Susan's backyard has three faucets: A, B and C. If she turns on all three of them, it takes 380 minutes to fill the pool. If she turns on only A and B, it takes 570 minutes, and if she turns on only B and C, it takes 760 minutes. If she turns on only A and C, how long will it take (in minutes) to fill the pool?

1/a + 1/b + 1/c = 1/380

1/a + 1/b = 1/570
1/b + 1/c = 1/760
solve for a,b,c and you get
1/a + 1/c = 1/760 + 1/1140 = 1/456

To find out how long it will take to fill the pool by turning on only faucets A and C, we can analyze the information given and solve for the time it takes.

Let's assign the following variables:
- A: the rate at which faucet A fills the pool
- B: the rate at which faucet B fills the pool
- C: the rate at which faucet C fills the pool

From the given information, we have the following equations:

1) A + B + C = 1/380
2) A + B = 1/570
3) B + C = 1/760

We want to find the time it takes to fill the pool with only faucets A and C, so we need to find the rate at which A and C can fill the pool together.

To do this, we can subtract equation 3) from equation 2):

(A + B) - (B + C) = 1/570 - 1/760
A - C = 1/570 - 1/760

Simplifying the right side:

A - C = (760 - 570) / (570 * 760)
A - C = 190 / (570 * 760)
A - C = 1 / (3 * 760)

Now, let's add this equation to equation 1) to find the rate at which A and C can fill the pool together.

(A + B + C) + (A - C) = 1/380 + 1 / (3 * 760)
2A = 1/380 + 1 / (3 * 760)

Simplifying the right side:

2A = (3 * 760 + 380) / (380 * 3 * 760)
2A = 3040 / (380 * 3 * 760)
2A = 1 / (380 * 3)

Divide both sides by 2:

A = 1 / (2 * 380 * 3)
A = 1 / (2 * 3 * 380)
A = 1 / 2 * 3 * 380
A = 1 / (6 * 380)

Now, we can substitute this value for A into equation 1) and solve for the time it takes to fill the pool with only faucets A and C.

1/ (6 * 380) + B + 1/ (6 * 380) = 1/380
B = 1/380 - 1/ (6 * 380) - 1/ (6 * 380)
B = (6 - 1 - 1) / (6 * 380)
B = 4 / (6 * 380)
B = 1 / (6 * 95)

To find the time it takes to fill the pool with only faucets A and C, we can add the rates of A and C:

(1 / (6 * 380)) + (1 / (6 * 95)) = (1 * 95 + 380) / (6 * 380 * 95)
= 475 / (6 * 380 * 95)
= 475 / 216600
= 0.00219184...

Rounding to the nearest minute, it will take approximately 0 minutes to fill the pool with only faucets A and C.

To determine how long it will take to fill the pool when only faucets A and C are turned on, we can use a system of equations.

Let's assign variables to represent the rate at which each faucet fills the pool. Let's say faucet A fills the pool at a rate of "x" per minute, faucet B fills the pool at a rate of "y" per minute, and faucet C fills the pool at a rate of "z" per minute.

Based on the given information, we can set up the following equations:

Equation 1:
x + y + z = 1/380 (since all three faucets take 380 minutes to fill the pool)

Equation 2:
x + y = 1/570 (since faucets A and B take 570 minutes to fill the pool)

Equation 3:
y + z = 1/760 (since faucets B and C take 760 minutes to fill the pool)

We can solve this system of equations to find the values of x, y, and z.

First, let's find the value of y using Equation 2:
x + y = 1/570
y = 1/570 - x

Next, let's find the value of z using Equation 3:
y + z = 1/760
z = 1/760 - y

Now, substitute the expressions for y and z into Equation 1:
x + (1/570 - x) + (1/760 - (1/570 - x)) = 1/380

Simplify the equation:
x + 1/570 - x + 1/760 + x - 1/570 = 1/380

Combine like terms:
1/570 + 1/760 = 1/380

Now, find the least common denominator (LCD) for 570, 760, and 380, which is 1. Multiply each denominator by the LCD:

1/570 * (380/380) + 1/760 * (380/380) = 1/380

Simplify:
380/216600 + 380/288800 = 1/380

Multiply both sides of the equation by the LCD to eliminate the denominators:
380 * 380 + 380 * 380 = 216600

Simplify:
144400 + 144400 = 216600

Combine like terms:
288800 = 216600

This equation is not true, meaning there is no solution. It seems there may be an error or inconsistency in the given information or calculations.

Therefore, we cannot determine how long it will take to fill the pool when only faucets A and C are turned on based on the given information.