Problem On Time & Work –
Example.1) A can do a piece of work in x days and B can do it in (x + 16) days. If both working together can do it in 15 days, calculate x.
1 1
Ans.) A’s 1 day’s work = -------- and B’s 1 day’s work = -----------
x (x + 16)
1 1
(A + B)’s 1 day’s work = [ -------- + ----------]
x (x + 16)
But, it is being given that both working together can do the work in 15 days
1
So, (A + B)’s 1 day’s work = -------
15
1 1 1
So, [-------- + ----------] = -------
x (x + 16) 15
(x + 16) + x 1
=> [---------------] = -------
x (x + 16) 15
=> 15 (2x + 16) = x² + 16x
=> 30x + 240 = x² + 16x
=> x² + 16x – 30x – 240 = 0
=> x² - 14x - 240 = 0
=> x² - (24 – 10)x – 240 = 0
=> x² - 24x + 10x – 240 = 0
=> x (x – 24) + 10 (x – 24) = 0
=> (x – 24)(x + 10) = 0
=> (x – 24) = 0 or (x + 10) = 0
=> x = 24 or x = - 10
=> x = 24 [x ≠ - 10, since number of days cannot be negative]
So, the required working days is 24 (Ans.)
Example.2) One pipe can fill a cistern in 3 hours less than the other. The two pipes together can fill it in 6 hours 40 minutes. Find the time that each pipe will take to fill the cistern.
Ans.) Let the pipes A & B take x hrs. and (x + 3) hrs. respectively to fill the cistern.
1
Part of the cistern filled by A in 1 hr = -------
x
1
Part of the cistern filled by B in 1 hr = ---------
(x + 3)
1 1
Part of the cistern filled by (A + B) in 1 hr = [------- + ---------]
x (x + 3)
Total time taken by both to fill the cistern = 6 hours 40 minutes
40
= 6 --------
60
2 20
= 6 ------ = -------- hrs.
3 3
1 3
Part of the cistern filled by both in 1 hour = -------- = -------
20/3 20
So, as per the given condition –
1 1 3
--------- + ---------- = --------
x (x + 3) 20
x + 3 + x 3
=> -------------- = -------
x (x + 3) 20
=> 20 (2x + 3) = 3 (x² + 3x)
=> 40x + 60 = 3x² + 9x
=> 3x² + 9x – 40x – 60 = 0
=> 3x² - 31x – 60 = 0
=> 3x² - (36 – 5)x – 60 = 0
=> 3x² - 36x + 5x – 60 = 0
=> 3x (x – 12) + 5 (x – 12) = 0
=> (x – 12) (3x + 5) = 0
=> (x – 12) = 0 or (3x + 5) = 0
=> x = 12 or x = - 5/3
=> x = 12 [ x ≠ - 5/3, since time cannot be negative]
Hence, the first pipe alone takes 12 hours to fill the cistern, while second alone takes (x + 3) = (12 + 3) = 15 hours to fill it. (Ans.)
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