Simultaneous equations question

Question

There is a pond with 10m³ of water. Every minute, xm³ of water is filled into the pond and the excess water just flows into a river. Every minute, using two pumps, ym³ of water for each pump is sucked up. The pond can be emptied in 30 mins. Using four pumps, the pond can be emptied in 12 mins.

Find out the values of x and y. For the pond to be emptied in under 7 mins, how many pumps would be necessary?

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Answer 1

10 + 30x cubic meters of water is removed by 2 pumps, whose work rate is y cubic meters per minute each.

10 + 30x = 2y*30

10 + 30x = 60y

10 + 12x = 4y*12

10 + 12x = 48y

The system is:

10 + 30x = 60y

10 + 12x = 48y

first equation says: (1/6) + (1/2)x = y

10 + 12x = 48( 1/6 + x/2)

10 + 12x = 8 + 24x

2 = 12x

x = 1/6

The pond is filled at 1/6 cubic meter per minute.

first equation says: (1/6) + (1/2)x = y

(1/6) + (1/2)(1/6) = y

(1/6) + 1/12 = y

3/12 = y

y = 1/4

each pump can empty 1/4 cubic meter per minute.

check:

after 30 minutes, there is 10 + 30(1/6) = 10 + 5 = 15 cubic meters of water in the pond.

THe pumps empty 30*1/4*2 = 30*1/2 = 15 cubic meters of water out of the pond, thus emptying it.

after 12 minutes, there is 10 + 12(1/6) = 10 + 2 = 12 cubic meters of water in the pond.

the pumps empty 12 * 1/4 * 4 = 12 cubic meters of water out of the pond, thus emptying it.

So after 7 minutes, there will be 10 + 7/6 = 67/6 = 11 and 1/6 cubic meters of water in the pond.

67/6 = N*(1/4)*7 where N is the number of pumps required to empty the pond.

67/6 = 7/4 * N

(4/7)*67/6 = N

268 / 42 = N

134/21 = N = 6 and 8/21

6 pumps is not enough, they will only empty 10.5 = 10 and 1/2 cubic meters.

7 pumps will empty 49/4 = 12 and 1/4 cubic meters.

Note that 134/21 * 7 * 1/4 = 134/12 = 67/6

7 pumps or more are required to empty the pond in 7 minutes or faster