ $13x = 52$
$\Rightarrow$ $x = 4$
Hence, age of Vamsi is $4$ years.

Deepak bought $12$ oranges for Rs $7.20$. Vimal bought x oranges more than Deepak's for Rs $9.60$. What is the value of x?

maths equations and simple functions further equations solving linear equations with variable on both sides solution of linear equations in one variable

  1. $2$

  2. $5$

  3. $4$

  4. $6$

Show answer
Correct Option: C
Explanation:

Deepak bought $12$ oranges for Rs.$7.20$

Thus, cost of one orange =Rs.$\dfrac{7.20}{12}$=Rs $0.60$

Vimal bought $x$ oranges more than Deepak's for Rs.$9.60$

Number of oranges Vimal bought=$\dfrac{9.60}{0.60}=16$

$x$= Number of oranges Vimal bought - Number of oranges Deepak bought

$x=16-12=4$

$x=4$