SOLUTION: A 90,000 gallon water tank can be filled in two hours by opening valve A alone and in two and a half hours by opening valve B alone. It can be emptied in three hours by opening val

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Question 503580: A 90,000 gallon water tank can be filled in two hours by opening valve A alone and in two and a half hours by opening valve B alone. It can be emptied in three hours by opening valve C alone. How long will it take to fill the tank under each of the following conditions? If an answer is not whole hours, include hours, minutes and seconds in your answer.
a. Valve B is kept closed. Valves A and C are opened
b. Valve C is kept closed. Valves A and B are opened
c. Valve A is kept closed. Valves B and C are opened
d. valve C is kept closed. Valve A is opened. Twenty minutes later valve B is also opened
e. Valves A and C are opened. Twenty minutes later valve B is also opened
f. Valve A is opened. Twenty minutes later valves B and C are also opened
g. Valve C is kept closed. Valve A is opened. Valve B is also opened twenty minutes later and is then closed after another ten minutes
h. Valve C is kept closed. Valve A is opened. Twenty minutes later valve B is also opened. Ten minutes after that, Valve A is closed
Found 2 solutions by xdragonfight, Edwin McCravy:
Answer by xdragonfight(116)   (Show Source): You can put this solution on YOUR website!
its G
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
A 90,000 gallon water tank can be filled in two hours by opening valve A alone and in two and a half hours by opening valve B alone. It can be emptied in three hours by opening valve C alone. How long will it take to fill the tank under each of the following conditions? If an answer is not whole hours, include hours, minutes and seconds in your answer.
To avoid so many fractions, let's use minutes:

A tank can be filled in 120 minutes by opening valve A alone and in 150 
minutes by opening valve B alone. It can be emptied in 180 minutes by 
opening valve C alone. 

Make this chart. Fill in 1's for the number of tanks filled, and -1 for C,
since to empty a tank is mathematically the sames as "filling -1 tanks".
Fill in the times required as stated in the problem: 


                               Number of      Time in       Rate in
                               Tanks filled     minutes       tanks/hour
       A only                        1            120         
       B only                        1            150         
       C only                       -1            180        
     


Now fill in the rates in tanks/hour by dividing the number of tanks
filled by the the number of hours


                               Number of      Time in       Rate in
                               Tanks filled     minutes       tanks/hour
       A only                        1            120         1/120
       B only                        1            150         1/150
       C only                       -1            180        -1/180

Now we'll bring in condition "a":  Fill in 1 for the number of tanks filled
x for the number of minutes required and fill in the rate by dividing
number of tanks filled by minutes.

                               Number of      Time in       Rate in
                               Tanks filled     minutes       tanks/hour
       A only                        1            120         1/120
       B only                        1            150         1/150
       C only                       -1            180        -1/180
  a. A&C only                        1             x            1/x                                              

We use

  , answer 360 minutes = 6 hours
--------------------------

Now we'll bring in condition "b":  Fill in 1 for the number of tanks filled
x for the number of minutes required and fill in the rate by dividing
number of tanks filled by minutes.

                               Number of      Time in       Rate in
                               Tanks filled     minutes       tanks/hour
       A only                        1            120         1/120
       B only                        1            150         1/150
       C only                       -1            180        -1/180
  b. A&B only                        1             x            1/x                                              

We use

  , answer 66 2/3 minutes or
1 hour, 6 minutes, 40 seconds  

---------------------

Now we'll bring in condition "c":  Fill in 1 for the number of tanks filled
x for the number of minutes required and fill in the rate by dividing
number of tanks filled by minutes.

                               Number of      Time in       Rate in
                               Tanks filled     minutes       tanks/hour
       A only                        1            120         1/120
       B only                        1            150         1/150
       C only                       -1            180        -1/180
  c. B&C only                        1             x            1/x                                              

We use

  , answer 900 minutes = 15 hours
--------------------------

Now we'll bring in condition "d", which is in two parts.
For the first 20 minutes fill in A's rate, 1/120 and 20 for the time in minutes:
Then fill in the "tanks filled" by multiplying rate by the time, getting 
20/120 or 1/6 of a tank filled.  Then put x for the number of minutes after the
first 20 minutes, and the sum of A's and B's rates 1/120 + 1/150 = 3/200.
Then to find the number of tanks filled we multiple the rate by the time,
and get (3/200)x  

                               Number of      Time in       Rate in
                               Tanks filled     minutes       tanks/hour
       A only                        1            120         1/120
       B only                        1            150         1/150
       C only                       -1            180        -1/1
d1. A only for 20 min              1/6            20          1/120 
d2. A&B only after 20 minutes    (3/200)x          x          3/200

We use

  , answer 500/9 minutes 
= 55 minutes 33 1/3 seconds.
Adding on the first 20 minutes, 75 minutes, 33 1/3 seconds or

1 hour, 15 minutes, 33 1/3 seconds. 


---------------------------

Now we'll bring in condition "e", which is also in two parts.
For the first 20 minutes fill in A&C's rate, 1/120-1/180 = 1/72, and
20 minutes for the time 
Then fill in the "tanks filled" by multiplying rate by the time, getting 
20/120 or 1/6 of a tank filled.  Then put x for the number of minutes after the
first 20 minutes, and the sum of A's,B's, and C's rates 
1/120 + 1/150 - 1/180 = 17/1800.
Then to find the number of tanks filled we multiple the rate by the time,
and get (3/200)x  

                               Number of      Time in       Rate in
                               Tanks filled     minutes       tanks/hour
       A only                        1            120         1/120
       B only                        1            150         1/150
       C only                       -1            180        -1/180
e1. A&C only for 20 minutes        5/18            20         1/72 
e2. A&B&C after 20 minutes    (17/1800)x            x        17/1800  

We use

  , answer 1300/17 minutes 
= 76 minutes 28 4/17 seconds.
Adding on the first 20 minutes, 96 minutes, 28 4/17 seconds or

1 hour, 36 minutes, 28 4/17 seconds. 


---------------------------
---------------------------

Now we'll bring in condition "f", which is also in two parts.
For the first 20 minutes fill in As rate, 1/120, and
20 minutes for the time 
Then fill in the "tanks filled" by multiplying rate by the time, getting 
20/120 or 1/6 of a tank filled.  Then put x for the number of minutes after the
first 20 minutes, and the sum of A's,B's, and C's rates 
1/120 + 1/150 - 1/180 = 17/1800.
Then to find the number of tanks filled we multiple the rate by the time,
and get (3/200)x  

                               Number of      Time in       Rate in
                               Tanks filled     minutes       tanks/hour
       A only                        1            120         1/120
       B only                        1            150         1/150
       C only                       -1            180        -1/180
f1. A for 20 minutes                1/6            20          1/120 
f2. A&B&C after 20 minutes    (17/1800)x            x        17/1800

 We use

  , answer 1500/17 minutes 
= 88 minutes 14 2/17 seconds.
Adding on the first 20 minutes, 108 minutes, 14 2/17 seconds or

1 hour, 48 minutes, 14 2/17 seconds. 


---------------------------
---------------------------

Now we'll bring in condition "g", which is also in three parts.
For the first 20 minutes fill in As rate, 1/120, and
20 minutes for the time 
Then fill in the "tanks filled" by multiplying rate by the time, getting 
20/120 or 1/6 of a tank filled.  
For the next 20 minutes fill in A&B's rate, 1/120+1/150 = 3/200, and
20 minutes for the time 
Then fill in the "tanks filled" by multiplying rate by the time, getting 
60/200 = 3/10 of a tank filled.


Then put x for the number of minutes after the
first 40 minutes, and the sum of A's and B's rates 
1/120 + 1/150 = 3/200.
Then to find the number of tanks filled we multiple the rate by the time,
and get x/120  

                               Number of      Time in       Rate in
                               Tanks filled     minutes       tanks/hour
       A only                        1            120         1/120
       B only                        1            150         1/150
       C only                       -1            180        -1/180
g1. A only for 20 minutes           1/6            20         1/120 
g2. A&B after 20 minutes           3/10            20         3/200
g3. A only after 40 minutes       x/120             x         1/120



f1. A for 20 minutes                1/6            20          1/120 
f2. A&B&C after 20 minutes    (17/1800)x            x        17/1800

 We use

  , 64 minutes
Adding on the first 40 minutes, 104 minutes or

1 hour, 44 minutes. 


---------------------------

You do the last one!

Edwin
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