Two pipes fill a storage tank in 9 hours. If the larger pipe fills the tank three times as fast as the smaller one, how long would it take the larger pipe to fill the tank alone?
In terms of your variable, what fraction of the tank is filled in 1 h by the larger pipe alone? By the smaller pipe alone?
Please explain step-by-step.
Let t = time required by the larger pipe alone
then
3t = time required by the smaller pipe alone
Let the completed job = 1; (a full tank)
Each pipe will fill a fraction of the tank, the two fractions add up to 1.
9/t + 9/3t = 1
27 + 9 = 3t
36 = 3t
12 = t
Large pipe :12 hours
Small pipe: 36 hours
To solve this problem, we can assign variables to represent the rates at which the two pipes fill the tank. Let's assume that the rate at which the smaller pipe fills the tank is x tanks per hour.
According to the problem, the larger pipe fills the tank three times as fast as the smaller one. This means that the rate at which the larger pipe fills the tank is 3x tanks per hour.
We are given that both pipes fill the tank in 9 hours together. To determine the fraction of the tank that each pipe fills in 1 hour, we can use the concept of rates.
Let's say that the fraction of the tank filled by the smaller pipe in 1 hour is y. This means that the rate of the smaller pipe can be represented as y tanks per hour.
Similarly, the fraction of the tank filled by the larger pipe in 1 hour can be represented as z. This implies that the rate of the larger pipe is z tanks per hour.
We can set up an equation using the given information:
Rate of the smaller pipe * time taken by both pipes = Rate of the larger pipe * time taken by both pipes
y * 9 = z * 9
Simplifying this equation, we have:
9y = 9z
Dividing both sides by 9, we get:
y = z
This means that the fraction of the tank filled by each pipe in 1 hour is equal.
Since the total fraction of the tank filled by both pipes in 1 hour is 1, and the fraction filled by each pipe individually is equal, we can conclude that the fraction of the tank filled by each pipe individually in 1 hour is 1/2.
Therefore, in terms of the variable, the fraction of the tank filled in 1 hour by the larger pipe alone is 1/2, and the fraction of the tank filled in 1 hour by the smaller pipe alone is 1/2 as well.
To find out how long it would take the larger pipe to fill the tank alone, we can calculate the time using the fraction of the tank filled by the larger pipe in 1 hour.
If the larger pipe fills 1/2 of the tank in 1 hour, it would take it 2 hours to fill the entire tank by itself.
Hence, the larger pipe would take 2 hours to fill the tank alone.