Question

Two pumps are being filled by two different pumps. The first vat has 121212 gallons is being filled at a rate of 191919 gallons per second. The second vat has 404040 gallons and is being filled at a rate of 151515 gallons per second. After how many seconds will both vats have the same amount of liquid?

Answer 1

Let \( t \) be the number of seconds after the filling starts.

### First Vat
The first vat starts with \( 121212 \) gallons and is being filled at a rate of \( 191919 \) gallons per second. Thus, the amount of liquid in the first vat after \( t \) seconds can be represented as:

\[
\text{Amount in first vat} = 121212 + 191919t
\]

### Second Vat
The second vat starts with \( 404040 \) gallons and is being filled at a rate of \( 151515 \) gallons per second. The amount of liquid in the second vat after \( t \) seconds can be represented as:

\[
\text{Amount in second vat} = 404040 + 151515t
\]

### Setting the Amounts Equal
We need to find \( t \) such that both vats contain the same amount of liquid:

\[
121212 + 191919t = 404040 + 151515t
\]

### Solving for \( t \)
Rearranging the equation:

\[
191919t - 151515t = 404040 - 121212
\]

This simplifies to:

\[
40304t = 282828
\]

Now, we can solve for \( t \):

\[
t = \frac{282828}{40304}
\]

Calculating:

\[
t = 7
\]

### Final Result
Thus, both vats will have the same amount of liquid after \( \boxed{7} \) seconds.