Working together, two pumps can drain a certain pool in 6 hours. If it takes the older pump 14 hours to drain the pool by itself, how long will it take the newer pump to drain the pool on its own?
in 6 hrs, the old pump drains 6/14 of the pool
... so the newer pump drains 8/14 in 6 hr
(8/14) / 6 = 1 / x ... x = 6 / (8/14)
adding up the amount each pump can do in an hour, we have
1/14 + 1/x = 1/6
To solve this problem, we can use the concept of work rates.
Let's denote the working rate of the older pump as "R_old" (amount of work done per hour) and the working rate of the newer pump as "R_new".
According to the information given, we are told that two pumps working together can drain the pool in 6 hours. This means that the combined working rate of the pumps is 1/6 of the pool per hour.
Additionally, we know that the older pump alone takes 14 hours to drain the pool. This implies that its working rate is 1/14 of the pool per hour.
Let's set up an equation to represent the combined working rate of both pumps:
1/14 + R_new = 1/6
To find R_new, we can subtract 1/14 from both sides of the equation:
R_new = 1/6 - 1/14
R_new = (7/42) - (3/42)
R_new = 4/42
R_new = 2/21
This tells us that the newer pump has a working rate of 2/21 of the pool per hour.
To determine how long it will take the newer pump to drain the pool on its own, we can use the formula:
Time = Amount of work / Rate
Since the pool represents the amount of work, and the working rate of the newer pump is 2/21 of the pool per hour, we have:
Time = 1 / (2/21)
Time = 21/2
Time = 10.5
Therefore, it will take the newer pump 10.5 hours to drain the pool on its own.
To solve this problem, we need to use the concept of work rates.
Let's denote the work rate of the older pump as "O" (in pool per hour) and the work rate of the newer pump as "N" (also in pool per hour).
Given that working together, the two pumps can drain the pool in 6 hours, we can set up the following equation using the formula:
1 pool = (O + N) * 6
Now, we are given that the older pump takes 14 hours to drain the pool by itself. Therefore, its work rate can be calculated as:
O = 1 pool / 14 hours
Now, we need to find the work rate of the newer pump, denoted as "N". To do this, we need to substitute the value of "O" into our equation:
1 pool = ((1 pool / 14 hours) + N) * 6
To simplify the equation, we can multiply both sides by 14 hours to get rid of the denominator:
14 hours * 1 pool = (1 pool + 14 N) * 6
Now, let's distribute the multiplication on the right side:
14 hours * 1 pool = 6 pool + 84 N
To isolate N, we need to eliminate the pool term on the right side. We can do this by subtracting "6 pool" from both sides:
14 hours * 1 pool - 6 pool = 84 N
Simplifying further:
-5 pool = 84 N
Finally, to find the work rate of the newer pump, we divide both sides by 84:
N = (-5 pool / 84)
The negative sign is irrelevant in this context since we are dealing with work rates. As a result, the newer pump can drain the pool at a rate of:
N = 5/84 pools per hour
To find out how long it will take the newer pump to drain the pool on its own, we can use the work rate:
Time = 1 pool / N
Substituting the value of N, we have:
Time = 1 pool / (5/84 pools per hour)
Simplifying:
Time = 84/5 hours
Thus, it will take the newer pump approximately 16.8 hours to drain the pool on its own.