Alice can paint a room in 6 hours. her assistant, tom, can paint the same room in 10 hours. how long will it take if the two of them work together? show all work.

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A simple way to find out:
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[tex]\text {Total time needed} = \dfrac{10 \times 6}{10 + 6} = \dfrac{60}{16} = 3 \dfrac{3}{4} \text{ hours} [/tex]

Answer: They will take 3 3/4 hours to paint the room together.


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A more detailed explanation:
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Alice:
6 hours = 1 room
1 hour = 1/6 of the room

Tom:
10 hours = 1 room
1 hour = 1/10 of the room

Alice and Tom together:
1 hour = 1/6 + 1/10 = 4/15 of the room

4/15 of the room = 1 hour
1/15 of the room  = 1 ÷ 4 = 1/4 hour
15/15 of the room = 1/4 x 15 = 15/4 = 3 3/4 hours

Answer: They will take 3 3/4 hours to paint the room together.