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A father and his son can paint a house in three days. Working alone, the father can paint the house in 4 days. How long would it take the son if he worked alone?

User Gladclef
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1 Answer

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We will investigate how to deal with proportions and ratios in real life situations.

We are given a time frame in which both father and son can perform paint job of the entire house as follows:


t_(dad-son)\text{ = 3 days}

We are told that if either of them works alone for a paint job of a house. The time taken would be as follows:


\begin{gathered} t_(dad)\text{ = 4 days} \\ t_{son\text{ }}\text{ = x days} \end{gathered}

One thing to notice here is the type of relation with the number of people working on a job and the time it takes to perform a paint job of a house. The more number of people work on it the less the time it will take. We will categorize this relationship as:


\text{Number of people }\propto\text{ }\frac{1}{time\text{ taken}}

This type of relationship between variables is called an inverse relationship!

We will convert the proportionality into an equation by changing the sign to " = " and multiplying a propotinality constant ( k ) on the right and side as follows:


\text{Number of people = }\frac{k}{time\text{ taken}}

Now we want to evaluate the proportionality constant ( k ) we will use the following data given:


\text{Father and son combined makes 2 people who completed the job in 3 days!}

Hence,


\begin{gathered} 2\text{ = }(k)/(3) \\ \\ k\text{ = 6 } \end{gathered}

Now we will determine the how much time it takes for 1 person in general to complete the paint job as follows:


\begin{gathered} \text{Number of people = }\frac{6}{time\text{ taken}} \\ \\ \text{Time taken = 1}\cdot6\text{ = 6 days} \end{gathered}

Now the above answer settles for a general person. This assumes the capabilities and aptitude of all people are identical. However, this is not the case! Father's capabilities to complete the job and son's capability to complete can not be identical.

Here we will assume the age of each person as a reference of comparison. We can predict that son would be quicker than his father due to young blood!

User Onkar Kole
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