Question

to do a piece of work Aryan takes 3 times as long as Akash and Amar together and Amar twice as long as Akash and Aryan together. If the three together can complete the work in 10 days, how long would each take by his/her self?

Can someone please answer this question?

Answer 1

Question is solved on sheet attached.

Answer 2

Let Aryan can complete the whole work by himself in x days.
Akash can complete the whole work by himself in y days
Amar can complete the whole work by himself in z days.
.
In 1 day Aryan can do $\frac{1}{x}$ part
In 1 day Akash can do $\frac{1}{y}$ part
In 1 day Amar can do $\frac{1}{z}$ part

 
In 1 day Akash and Amar together can complete$( \frac{1}{y}+ \frac{1}{z}) = \frac{z+y}{yz}$ part of the work.
Therefore, Akash and Amar together can complete whole work in $\frac{yz}{z+y}$


In 1 day Akash and Aryan together can complete$( \frac{1}{y}+ \frac{1}{x}) = \frac{x+y}{xy}$ part of the work.
Therefore, Akash and Aryan together can complete whole work in $\frac{xy}{x+y}$


In 1 day Aryan, Akash and Amar together can complete$( \frac{1}{x}+ \frac{1}{y}+ \frac{1}{z} )= \frac{yz+xz+xy}{xyz}$
Therefore, Aryan, Akash and Aryan together can complete whole work in $\frac{xyz}{yz+xz+xy}$


According to question:
Aryan = 3 times (Akash and Amar together)
x=$3*\frac{yz}{z+y}$
Multiplying both sides by (z+y):
[tex]x(z+y)=3yz \\ xz+xy=3yz[/tex].......(i)


Amar = 2 times ( Akash and Aryan together)
z=$2* \frac{xy}{x+y}$
Multiplying both sides by (x+y):
[tex]z(x+y)=2xy \\ xz+yz=2xy[/tex].......(ii)


Aryan, Akash and Amar together= 10 days
$\frac{xyz}{yz+xz+xy} =10$
Multiply both sides by (yz+xz+xy):
$xyz=10(yz+xz+xy)$......(iii)


Plug in xz+xy=3yz from (i) into (iii):
xyz=10(yz+xz+xy)
xyz=10(yz+3yz)
xyz= 10(4yz)
xyz=40yz
Dividing both sides by yz:
x=40



Plug in xz+yz=2xy from (ii) into (iii):
xyz=10(yz+xz+xy)
xyz=10(2xy+xy)
xyz= 10(3xy)
xyz=30xy
Dividing both sides by xy:
z=30


Plug in x=40 and z=30 in (i):
xz+xy=3yz
(40)(30)+(40)y=3y(30)
1200+40y=90y
Subtract 40y from both sides:
1200=50y
Divide both sides by 50:
24=y
y=24


Answer :
Aryan  =  x days = 40 days
Akash  =  y days = 24 days
Amar    =  z days = 30 days