Answer
Verified
479.7k+ views
Hint:
Consider the time taken by B as ‘x’. A takes 6 days less than B, so time taken by A is (x - 6). So work done by \[B=\dfrac{1}{x}\]and \[A=\dfrac{1}{x-6}\]. Total work done by (A + B) is \[\dfrac{1}{4}\]. Solve the equation formed to get the time taken by B to finish the work.
Complete step-by-step answer:
Suppose B alone takes x days to finish the work. Then A alone can finish the work in (x - 6) days.
Now we know A’s one day work\[=\dfrac{1}{x-6}\].
B’s work per day\[=\dfrac{1}{x}\].
The total work done by A and B in one day\[=\dfrac{1}{x}+\dfrac{1}{x-6}-(1)\]
\[A+B=\dfrac{1}{x}+\dfrac{1}{x-6}\].
It’s said that A and B can finish a work together in 4 days.
\[\therefore A+B=\dfrac{1}{4}-(2)\]
Now, Equating equation (1) and equation (2),
\[\dfrac{1}{x}+\dfrac{1}{x-6}=\dfrac{1}{4}\]
Simplifying the LHS, \[\dfrac{\left( x-6 \right)+x}{x\left( x-6 \right)}=\dfrac{1}{4}\]
\[\Rightarrow \dfrac{2x-6}{x\left( x-6 \right)}=\dfrac{1}{4}\]
Now cross multiplying them, then the equation becomes,
\[4\left( 2x-6 \right)=x\left( x-6 \right)\]
Opening the brackets and simplifying them,
\[\begin{align}
& 8x-24={{x}^{2}}-6x \\
& \Rightarrow {{x}^{2}}-6x-8x+24=0 \\
& {{x}^{2}}-x\left( 6+8 \right)+24=0 \\
& {{x}^{2}}-14x+24=0-(3) \\
\end{align}\]
Equation (3) is similar to the general quadratic equation, \[a{{x}^{2}}+bx+c=0\].
Comparing both general equation and equation (3), we get the values of constants a = 1, b = -14, c = 24.
Now, applying the above value in the quadratic formula,
\[\begin{align}
& \dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}=\dfrac{-\left( -14 \right)\pm \sqrt{{{\left( -14 \right)}^{2}}-4\times 1\times 24}}{2\times 1} \\
& =\dfrac{14\pm \sqrt{196-96}}{2}=\dfrac{14\pm \sqrt{100}}{2}=\dfrac{14\pm 10}{2} \\
\end{align}\]
The roots are \[\left( \dfrac{14+10}{2} \right)\]and \[\left( \dfrac{14-10}{2} \right)\]= 12 and 2.
The value of x cannot be less than 6.
If B takes the time of 2 days, then the time taken by A becomes, \[x-6=2-6=-4\]days.
i.e. time taken cannot be negative.
\[\therefore x=12\].
i.e. It would take B a period of 12 days to finish the job.
Note:
If we were asked to find the time taken by A to finish the work alone then we got x=12 i.e. the time taken by B alone to do a work. So, time taken by A is 12 – 6 = 6 days. As A takes 6 days less than the time taken by B. If we are adding time taken by (A + B) together\[=\dfrac{1}{6}+\dfrac{1}{12}=\dfrac{12+6}{12\times 6}=\dfrac{1}{4}\].
Consider the time taken by B as ‘x’. A takes 6 days less than B, so time taken by A is (x - 6). So work done by \[B=\dfrac{1}{x}\]and \[A=\dfrac{1}{x-6}\]. Total work done by (A + B) is \[\dfrac{1}{4}\]. Solve the equation formed to get the time taken by B to finish the work.
Complete step-by-step answer:
Suppose B alone takes x days to finish the work. Then A alone can finish the work in (x - 6) days.
Now we know A’s one day work\[=\dfrac{1}{x-6}\].
B’s work per day\[=\dfrac{1}{x}\].
The total work done by A and B in one day\[=\dfrac{1}{x}+\dfrac{1}{x-6}-(1)\]
\[A+B=\dfrac{1}{x}+\dfrac{1}{x-6}\].
It’s said that A and B can finish a work together in 4 days.
\[\therefore A+B=\dfrac{1}{4}-(2)\]
Now, Equating equation (1) and equation (2),
\[\dfrac{1}{x}+\dfrac{1}{x-6}=\dfrac{1}{4}\]
Simplifying the LHS, \[\dfrac{\left( x-6 \right)+x}{x\left( x-6 \right)}=\dfrac{1}{4}\]
\[\Rightarrow \dfrac{2x-6}{x\left( x-6 \right)}=\dfrac{1}{4}\]
Now cross multiplying them, then the equation becomes,
\[4\left( 2x-6 \right)=x\left( x-6 \right)\]
Opening the brackets and simplifying them,
\[\begin{align}
& 8x-24={{x}^{2}}-6x \\
& \Rightarrow {{x}^{2}}-6x-8x+24=0 \\
& {{x}^{2}}-x\left( 6+8 \right)+24=0 \\
& {{x}^{2}}-14x+24=0-(3) \\
\end{align}\]
Equation (3) is similar to the general quadratic equation, \[a{{x}^{2}}+bx+c=0\].
Comparing both general equation and equation (3), we get the values of constants a = 1, b = -14, c = 24.
Now, applying the above value in the quadratic formula,
\[\begin{align}
& \dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}=\dfrac{-\left( -14 \right)\pm \sqrt{{{\left( -14 \right)}^{2}}-4\times 1\times 24}}{2\times 1} \\
& =\dfrac{14\pm \sqrt{196-96}}{2}=\dfrac{14\pm \sqrt{100}}{2}=\dfrac{14\pm 10}{2} \\
\end{align}\]
The roots are \[\left( \dfrac{14+10}{2} \right)\]and \[\left( \dfrac{14-10}{2} \right)\]= 12 and 2.
The value of x cannot be less than 6.
If B takes the time of 2 days, then the time taken by A becomes, \[x-6=2-6=-4\]days.
i.e. time taken cannot be negative.
\[\therefore x=12\].
i.e. It would take B a period of 12 days to finish the job.
Note:
If we were asked to find the time taken by A to finish the work alone then we got x=12 i.e. the time taken by B alone to do a work. So, time taken by A is 12 – 6 = 6 days. As A takes 6 days less than the time taken by B. If we are adding time taken by (A + B) together\[=\dfrac{1}{6}+\dfrac{1}{12}=\dfrac{12+6}{12\times 6}=\dfrac{1}{4}\].
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Change the following sentences into negative and interrogative class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE