Filling a swimming pool

We need to know the rate at which each hose delivers water to the pool. Let $V$ be the volume of water in the pool, $f_l$ be the flow rate of water through the large hose, and $f_s$ be the rate through the small one. We'll use the equation of rate:

$Flowrate=\frac{Volume}{Time} \Rightarrow f = \frac{V}{t}$

If 5 hoses deliver $V$ in 2 hours, then:

$5\times f_l = \frac{V}{2} \Rightarrow \boxed{f_l = \frac{V}{10}}$

By similar reasoning, with 3 hoses delivering $V$ in 5 hours,

$3\times f_s = \frac{V}{5} \Rightarrow \boxed{f_l = \frac{V}{15}}$

So, with 2 large hoses and 1 small hose, the total fill rate ( $f_T$ ) is

$f_T = 2f_l+f_s=2 \frac{V}{10} + \frac{V}{15} = \frac{3V+V}{15} = \frac{4V}{15}$

Then, the time taken for filling the pool is

$t = \frac{V}{f_T}=V\times \frac{15}{4V}=\frac{15}{4} h$ ; that is, $3\frac{3}{4}$ hours or $\boxed{225 min}$

5 large hoses take 2 hours so 1 large hose will take 5 times more time than this which is equal to 120 *5=600 minutes so two hoses will take half of this time i.e. 300 minutes. 3 small hoses takes 5 hours so 1 small hose will take 3 times more time than this = 300 * 3= 900 minutes now we want to know how much time 2 large hoses and 1 small hose will take together so,

1/300 + 1/900 = 4/900 => 900/4 minutes = 225 minutes

1 large hose takes 2x5=10 hours to fill the pool

1 small hose takes 3x5=15 hours to fill the pool

Therefore 1 large hose is equivalent to 1.5 small hoses

2 large hoses plus 1 small hose is equivalent to 4 small hoses so will take 15/4 hours to fill the pool

=3.75hours or 225minutes.

Let the volume of the swimming pool be V .

Given that 5 identical large hoses fill the swimming pool in 2 hours (120 minutes), thus the rate of 5 identical large hoses filling the swimming pool can be written as ${ k }_{ L }=\frac { V }{ 120 }$ .

Given also that 3 identical small hoses fill the swimming pool in 5 hours (300 minutes), the rate of 3 identical small hoses filling the same swimming pool can be written as ${ k }_{ S }=\frac { V }{ 300 }$ .

Since we are finding how much time will it take to fill the swimming pool when using 2 large hoses and 1 small hose, we have,

$\frac { 2 }{ 5 } { k }_{ L }+\frac { 1 }{ 3 } { k }_{ S }=\frac { V }{ x }$

where x is the time taken when 2 large hoses and 1 small hose is used to fill up the swimming pool.

Therefore,

$\frac { 2 }{ 5 } (\frac { V }{ 120 } )+\frac { 1 }{ 3 } (\frac { V }{ 300 } )=\frac { V }{ x }$

Simplifying everything,

$\frac { 4V }{ 900 } =\frac { V }{ x}$

Gives, $x=\frac { 900 }{ 4 } =\boxed{225\quad minutes}$ .

For understanding what's going on , let be the volume of the swimming pool $1 m^{3}$

so if it takes 5 huge tubes T and 2 hours to full the swimming pool , $T = \frac{m^{3}}{10hour}$ in facts T*2 hours * 5 tubes = $1 m^{3}$

and if it takes 3 small tubes t and 5 hours to full the same swimming pool, $t = \frac{m^{3}}{15hour}$ in facts t*5 hours * 3 tubes = $1 m^{3}$

Than $2T + t = \frac{4 m^{3}}{15 hour}$ so it takes 15/4 hours to full a $1 m^{3}$ swimming pool.

15/4 hours is 3 hours and 3/4*60 minutes , or 225 minutes.

2 large hoses fills pool in 5 hrs. (300 min.) 1 small hose fills pool in 15 hrs. (900 min.) With 2 large and 1 small during 900 min they fill 4 pools; so, 1 pool is filled in 225 min.

2 large hose = 1/5 per 60min 1 small hose = 1/15 per 60min

1/5x + 1/15x = 60min

3/15x + 1/15x = 60min

4/15x = 60min

x = 60min * 15 / 4

x = 225min

Say that the swimming pool is 1000 liters because 1000 can be divided with 2 and 5. Large hoses is marked with x and small hoses is marked with y.

5x=5x*2=1000

=1000/2

x=500/5

=100

3y=3y*5=1000

=1000/5

y=200/3

y=66.66

1000/(2x+y)

=1000/(200+66.66)

=1000/266.66

=3.75 hours

=3 hours 45 minutes

The easy way:

First, let us assume that swimming pool capacity is 30 kilo litres (because 30 is a multiple of 2, 3 and 5)

Now: It takes 2 hours for 5 large hoses to fill 30 kl, that means 6 kl by one hose in 2 hours which gives $\boxed{3 kl per hose per hour}$ Similarly, 3 small hoses in 5 hours gives us $\boxed{2 kl per hose per hour}$

We have 2 large hoses and 1 small hose, so we can fill 8 kl per hour ( $2 \times 3 + 1 \times 2$ ). In order to fill 30 kl. time taken = $\frac{30}{8}$ = 3 hours 45 minutes = $\boxed{225 minutes}$