Answer: The time taken by Jennifer to complete this job on her own is 6 hours.
Step-by-step explanation:
Let x be the hour taken by Jennifer in the entire work when she works alone,
⇒ The time taken by Carl when he works alone = (x - 3) hours ( Because, Carl can paint a room 3 hours faster than Jennifer can ),
The Work done by Jennifer in one day = \(\frac{1}{x}\)
While, the The Work done by Carl in one day = \(\frac{1}{x-3}\)
The total work done by both in one day = \(\frac{1}{x}+\frac{1}{x-3}\)
According to the question,
If they work together, they can complete the job in 2 hours.
⇒ The total work done by both in one day when they work together = \(\frac{1}{2}\)
\(\implies \frac{1}{x}+\frac{1}{x-3} = \frac{1}{2}\)
\(\frac{x-3+x}{x^2-3x}=\frac{1}{2}\)
\(\frac{2x-3}{x^2-3x}=\frac{1}{2}\)
\(4x-6=x^2-3x\)
\(x^2-7x+6=0\)
\(x^2-6x-x+6=0\)
\(x(x-6)-1(x-6)=0\)
\((x-1)(x-6)=0\)
If x-1 = 0 ⇒ x = 1 ( But the time taken by carl = 1-3 = -2, which can not be possible ),
If x-6=0 ⇒ x = 6,
Hence, the time taken by Jennifer to complete this job on her own = 6 hours.
