Question

The Bubna dam has four inlets. Through the first three inlets, the dam can be filled in 12
minutes; through the second, the third and the fourth inlet, it can be filled in 15 minutes, and
through the first and the fourth inlet, in 20 minutes. How much time (in minutes) will it take all
the four inlets to fill up the dam ?​

Answer 1

Given:

Through the first three inlets, the dam can be filled in 12  minutes

Through the second, the third and the fourth inlet, dam can be filled in 15 minutes

Through the first and the fourth inlet, in 20 minutes

To find:

Time taken by all four inlets to fill the dam

Solution:

Let the four inlets be denoted as "A", "B", "C" & "D".

Therefore,

In 1 minute, the fraction of dam filled by the inlets A, B & C will be,

$\bold{\frac{1}{A} +\frac{1}{B} +\frac{1}{C} = \frac{1}{12}}$ ....... (i)

In 1 minute, the fraction of dam filled by the inlets B, C & D will be,

$\bold{\frac{1}{B} +\frac{1}{C} +\frac{1}{D} = \frac{1}{15}}$ ....... (ii)

In 1 minute, the fraction of dam filled by the inlets A & D will be,

$\bold{\frac{1}{A} + \frac{1}{D} = \frac{1}{20}}$ ....... (iii)

Now, on adding equations (i), (ii) & (iii), we get

$[\frac{1}{A} +\frac{1}{B} +\frac{1}{C}]+[ \frac{1}{B} +\frac{1}{C} +\frac{1}{D}]+[ \frac{1}{A} + \frac{1}{D}] = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}$

⇒ $2\times[\frac{1}{A} +\frac{1}{B} +\frac{1}{C}+ \frac{1}{D}] = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}$

⇒ $2\times[\frac{1}{A} +\frac{1}{B} +\frac{1}{C}+ \frac{1}{D}] = \frac{5\:+\:4\:+\:3}{60}$

⇒ $2\times[\frac{1}{A} +\frac{1}{B} +\frac{1}{C}+ \frac{1}{D}] = \frac{12}{60}$

⇒ $[\frac{1}{A} +\frac{1}{B} +\frac{1}{C}+ \frac{1}{D}] = \frac{12}{120}$

⇒ $\bold{[\frac{1}{A} +\frac{1}{B} +\frac{1}{C}+ \frac{1}{D}] = \frac{1}{10}}$ ←  the fraction of dam filled by the inlets A, B, C & D in 1 minute

Thus, to fill up the dam all the 4 inlets will take 10 minutes.

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Answer 2

Answer:

10 mins

Step-by-step explanation:

Let the time taken by the inlets individually be i1 ,i2 ,i3 ,i4 .

Then in a minute each of them can fill :- 1/i1 ,1/i2 ,1/i3 ,1/i4 .

we have to find    1/ (1/t1 +1/t2 +1/t3 +1/t4)

formula used :- time * (time taken to complete 1 unit of work individually) =1 [1 = 1 unit = 100%]

Equations:-

1.      12 * ( 1/i1 +1/i2 +1/i3 ) = 1

2.     15 * ( 1/i2 +1/i3 +1/i4 ) = 1

3.     20 *( 1/i1 +1/i4 ) =1

4.    12/i1  = 1 - ( 12/i2 + 12/i3 ).......*5

5.    15/i4 = 1 - ( 15/i2 + 15/i3 ).......*4

add 4. and 5.

60 * ( 1/i1 + 1/i4 ) = 9 - 120 * ( 1/i2 +1/i3 )

60/20 = 9 - 120 * ( 1/i2 +1/i3 )

1/i2 + 1/i3 =1/20

Thus,

1/i1 + 1/i2 +1/i3 +1/i4 = 1/20 + 1/20 =2/20 =1/10

time =10 mins