Question

Arul, Madan and Ram working together can clean a store in 6 hours. Working alone, Madan takes twice as long to clean the store as Arul does. Ram needs three times as long as Arul does. How long would it take each if they are working alone?

Answer 1

Let the time taken by Arul be “x” hours

Let the time taken by Madan be “y” hours

Let the time taken by Ram be “z” hours

By the given first condition

`1/x + 1/y + 1/z = 1/6`

Again by the given second condition

`1/x = 2 xx 1/y`

`1/x - 2/y` = 0

By the given third condition

`3 xx 1/z = 1/x`

`- 1/x + 3/z` = 0

Let `1/x` = a, `1/y` = b, `1/z` = c

a + b + c = `1/6`

6a + 6b + 6c = 1  ...(1)

a – 2b = 0  ...(2)

– a + 3c = 0  ...(3)

(1) × 1   ⇒    6a + 6b + 6c = 1  ....(1)
(2) × 3   ⇒    3a – 6b + 0  =  0  ....(2)  
(1) + (2) ⇒    9a + 6c = 1          ....(4)
(3) × (2) ⇒ – 2a + 6c = 0          ....(3) 
                            (–)    (–)    (–)            
(4) – (3) ⇒  11a = 1 ⇒ a = `1/11`

Substitute the value of a = `1/11` in (2)

`1/11 - 2"b"` = 0

`1/11` = 2b ⇒ b = `1/22`

Substitute the value of a in (3)

`- 1/11 + 3"c"` = 0 ⇒ 3c = `1/11` ⇒  c = `1/33`

But `1/x` = a `1/x = 1/11` x = 1

`1/y` = b `1/y = 1/22` y = 22

`1/z` = c `1/z = 1/33` y = 33

Arul take 11 hours, Madan take 22 hours and Ram takes 33 hours.