Question

Raj can build a house alone in \[16days\] but Suraj alone can build it in \[12days\]. Raj and Suraj work on alternate days. If Raj works on the first day, the house will be built in how many days?

Answer 1

Hint: We have to find the number of days taken by both Raj and Suraj to complete a piece of work together working alternatively. We solve this question using the concept of solving linear equations. We will first find the amount of work in which both Raj and Suraj alone can complete the piece of work in one day. Then to calculate the number of days in which both Raj and Suraj can together complete the work, alternatively we will add the amount of work done by both Raj and Suraj alone for one day. Adding the reciprocals would give us the amount of work done by both in one day. Then we will keep on adding the work done by both such that we obtain the total number of days in which the house can be built by both Raj and Suraj.

Complete step-by-step solution:
Given:
Raj can build a house alone in \[16days\] but Suraj alone can build it in \[12days\].
So , the work done in one day by Raj is given as :
\[\text{Work done in one day by Raj} = \dfrac{1}{{16}}\]
Similarly , the work done in one day by Suraj is given as :
\[\text{Work done in one day by Suraj} = \dfrac{1}{{12}}\]
So, as both work alternatively the work done in \[2days\] can be given as:
\[\text{Work done in}{\text{ 2 days = }}\dfrac{1}{{16}} + \dfrac{1}{{12}}\]
\[\text{Work done in}{\text{ 2 days = }}\dfrac{7}{{48}}\]
So, the work done in \[{\text{12 days}}\] can be given as:
\[\text{Work done in}{\text{ 12 days = }}\dfrac{7}{{48}} \times 6\]
\[\text{Work done in}{\text{ 12 days = }}\dfrac{{42}}{{48}}\]
Now, on work done on the \[{13^{th}}day\] can be given as:
\[\text{work done in}{\text{ 12 days + work done in 1}}{{\text{3}}^{th}}{\text{ day = }}\dfrac{{42}}{{48}} + \dfrac{1}{{16}}\]
\[\text{work done in}{\text{ 13 days = }}\dfrac{{45}}{{48}}\]
Now, on the \[{14^{th}} \text{day}\] Suraj will work and the work left to build the house can be given as:
\[Work left = 1 - work{\text{ done in 13 days}}\]
Putting the values, we get the work left as:
\[\text{Work left} = 1 - \dfrac{{45}}{{48}}\]
\[\text{Work left} = \dfrac{3}{{48}}\]
As we know that Suraj can complete \[\dfrac{1}{{12}}\] work in one day. So, the time to complete \[\dfrac{3}{{48}}\] of the work can be given as:
\[1day = \dfrac{1}{{12}}\text{of the work}\]
\[\dfrac{3}{{48}} \text{of the work} = \dfrac{3}{{48}} \times 12{\text{ days}}\]
\[\dfrac{3}{{48}} \text{of the work} = \dfrac{3}{4}{\text{days}}\]
The total number of days to complete the work can be given as:
\[\text{Total days} = \left( {13 + \dfrac{3}{4}} \right)days\]
\[\text{Total days} = \dfrac{{55}}{4}days\]
Hence, the total number of days for both Raj and Suraj to complete the work alternatively is \[\dfrac{{55}}{4}days\].

Note: On adding the work done by both alone in one day, we get the value of the work completed in two days. We multiplied the work done by \[6\], as \[6\] was the greatest possible number such that after the multiplication the value of the numerator would not exceed the denominator. As we are working in reciprocals the complete work is to be considered as \[1\] only. We can’t exceed the value \[1\] hence we multiplied the number of days by \[6\].