Question

Together, Juanita and Mark can make preparations to cater a dinner in 6 hours. Alone, Juanita can do the job 2 hours faster than Mark. Find the time that each person takes to prepare to cater a meal.

Answer 1

Answer:  

Time taken by Juanita and mark to cater meals is 11.08 hours or 11 hours 4 minutes and 13.08 hours or 13 hours 4 minutes respectively.

Solution:

Given, Together, Juanita and Mark can make preparations to cater a dinner in 6 hours.  

Alone, Juanita can do the job 2 hours faster than Mark.  

We need to find the time that each person takes to prepare to cater a meal.

Let the time taken by Juanita be x hours.

Then time taken by mark becomes x + 2 hours.

Now, work done by Juanita in 1 hour = $\frac{1}{x}$ [by assuming total work done as 1 unit]

And, work done by mark in 1 hour = $\frac{1}{(x+2)}$

Together, in one hour they can complete  

$\frac{1}{x} + \frac{1}{(x+2)}$ work, which equals to $\frac{1}{6}$

[ as given that together they completed in 6 hours ]

$\begin{array}{l}{\frac{1}{x}+\frac{1}{x+2}=\frac{1}{6}} \\\\ {\frac{x+2+x}{x(x+2)}=\frac{1}{6}} \\\\ {6(2 x+2)=x^{2}+2 x} \\\\ {12 x+12=x^{2}+2 x} \\\\ {x^{2}+2 x-12 x-12=0} \\\\ {x^{2}-10 x-12=0}\end{array}$

Now, by using quadratic formula,

$\begin{array}{l}{x=\frac{-(-10) \pm \sqrt{(-10)^{2}-4 \times 1 \times(-12)}}{2 \times 1}} \\\\ {x=\frac{10 \pm \sqrt{100+48}}{2}} \\\\ {x=\frac{10+\sqrt{148}}{2}, \frac{10-\sqrt{148}}{2}} \\\\ {x=\frac{10+2 \sqrt{37}}{2^}}, \frac{10-2 \sqrt{37}}{2}}\end{array}$

$x=5+\sqrt{37}, 5-\sqrt{37}$

We can neglect $5-\sqrt{37}$ as time cannot be negative.

$x = 5+\sqrt{37}=5+6.0827$

= 11.08 approximately

So, Juanita completes work alone in 11.08 hours or 11 hours 4 minutes approximately.

So, mark completes work in 11.08 + 2 = 13.08 hours or 13 hours 4 minutes approximately.

Hence, time taken by Juanita and mark to cater meals is 11.08 hours or 11 hours 4 minutes and 13.08 hours or 13 hours 4 minutes respectively.

Answer 2

If it takes them 6 hours to do it together, then Juanita can do it in 2 hours, and Mark can do it in 4 hours (since 2 + 4 = 6).