A hose can fill a swimming pool in 4 hours. Another hose needs 9 more hours to fill the pool than the two hoses combined. How long would it take the second hose to fill the pool?
Lets think of the situation. It doesn't tell you how big the pool but you can make up a value. To make it easier use a number that it a multiple of both 4 and 9. In this case it would be 36.
Okay say the pools need 36 gallons of water to be full. To find the rate for each hose divide
total amount needed/time
Hose 1: 36/4=9
Hose 2: 36/9=4
Hose 1 rate is 9 gallons per hour and Hose 2 rate is 4 gallons per hour
Add the 2 together 9+4=13 gallons per hour
Now divide 36 by 13
36/13= 2.77 Hours
Is this clear? I know it's long winded but I wanted to explain the full process.
that helps a lot thank you
or,
1/4 + 1/(x+9) = 1/x
x = 3
so, the 2nd hose takes 12 hours
check: in 1 hour, the two hoses fill
1/4 + 1/12 = 1/3 of the pool.
To solve this problem, we can break it down into smaller steps. Let's assign some variables to the given information:
Let's say the first hose can fill the pool in x hours.
Therefore, the rate at which the first hose can fill the pool is 1 pool per x hours (since it can fill the entire pool in x hours).
The second hose takes 9 more hours to fill the pool than the two hoses combined.
So, the second hose takes (x + 9) hours to fill the pool.
We need to find the time it takes the second hose to fill the pool, so we need to find the value of x.
First, let's find the rates at which each hose fills the pool. The rate is given by 1 pool divided by the time in hours.
For the first hose:
Rate of the first hose = 1 pool / x hours = 1/x pools per hour.
For the second hose:
Rate of the second hose = 1 pool / (x + 9) hours = 1/(x + 9) pools per hour.
Now, we can combine their rates to find the rate at which they both fill the pool together.
Rate of both hoses = Rate of the first hose + Rate of the second hose
= 1/x + 1/(x + 9) pools per hour.
We know that the time it takes to fill the pool when both hoses are used is 4 hours, so the rate of both hoses is:
Rate of both hoses = 1/4 pools per hour.
Now, we can set up an equation and solve for x:
1/x + 1/(x + 9) = 1/4
To solve the equation, we need to find a common denominator:
(4(x + 9) + 4x)/(x(x + 9)) = 1/4
(4x + 36 + 4x)/(x^2 + 9x) = 1/4
8x + 36 = (x^2 + 9x)/4
Multiply both sides of the equation by 4 to eliminate the fraction:
32x + 144 = x^2 + 9x
Rearrange and combine like terms:
x^2 + 9x - 32x - 144 = 0
x^2 - 23x - 144 = 0
Now we can factor the quadratic equation:
(x - 32)(x + 9) = 0
Setting each factor to zero gives:
x - 32 = 0 or x + 9 = 0
Solving for x:
x = 32 or x = -9
Since time cannot be negative, we take the positive value:
x = 32
So, the first hose can fill the pool in 32 hours.
To find the time it takes for the second hose to fill the pool, we substitute the value of x into the equation:
Time for the second hose = x + 9 = 32 + 9 = 41
Therefore, the second hose would take 41 hours to fill the swimming pool.