Question: Samuel can paint a house in 5 days and Rose can paint the same house in 7 days. If they work together, how long will it take them to paint the house?

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Solution
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There is a similar question here http://samimath.com/word-problems-2/ , I advise that you read that first.

I want to use this question to explain another method of solving this kind of question. I will use the method I used in the former question first then I will use another method.

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First Method
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For Samuel:

5 days = 1 house

Therefore,
1 day = 1/5 house

For Rose:
7 days = 1 house

Therefore,
1 day = 1/7 house


If the work together, for 1 day they will both complete (1/5 +1/7)house:
1 day = 1/5 + 1/7


Adding the fraction by using the
butterfly method,

1 day = 12/35 house.

12/35 house = 1 day

*Multiplying both sides by 35/12 gives:

1 house = 35/12 days

So, it will take them 35/12 days to paint the house together
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Second Method
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Since we are to find the number of days it will take both of them to paint the house, we can first find the number of houses each of them can paint if they work for the same number of days.
We will consider the LCM of 5 and 7 which is 35.

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For Samuel:
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5 days = 1 house
For 5 X 7 days = 1 X 7 houses,

That is,
For 35 days Samuel will paint 7 houses.

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For Rose:
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7 days = 1 house
For 7 X 5 days = 1 X 5 houses,

That is, for 35 days Rose can paint 5 houses.
So if they work together for 35 days they will both paint 7+5 houses:
35 days = 12 houses

That is,
12 houses = 35 days
Dividing both side by 12 gives:
1 house = 35/12 days

So they will both paint the house in 35/12 days if they work together.

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General formular for solving combined rate questions
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There is a general formular for solving questions that involve combined rates....click on this link to continue reading http://samimath.com/word-problem-3/


Tutorial on the butterfly method is here: http://samimath.com/easiest-method-of-adding-or-subtracting-fractions-the-butterfly-method/

NOTE: If you have questions, All questions should be ask here http://samimath.com/forums/math-forum/

About 2 days and 22 hours.

It's a case of using the given data to estimate how much of the work each would do in a day individually; then putting each of their daily work together and figuring out how many days it would take both efforts to finish the house.

Maths is all about pictures; draw the picture and the logic becomes clear, even without putting a word down.

Abstract Algebra, Real Analysis, Mathematical Methods, Topology and Lebesque Measure had always defied pictures though