One pipe can fill a cistern in 1 1/2 hours while a second pipe can fill it in 2 1/3 hrs. Three pipes working together fill the cistern in 42 minutes. How long would it take the third pipe alone to fill the tank?
1/90 + 1/140 + 1/x = 1/42
To solve this problem, we need to determine the individual rates at which each pipe fills the cistern per hour. Then we can calculate the rate at which all three pipes fill the cistern together.
Let's start by finding the rate of the first pipe. We know that the first pipe can fill the cistern in 1 1/2 hours, or 1 hour and 30 minutes. In decimal form, this is 1.5 hours. Therefore, the rate of the first pipe is 1 cistern per 1.5 hours or 1/1.5 cisterns per hour.
Next, we will find the rate of the second pipe. The second pipe fills the cistern in 2 1/3 hours, which is 2.33 hours. Therefore, the rate of the second pipe is 1 cistern per 2.33 hours or 1/2.33 cisterns per hour.
Now, let's find the combined rate of all three pipes working together. We are given that when all three pipes work together, they can fill the cistern in 42 minutes. Let's convert this to hours by dividing 42 minutes by 60 (since there are 60 minutes in an hour). 42 minutes is equal to 0.7 hours.
Since we want to find the rate at which all three pipes together fill the cistern, we need to find the combined rate. Let's denote the combined rate as R.
R = 1/1.5 + 1/2.33
To add the fractions, we need to find a common denominator, which is the least common multiple (LCM) of 1.5 and 2.33. The LCM of 1.5 and 2.33 is 4.65.
Now, let's rewrite the fractions with the common denominator of 4.65.
R = (1/1.5) * (4.65/4.65) + (1/2.33) * (4.65/4.65)
R = 3.1/4.65 + 2/4.65
R = (3.1 + 2)/4.65
R = 5.1/4.65
R ≈ 1.097 cisterns per hour
Now that we know the combined rate of all three pipes, we can find the rate of the third pipe alone. Let's denote the rate of the third pipe as T.
T = R - (1/1.5) - (1/2.33)
T = 1.097 - 0.666 - 0.429
T ≈ 0.002 cisterns per hour
Since the rate of the third pipe alone is approximately 0.002 cisterns per hour, we can find the time it takes for the third pipe to fill the tank alone by taking the reciprocal of the rate:
Time = 1 / (0.002 cisterns per hour)
Time ≈ 500 hours
Therefore, it would take the third pipe approximately 500 hours to fill the tank on its own.