Question

Two pumps are filling large vats of liquid. One vat is empty, and the pump is filling it at a rate of 6 gallons per minute. The other is already filled with 10 gallons, and it is continuing to fill at a rate of 4 gallons per minute. Write an equation to show when the two vats will have an equal amount of liquid. Use x as your variab

Answer 1

Let \( x \) represent the number of minutes that have passed.

For the first vat, which starts empty and is filling at a rate of 6 gallons per minute, the amount of liquid in the vat after \( x \) minutes can be expressed as:

\[ \text{Amount in first vat} = 6x \]

For the second vat, which starts with 10 gallons and continues to fill at a rate of 4 gallons per minute, the amount of liquid in this vat after \( x \) minutes can be expressed as:

\[ \text{Amount in second vat} = 10 + 4x \]

To find when the two vats will have an equal amount of liquid, we can set the two expressions equal to each other:

\[ 6x = 10 + 4x \]

Now, we can solve this equation for \( x \):

1. Subtract \( 4x \) from both sides:

\[ 6x - 4x = 10 \]

2. Simplify:

\[ 2x = 10 \]

3. Divide by 2:

\[ x = 5 \]

So, after 5 minutes, both vats will have an equal amount of liquid.

To summarize, the equation representing when the two vats will have equal amounts of liquid is:

\[ 6x = 10 + 4x \]

And they will be equal after \( x = 5 \) minutes.