ElectricPavlov

x = the number 

5x / ( x+2) = 6       multiply both sides of the equation by (x+2) to get 

5x = 6x+ 12 

x = - 12    

a)   Two numbers 

      x and 6x 

      1/x   + 1/6x = 7/3     <==== given ,        multiply entire  equation by x to get

      1 + 1/6 = 7/3 x 

        7/6 = 7/3 x 

          x = 7/6 * 3/7 = 1/2 

                  the other number is   3

b)    16 m * 9 d  / 10 km =   14.4 man days per km 

            25 km will require      25 * 14.4 = 360 man days         if you have 20 men then 18 days required 

a b^4 = 48 a^2 b 

b^3 = 48 a

b^3 / 48 = a 

Sub into second equation 

  b^6 / 2304 * b^5  =  6 b^9 /110592 * b 

b^11/ 2304 = 6b^10 / 110592 

b = 1/8     then      a = 1/24576

You can use the Law of sines :

sin (70) / 25  =  sin C / 26        then sin C = .97728     C = ARCSIN ( .97728 ) = 77.76 degrees 

Now, you should know there are 180 degrees in the sum of the interior angles of a triangle :

Then angle B = 180 - 70 - 77.76 = 32.236 degrees      the sin of which is .5334 

Using an online  Sigma calculator 

\(\sum_{1}^{∞}\)1/(n(n+1)    =   1 

Original volume 1000 cc      200  is water    ( 1/5 is water) 

    draining 150  removes  1/5 * 150 = 30 cc water   now there is 170 water 

       add 150 water     now there is 320 water in 1000 cc 

          draining 250 removes   320 /1000 * 250 = 80 cc water      Now there is 240 water 

              adding 250 cc water makes 490 water in 1000 cc 

5 workers * x days      <====== are needed  to complete the job 

   Add 1 worker and 15 less days are required : 

6 workers * (x-15) = 5 * x 

6x - 90 = 5x 

x =90 days for 5 workers to complete the job 

so 5 workers * 90 days = 450 worker days required for the job 

450 worker days / w  =  90 - 40  days 

w = workers needed = 450 / 50 =  9 workers needed      or   4 additional workers above the original number of 5 

You can multiply/divide both sides by a  POSITIVE  number.

You can multiply/divide both sides by a NEGATIVE number if you flip the inequality sign.

if a < b and b < c and a < b and c < d and 0 < a < b and 0 < c < d and n is positive or negative then ac <  bd          a^n  b^n