Two pipes can be used to fill a pool. Working together, the two pipes can fill the pool in 2 hours. The larger pipe can fill the pool in 3 hours less time than the smaller pipe can alone. Find the time to the nearest tenth of an hour it takes for the smaller pipe working alone to fill the pool. Please help ASAP!!! :(

T = large time , t = small time

t - 3 = T

in 2 hr they each fill some fraction of the pool ... (2 / t) + (2 / T) = 1

(2 / t) + [2 / (t - 3)] = 1 ... 2(t - 3) + 2 t = t(t - 3) ... 4t - 6 = t^2 - 3t

t^2 - 7t + 6 = 0 ... solve for t

Let's assume that the smaller pipe takes x hours to fill the pool alone.

Since the larger pipe can fill the pool in 3 hours less time than the smaller pipe, it would take (x - 3) hours for the larger pipe to fill the pool alone.

To find the rate at which each pipe fills the pool, we can set up the following equation:

1/(x-3) + 1/x = 1/2

Now let's solve for x.

Multiply the equation by 2x(x - 3) to eliminate the denominators:

2x + 2(x - 3) = x(x - 3)

Distribute and simplify:

2x + 2x - 6 = x^2 - 3x

Combine like terms:

4x - 6 = x^2 - 3x

Rearrange the equation:

x^2 - 7x + 6 = 0

Factor the quadratic equation:

(x - 6)(x - 1) = 0

Set each factor equal to zero and solve:

x - 6 = 0 or x - 1 = 0

x = 6 or x = 1

We can ignore the solution x = 6 since it takes less time for the larger pipe to fill the pool than it does for both pipes together. Therefore, the smaller pipe takes 1 hour to fill the pool alone.

To solve this problem, let's assume that it takes x hours for the smaller pipe to fill the pool alone.

Since the larger pipe can fill the pool in 3 hours less time than the smaller pipe can alone, the larger pipe takes (x - 3) hours to fill the pool alone.

Now, we can consider their combined rate of filling the pool. The rate at which the smaller pipe fills the pool is 1 pool/x hours (1 pool per x hours), and the rate at which the larger pipe fills the pool is 1 pool/(x - 3) hours.

The combined rate of both pipes working together is 1 pool/2 hours.

Using the idea that rates are additive, we can create the equation:

1/x + 1/(x - 3) = 1/2

To solve this equation, we need to find a common denominator. In this case, it is 2x(x - 3).

2(x - 3) + 2x = x(x - 3)

Simplifying the equation:

2x - 6 + 2x = x² - 3x

4x - 6 = x² - 3x

Rearranging the equation to form a quadratic equation:

x² - 3x - 4x + 6 = 0

x² - 7x + 6 = 0

Now, we can solve this quadratic equation by factoring or using the quadratic formula.

(x - 6)(x - 1) = 0

x = 6 or x = 1

Since the time to fill the pool cannot be negative, we discard the solution x = 1.

Therefore, it takes the smaller pipe approximately 6 hours (to the nearest tenth of an hour) to fill the pool alone.