SOLUTION: A and B do a work together in five days If A do the work with double speed and B with half the speed then they do the work in 4 days together find A alone do the work in how much t

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Question 672824: A and B do a work together in five days If A do the work with double speed and B with half the speed then they do the work in 4 days together find A alone do the work in how much time.
Found 2 solutions by ankor@dixie-net.com, Edwin McCravy:
Answer by ankor@dixie-net.com(22740)   (Show Source): You can put this solution on YOUR website!
A and B do a work together in five days
If A do the work with double speed and
B with half the speed then they do the work in 4 days together
find A alone do the work in how much time.
:
let a = A's time alone
let b = B's time alone
then
.5a = A's time when working twice as fast
and
2b = B's time when working half as fast
:
Let the completed job = 1
:
Write a shared work equation for each scenario
:
+ = 1
get rid of the denominators, mult by ab
5b + 5a = ab
and
+ = 1
multiply by 2ab to get rid of the denominators
4b(4) + 4a = 2ab
16b + 4a = 2ab
:
Use elimination here, mult the 1st equation by 4, mult the 2nd eq by 5
then subtract the 1st from the 2nd
80b + 20a = 10ab
20b + 20a = 4ab
------------------subtraction eliminates a
60b + 0 = 6ab
divide both sides by b
60 = 6a
a = 60/6
a = 10 days working alone
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
A and B do a work together in five days If A do the work with double speed and B with half the speed then they do the work in 4 days together find A alone do the work in how much time.

Here's a slightly different approach, where we add their rates in
jobs per day working alone to get their combined rate working together. 
We let x = A's time in days working alone and y = B's time in days
working alone.  In each case 1 job is done.  Double speed means half
the time and half speed means twice the time.  We get the rates in
jobs per day by dividing jobs by days: 


                                      Jobs       Time    Rate in  
                                      done     in days   Jobs/day
A working alone at regular speed        1          x        1/x
B working alone at regular speed        1          y        1/y
A working alone at double speed         1         x/2       1/(x/2) = 2/x
B working alone at half-speed           1         2y        1/(2y)
A&B together at regular speed           1          5        1/5   
A at double speed & B at half speed     1          4        1/4

 The two equations come from

              +  = 

                        +  = 

and 

               +  =     
 

                        +  = 

The system is:

                        +  = 
                        +  = 

A slightly different way of dealing with systems with the variables
in the denominators is to eliminate WITHOUT clearing of fractions to
avoid getting xy terms (or ab terms in the other tutor's solution).


Multiply the first equation by -2

                        -  = 
                        +  = 

Add the equations and get:

                        +   =  +            

Now we clear of fractions.  Multiply through by 20y
                       
                       -40 + 10 = -8y + 5y
                            -30 = -3y
                             10 = y

Substitute in

                        +  = 
                        +  =  

Multiply through by 10x

                         10 + x = 2x
                             10 = x     


So it takes them each 10 days working alone.

When the variables are in the denominators it's better to
eliminate WITHOUT clearing of fractions, and wait till
after eliminating to clear of fractions:

Edwin
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