Question 16595
Question #1.

If details 1 and 2 (D1 + D2) together can load the plane in 6 hours, then they can load 1/6 of the plane in 1 hour.

But, since D2 takes 2 hours longer to load than D1, then in 1 hour, working together, their loading rates would be:

{{{1/t + 1/(t+2)}}} where t is the loading time in hours, and this is equal to 1/6.

So we have:

{{{1/t + 1/(t+2) = 1/6}}} Simplify and solve for t, the loading time in hours.

{{{((t+2) + t)/t(t+2) = 1/6}}} 
{{{(2t+2)/(t^2+2t) = 1/6}}} Multiply both sides by {{{(t^2+2t)}}}
{{{2t+2 = (t^2+2t)/6}}} Multiply both sides by 6.
{{{12t+12 = t^2+2t}}} Put this into standard quadratic form.
{{{t^2-10t-12 = 0}}} Solve for t using the quadratic formula: {{{t=(-b+-sqrt(b^2-4ac))/2a}}}
{{{t=(-(-10)+-sqrt((-10)^2-4(1)(-12)))/2(1)}}}
{{{t=(10+-sqrt(148))/2}}}
{{{t=5+-sqrt(37)}}}
{{{t = 5+6.1}}} or {{{t = 5-6.1}}} Discard the negative result.

t = 11.1 hours This is the time taken by detail 1 to load the C-130
t+2 = 13.1 hours This is the time taken by detail 2 to load the C-130.

Check:
In 1 hour, detail 1 can load 1/11 of the plane.
In 1 hour, detail 2 can load 1/13 of the plane.

Together, they can load 1/11 + 1/13 = 24/143 of the plane.

It will take them 143/24 hours to load the plane together
143/24 hours= 5.958 hours This is approximately 6 hours. It's not exactly 6 because we had to take the square root of 37 resulting in an irrational number.  

Problem #2.

If it takes the cranberry juice pipe 2 hours to fill the tank, then it can fill 1/2 ths tank in 1 hour.
If it takes the apple juice pipe 3 hours to fill the tank, then it can fill 1/3 of the tank in 1 hour.

So, together,the pipes can fill (1/2 + 1/3 = 5/6) of the tank in 1 hour.
Therefore, it would take them 6/5 hours to completely fill the tank working together.
6/5 hours = 1 hour and 12 minutes.

Problem #3.

Let n be the number.

{{{(2n+1)/((n/2)-1) = 5}}} Simplify.
{{{(2n+1)/((n-2)/2) = 5}}}
{{{2(2n+1)/(n-2) = 5}}} Multiply both sides by (n-2)
{{{4n+2 = 5n-10}}} Subtract 4n from both sides.
{{{2 = n-10}}} Add 10 to both sides.
{{{12 = n}}}

The number is 12.