Tag: solving linear equations with variable on both sides

Questions Related to solving linear equations with variable on both sides

Pipes A and B can fill a tank in $18$ minutes and $12$ minutes respectively. If both the pipes are opened simultaneously, how long will they take to fill the tank?

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

  1. $30$ minutes

  2. $20$ minutes

  3. $10 $ minutes

  4. $7\dfrac{1}{5}$ minutes

Show answer
Correct Option: D
Explanation:
Let $'V'$ be total volume of tank
Time taken by $A$ to fill tank $=18 min$
Speed of $A=\dfrac{V}{18}$
Time taken by $B$ to fill tank $=12 min$
Speed of $B=\dfrac{V}{12}$
$\therefore$  Total time taken by $A$  and  $B$  to fill tank when opened
Simultaneously $=\dfrac{V}{(\dfrac{V}{12}+\dfrac{V}{18})}$
$=\dfrac{18\times 12}{30}$
Time taken $= 7.2 min$

The sum of three non-zero prime numbers is $100$. One of them exceeds the other by $36$. Find the largest number.

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

  1. $73$

  2. $91$

  3. $67$

  4. $57$

Show answer
Correct Option: C
Explanation:
As we know that the sum of three odd numbers cannot be even.
$\therefore$ one of the prime is even 
Since $2$ is the only prime number which is even.
$\therefore$ one of the three prime numbers is $2$.
Let one of the other prime numbers is ${p} _{1}$ then the third prime number will be ${p} _{1} + 36$. 
Now according to question,
$2 + {p} _{1} + {p} _{1} + 36 = 100$
$2{p} _{1}=62$
$p1=31$

Hence, the three prime numbers are $2, 31$ and $67$ and the largest among them is $67$.
Hence, $67$ is the correct answer.

The sum of two numbers is $45$ and their difference is $11$. What are the two numbers?

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

  1. $28$ and $17$

  2. $27$ and $18$

  3. $25$ and $20$

  4. $22$ and $23$

Show answer
Correct Option: A
Explanation:

Let the numbers be $a$ and $b$
Given that 

$a+b=45$ ....(1)
$a-b=11$ ....(2)
Adding these two equations, we get
$2a=56$
$\Rightarrow a=28$
Substituting value of $a$ in equation (1), we get
$28+b=45$
$\Rightarrow b=45-28$
$\Rightarrow b=17$
We get $a = 28$ and $b=17$

The number of solution(s) of the equation $[x]+2{-x}=3x$, is$/$are (where $[]$ represents the greatest integer function and ${ x}$ denotes the fractional part 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. $1$

  2. $2$

  3. $3$

  4. $0$

Show answer
Correct Option: A
Explanation:

Given; $[x]+2\left{-x\right}=3x$

$\left{-x\right}+[-x]=-x     (\because  \left{a\right}+[a]=a)$
$\therefore  \left{-x\right}=-x-[-x]$
We know that,  $[-x]=-1-[x]$
$[x]-2x-2[-x]=3x$
$[x]-2x+2+2[x]=3x$
$3[x]=5x-2$
L.H.S is integer
$\therefore$ R.H.S must be integer
$\therefore$  $x$must be integer
As $x$ is integer , $[x]=x$
$3x=5x-2$
$x=1$
Only one solution is possible.

A number consists of two digits whose sum is 9. If 27 is added to the number, its digits are interchanged. Which of the given steps is CORRECT to find the number?
Step 1 : Let the units digit be x
Step 2 : Then, ten's digit = (9 - x)
$\therefore$  Number = 10 x (9 - x) + x
$\Rightarrow$  90 - 10x + x = (90 - 9x)
Step 3 : Adding 27 to the number 90 - 9x, we get 117 - 9x
Step 4 : Number with digits interchanged is 10x + (9 - x) = 9x + 9
Step 5 : 117 - 9x = 9x + 9
Step 6 : Therefore unit's digit = 6 and ten's digit = 3
Step 7 : Hence the number = 36.

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

  1. Only Step 4

  2. Both Step 1 and Step 2

  3. Step 1, 2, 3 and 4

  4. All steps are correct

Show answer
Correct Option: D
Explanation:

In given question we have to find the number.

$\Rightarrow$  To find the numbers $7$ steps are given.
$\Rightarrow$  All $7$ steps are correct to find the required  number.
$\therefore$   Correct answer is option $D.$

Peter's age in $10$ years will be $12$ less than $4$ times his current age. What is Peter's current age (in years)?

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

  1. $7.33$

  2. $7.71$

  3. $6.04$

  4. $6.49$

Show answer
Correct Option: A
Explanation:

Let Peter's current age be $x$ years

According to question,

$\Rightarrow$$(x+10)=4x-12$

$\Rightarrow$$4x-(x+10)=12$

$\Rightarrow$$4x-x-10=12$

$\Rightarrow$ $3x=22$


$\Rightarrow$$x=\cfrac { 22 }{ 3 } =7.33$


Peter's age $=7.33$ years

Arvind has Piggybank. It is full of one-rupee and fifty paise coins. It contains $3$ times as many fifty paise coins as one rupee coins. The total amount of money in the bank is $Rs\ 35$. How many one-rupee and fifty paise coins are there in the bank ?(respectively)

maths equations and rearranging formulae further equations solving linear equations with variable on both sides equations and simple functions

  1. $14,\ 42$

  2. $15,\ 42$

  3. $14,\ 52$

  4. None of these

Show answer
Correct Option: A
Explanation:

Let no. of fifty paise coins be $x$ and no.of one Rupees coins be $y$

then $x=3y$

According to question 
$0.50 \times x+1 \times y=35\$
$\Rightarrow 0.5x+y=35\$
$\Rightarrow 0.5 \times 3y+y=35\$
$1.5y+y=35\$
$\Rightarrow 2.5y=35\$
$\Rightarrow y=\dfrac{350}{25}\$
$\Rightarrow y=14\$
So,
    $x=3y\$
$=3 \times 14\$
$=42$

Therefore no.of fifty paise coins is $42$ ans no.of one Ruppee coin is $14$.

Sum of the ages of three friends $x$ years ago was $y$ years. Then what will be the sum of their ages now?

maths equations and rearranging formulae further equations solving linear equations with variable on both sides equations and simple functions

  1. $3x+y$

  2. $x+3y$

  3. $3x-y$

  4. $x-3y$

Show answer
Correct Option: A
Explanation:

Let their present age be $a,b,c$


Sum of their present age $=a+b+c$

Their ages $x$ years ago $a-x,b-x,c-x$

Sum $=a+b+c-3x$

Given $a+b+c-3x=y$

$a+b+c=y+3x=3x+y$


So option $A$ is correct.