Question

A and B can do a piece of work in 15 days. If A’s one day work is \[1\dfrac{1}{2}\] times the one day work of B. Find how many days can each do the work?

Answer 1

Hint: Here we will first assume the A alone can do work in x days and B alone can do the work in y days.
Then one day work of A is given by:- \[\dfrac{1}{x}\]
One day work of B is given by:- \[\dfrac{1}{y}\]
Using the data given in the question we will form the linear equations in terms of x and y and then find the value for x and y.

Complete step-by-step answer:
Let us assume that A alone can do work in x days
B alone can do the work in y days
Then we know that:-
One day work of A is given by:- \[\dfrac{1}{x}\]
One day work of B is given by:- \[\dfrac{1}{y}\]
Now it is given that A’s one day work is \[1\dfrac{1}{2}\] times the one day work of B
Hence,
\[\dfrac{1}{x} = 1\dfrac{1}{2}\left( {\dfrac{1}{y}} \right)\]
Solving it further we get:-
\[\dfrac{1}{x} = \dfrac{3}{2}\left( {\dfrac{1}{y}} \right)\]
\[ \Rightarrow y = \dfrac{{3x}}{2}\] ……………………………(1)
Now it is given that:-
A and B can do piece of work in 15 days.
Hence one day work of them working together is given by:-
\[\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{{15}}\]…………………………………. (2)
Now we will solve equations 1 and 2 to get the values of x and y.
Hence from equation 1 we get:-
Taking reciprocal of both the sides we get:-
\[\dfrac{1}{y} = \dfrac{2}{{3x}}\]
Putting this value in equation 2 we get:-
\[\dfrac{1}{x} + \dfrac{2}{{3x}} = \dfrac{1}{{15}}\]
Taking LCM we get:-
\[\dfrac{{\left( 1 \right)\left( 3 \right) + \left( 2 \right)\left( 1 \right)}}{{3x}} = \dfrac{1}{{15}}\]
Simplifying it further we get:-
\[\dfrac{{3 + 2}}{{3x}} = \dfrac{1}{{15}}\]
\[ \Rightarrow \dfrac{5}{x} = \dfrac{1}{5}\]
Solving for x we get:-
\[x = 5 \times 5\]
\[ \Rightarrow x = 25\]
Now Putting this value in equation 1 we get:-
\[y = \dfrac{{3\left( {25} \right)}}{2}\]
Solving it further we get:-
\[y = \dfrac{{75}}{2}\]

Hence A alone can do work in 25 days and B alone can do work in \[\dfrac{{75}}{2}\]days.

Note: In such types of questions we form two linear equations with two unknown variables with the help of the given information in the question and then solve them to get the values of x and y.
Also, students should note that when a single person works then it takes more days to complete the work than when two persons do the same work.
Time is directly proportional to work. With increase in time, work will also increase.