There are two different cats, the first vat has 12 gallons and is being filled at a rate of 3 gallons per minute. The second vat has 108 gallons and is leaking fluid at 7 gallons per minute. After how many seconds when will both bats have the same amount of liquid?

Answer 1

We can set up an equation to solve this problem. Let's assume $t$ minutes have passed.
The amount of liquid in the first vat after $t$ minutes will be $12 + 3t$ gallons.
The amount of liquid in the second vat after $t$ minutes will be $108 - 7t$ gallons.
To determine when both vats have the same amount of liquid, we set the two expressions equal to each other and solve for $t$:
$12 + 3t = 108 - 7t$
Adding $7t$ to both sides: $12 + 10t = 108$
Subtracting $12$ from both sides: $10t = 96$
Dividing by $10$: $t = 9.6$ minutes.
Since there are 60 seconds in 1 minute, we multiply $9.6$ by $60$ to convert it into seconds.
$9.6 \times 60 = \boxed{576}$ seconds. Answer: \boxed{576}.