It takes 12 hours for a single hose to fill a large vat. when a second hose is added, the vat can be filled in 4 hours. how many hours are required for the second hose working alone to fill the vat?

6 hours

The two hoses together take 1/3 the time (4/12 = 1/3), so the two hoses together are equivalent to 3 of the first hose.

That is, the second hose is equivalent to 2 of the first hose. Two of the first hose could fill the vat in half the time one of them can, so 6 hours.

The second hose alone can fill the vat in 6 hours.

The first hose's rate of doing work is ...

... (1 vat)/(12 hours) = (1/12) vat/hour

If h is the second hose's rate of doing work, then working together their rate is ...

... (1/12 vat/hour) + h = (1/4 vat/hour)

... h = (1/4 - 1/12) vat/hour = (3/12 -1/12) vat/hour = 2/12 vat/hour

... h = 1/6 vat/hour

so will take 6 hours to fill 1 vat.

here is your mark brainliest