CPhill

See if this might help you :  https://www.quora.com/A-rod-is-broken-into-three-parts-what-is-the-probability-that-the-three-parts-can-be-arranged-to-form-a-triangle

[ I think that Melody actually may have answered this one before ]

We have

10000  = P  ( 1 + .06)^5       divide both sides by (1 + .06)^5

10000 / (1 + .06)^5   =  P  ≈  $7472.58

sqrt (4x^2) - sqrt (x^2)  = 6       we can write

2sqrt (x^2)  - sqrt (x^2)  =  6

sqrt (x^2) = 6     square both sides

x^2  =  36      take both roots

x = ± 6

LOL!!!...it would be hard to intuit this if some work weren't done  !!!!

Multiply the  first equation through by -2  and we have

-2x - 2y - 6z  = -20       add this to the second equation  and we get

-6x - z  = -13    multiply this through by  - 1

6x + z  =  13      (1)

So..notice that if k  = 6 we have

6x + z  = 3   (2)

But  (1)  and (2)  are parallel lines that never intersect.....so no solution is possible when  k = 6

This may be a little difficult to solve algebraically  [ although there may be a way to do so ]

Look at the graph here :  https://www.desmos.com/calculator/oznsn0uwpe

Because of the presence of the xy term,  we have a rotated conic (ellipse??)  

We are looking for all the lattice points  (points with integer coordinates ) that this graph passes through

They are the  same ones I enumerated in my previous answer.......

We don't really  need any proportions here

1.

Cost of book  +  5% * cost of book  = final price

25  +  .05 (25)  =   $26.25

2.

Amt borrowed  +  8.5% * Amt borrowed  = Amt Paid Back

1500 +  .085 (1500)   = $1627.50

All of the members of our spray-painting team paint at the same speed. If 12 team members can paint a 2400 square foot wall in 45 minutes, then how many minutes would it take for 9 team members to paint a 3600 square foot wall

Each minute   ...8 workers can paint       2400 /  [ 45  ]  = 160 / 3  sq ft /min

So....1 worker can paint 1/8 of this in one minute  = 160 /3 *  (1/8) =    20/3 sq ft / min

So....9  workers can paint    9 * (20 / 3)  =   180 / 3  =   60 sq ft / min

So....to paint  3600 square ft will take the 9 workers       3600 / 60   =  60 min 

All of the members of our spray-painting team paint at the same speed. If 8 team members can paint a 5000 square foot wall in 30 minutes, then how many minutes would it take the 8 members to paint an 8000 square foot wall?

Each minute.....      5000/ 30   =  500/ 3 sq ft is being painted  by the 8 people

So...the number of minutes it would take them to paint 8000sq ft  =

8000 / [ 500/3]   =  24000/ 500  =  48 min 

Another way to see this is  that    5/8  of the work is done in 30 min

So

30 /  (5/8)   =    30 * 8  /  5  =    240  / 5   =   48 min

All of the members of our spray-painting team paint at the same speed. If 5 team members can paint a certain wall in 27 minutes, then how many minutes would it take 15 of our team members to paint the same wall?

We have  three times the labor  with 15 team members as with 5.....so...it should take 1/3 as long  = 9 minutes

Mathematically

5 * 27  =  x * 15     divide both sides by 15

5 * 27 / 15   =  x

5 * 9 / 5   =    x   =    9 (mnutes)

Tina lives 30 miles away from her work, and she normally drives at 45 miles per hour. If she wants to get to work 10 minutes earlier, how fast should she drive to work, in miles per hour?

Distance / Rate   =  Time   ...so.....

30 mi / 45 miles per hour   =   2/3 hrs to get to work  =  40 minutes

So...10 minutes less is 30 minutes...so

Distance / Time  = Rate

30 mi  / 30 min   =   30 mi / (1/2) hr  =   60 mi/ hr