Question

15. A bath tub is fitted with three taps. The first two taps, working together, can fill the bath tub in the same time as the time taken by the third tap alone. The second tap fills the bath tub 5 hours faster than the first tap and 4 hours slower than the third tap. Find the time taken by the third tap to fill the tub. (a) 12 hours (b) 10 hours (c) 8 hours (d) 6 hours​

Answer 1

Solution :-

Let us assume that, the first pipe alone can fill the bath tub in x hours.

we have given that, The second tap fills the bath tub 5 hours faster than the first tap and 4 hours slower than the third tap.

So,

→ Second pipe alone can fill the bath tub in = (x - 5) hours .

→ Third pipe alone can fill the bath tub in = (x - 5) - 4 = (x - 9) hours.

Now, we have given that, The first two taps, working together, can fill the bath tub in the same time as the time taken by the third tap alone.

So,

→ Efficiency of First Pipe + Efficiency of second Pipe = Efficiency of third Pipe .

→ (1/x) + (1/x - 5) = 1/(x - 9)

→ (x - 5 + x) / x(x - 5) = 1/(x - 9)

→ (x - 9)(2x - 5) = x(x - 5)

→ 2x² - 5x - 18x + 45 = x² - 5x

→ 2x² - x² - 5x + 5x - 18x + 45 = 0

→ x² - 18x + 45 = 0

→ x² - 15x - 3x + 45 = 0

→ x(x - 15) - 3(x - 15) = 0

→ (x - 15)(x - 3) = 0

→ x = 15 or 3.

Now, We can not take the value x = 3 because than time of third pipe (x - 9) becomes negative which is not possible .

Therefore,

→ The first pipe alone can fill the bath tub in = 15 hours.

Hence,

→ The time taken by the third tap to fill the tub = 15 - 9 = 6 hours. (Ans.)