Question

A contractor estimates that 3 persons could rewire Jasminder's house in 4 days. If he uses 4 people instead of three, how long should they take to complete the job?

Answer 1

Hint: In this question, first of all, find whether the number of workers and the time taken is directly proportional or inversely proportional to each other. Accordingly, write the equation and find the value of the constant. Now, use the same equation to find the value of the time taken.

Complete step-by-step answer:
Here, we are given that 3 persons could rewire Jasminder’s house in 4 days. We need to find the time to complete the job if he uses 4 people instead of three. Before proceeding with this question, we must know the concept of direct and inverse proportion. A direct and inverse proportion is used to show the quantities and the amounts are related to each other. They are also mentioned as directly proportional or inversely proportional. The symbol used to denote the proportionality is \[\propto \].
For example, if we say ‘a’ is proportional to ‘b’, then it is represented as \[a\propto b\] , and if we say a is inversely proportional to b, then it is denoted as \[a\propto \dfrac{1}{b}\].
Direct proportion: Two quantities a and b are said to be in direct proportion if they increase or decrease together. In other words, the ratio of their corresponding values remains constant. This means that,
\[\dfrac{a}{b}=k\]
where k is a positive number, then the quantities a and b are said to vary directly.
Inverse Proportion: Two quantities a and b are said to be in inverse proportion if there is an increase in the quantity a, there will be a decrease in the quantity b and vice versa. In other words, the product of their corresponding values should remain constant. This means that,
\[a\times b=k\]
where k is a positive number, then the quantities a and b are said to vary inversely.
Now, let us consider our question. We know that more workers on a job would reduce the time to complete the task. Here, also if more persons wound rewire Jasminder’s house, the job would be completed in less number of days. So here, we can say that
\[\text{Persons required (P)}\propto \dfrac{1}{\text{Days to complete the job (D)}}\]
\[\Rightarrow P\propto \dfrac{1}{D}\]
\[\Rightarrow P=\dfrac{k}{D}\]
where k is constant of proportionality.
Let us first find the value of k. Here, we are given that initially, there were 3 persons and the time taken was 4 days. So, we get,
\[3=\dfrac{k}{4}\]
By multiplying 4 on both the sides, we get,
k = 12.
Hence, we get the relation,
\[P=\dfrac{12}{D}\]
Now, instead of three, if there were 4 persons, then the time taken could be found by substituting the respective values in the above equation. We get,
\[4=\dfrac{12}{D}\]
By cross – multiplying the above equation, we get,
\[D=\dfrac{12}{4}=3\text{ days}\]
Hence, the time taken by 4 persons to complete the job is 3 days.

 Note: Here, students must note that in these types of questions, first of all, it is very important to find what are the two quantities and whether they are directly proportional or inversely proportional and accordingly writing the relation between them in the form of an equation. Also, remember the value of k that is constant will remain the same, and values of the quantities will change.