If Sally can paint a house in 4 hours, and John can paint the same house in 6 hours, how long will it take for the to pain the house together?

In 24 hours, both of them can paint 6+4 houses, or 10 houses.

So the combined rate is 10houses/24hrs, or one house per 2.4 hours.

From the information I have they are saying it should be 2 hours and 24 minutes. Can you explain?

To solve this problem, we can use the concept of work rates.

The work rate is the amount of work done per unit of time. In this case, Sally can paint 1 house in 4 hours, so her work rate is 1 house/4 hours. Similarly, John can paint 1 house in 6 hours, so his work rate is 1 house/6 hours.

To find out how long it will take for them to paint the house together, we need to combine their work rates. Since they are working together, their work rates can be added.

So, Sally and John's combined work rate is 1/4 + 1/6 houses per hour.

To add these fractions, we need to find a common denominator. The least common multiple of 4 and 6 is 12.

Now, let's convert the fractions to have the same denominator:

1/4 = 3/12 (multiply numerator and denominator by 3)
1/6 = 2/12 (multiply numerator and denominator by 2)

Now we can add the fractions:

3/12 + 2/12 = 5/12

The combined work rate is 5/12 houses per hour.

This means that working together, Sally and John can paint 5/12 of the house in one hour.

To find the time it will take to paint the entire house, we can set up a ratio using the work rates:

5/12 = 1/t

where t is the time it will take for them to paint the house together.

To solve for t, we can cross multiply:

5t = 12

t = 12/5

So, it will take them 12/5 hours, or 2.4 hours, to paint the house together.