Question

Two people can paint a house in $14$ hours. Working individually, one of the people takes $2$ hours more than it takes the other person to paint the house. How much time will each person take while working individually to paint the house ? (approx.)
Choose the correct option ,
A) $29$ hrs
B) $30$ hrs
C) $28$ hrs
D) None of these

Answer 1

Hint: For solving the particular problem we have to let that it will take $x$ hrs long when one person is working individually to paint the house . and must consider that Working individually, one of the people takes $2$ hours more than it takes the other person to paint the house.

Complete step-by-step solution:
It is given that ,
Two people can paint a house together in $14$ hours.
It will take $x$ hrs long when one person is working individually to paint the house .
And one of the people takes $2$ hours more than it takes the other person to paint the house when they work individually.
According to the given statement . we can write ,
$ \Rightarrow \dfrac{1}{x} + \dfrac{1}{{x + 2}}$
According to the question ,
$ \Rightarrow \dfrac{1}{x} + \dfrac{1}{{x + 2}} = \dfrac{1}{{14}}$
$
   \Rightarrow \dfrac{{2x + 2}}{{{x^2} + 2x}} = \dfrac{1}{{14}} \\
   \Rightarrow 28x + 28 = {x^2} + 2x \\
   \Rightarrow {x^2} - 26x - 28 = 0 \\
 $
For finding roots of the original equation, we have to use quadratic formula i.e.,
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Now identify $a,b,c$ from the original equation given below,
$ \Rightarrow {x^2} - 26x - 28 = 0$
Now compare the coefficient ,
$
  a = 1, \\
  b = - 26, \\
  c = - 28. \\
 $
Put these values into the formula of finding the roots of quadratic equations,
$
   \Rightarrow \dfrac{{ - 26 \pm \sqrt {{{(26)}^2} - 4 \times 1 \times ( - 28)} }}{{2 \times 1}} \\
   \Rightarrow \dfrac{{26 \pm \sqrt {788} }}{2} \\
   \Rightarrow \dfrac{{26 \pm 28.07}}{2} \\
 $
After simplifying and by evaluating exponents and square root of the above equation we get the following simplified expression,
$x = \dfrac{{26 \pm 28.07}}{2}$
To find the roots of the equations , separate the particular equation into its corresponding parts : one part with the plus sign and the other with the minus sign.
$
  {x_1} = \dfrac{{26 + 28.07}}{2} \\
  {x_2} = \dfrac{{26 - 28.07}}{2} \\
 $
Simplify and then isolate the variable to find its corresponding solutions!
$
  {x_1} = 27.03 \approx 27 \\
  {x_2} = - 1.03 \\
 $
Since $x$ cannot be negative, therefore rejecting the negative value and accepting the other value that is $27$ .
Now, working individually to paint the house person takes $27 + 2 = 29$ hrs.

Note: For finding roots of the original equation, we have to use quadratic formula i.e.,
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
And have to identify $a,b,c$ from the original equation .
We have to reject the negative value. since time cannot be negative.