VerifiedHint: We use the unitary method for indirect variation and find the number of hours P and Q will take separately to complete the work by multiplying given hours and days. We find the part of the park they can complete in 1 hour separately and then together. We take reciprocal of the part of the work they can complete in 1 hour to get the number of hours they will take to complete the whole work. We divide it by 8 to get the number of days.
Complete answer:
We know that in unitary method if one quantity $a$ decreases with increase in other quantity $b$ (called indirect variation) then we multiply both the quantities to get the unit value. Here the number of days decreases with increase in the number of hours and the unit value is the whole work. \[\]
We are given the question that P can complete a work in 12 days working 8 hours a day. So the number of hours P takes to complete the work is $12\times 8=96$.So the part or fraction of the total work that P can complete in 1 hour is $\dfrac{1}{96}$.\[\]
We are also given the question that Q can complete the same work in 8 days working 10 hours a day. So the number of hours Q takes to complete the work is $8\times 10=80$. So the part or fraction of the total work that Q can complete in 1 hour is $\dfrac{1}{80}$.\[\]
So the amount of work that P and Q can do in 1 hour is
\[\dfrac{1}{96}+\dfrac{1}{80}=\dfrac{80+96}{96\times 80}=\dfrac{176}{76680}=\dfrac{11}{480}\]
So if P and Q work together the number of hours they will take to complete the work is
\[\dfrac{1}{\dfrac{1}{96}+\dfrac{1}{80}}=\dfrac{1}{\dfrac{11}{480}}=\dfrac{480}{11}\]
They work 8 hours per day. So the number of days P and Q will take to complete the work if they work together is
\[\dfrac{\dfrac{480}{11}}{8}=\dfrac{480}{11}\times \dfrac{1}{8}=\dfrac{60}{11}=5\dfrac{1}{11}\]
So they will take total $5\dfrac{5}{11}$ days and hence the correct option is A.\[\]
So, the correct answer is “Option A”.
Note: We must be careful of the confusion between indirect and direct variation where $a$ increases with increase in $b$ .The ratio $\dfrac{a}{b}$ is constant in direct variation and product $ab$ is constant in indirect variation. Another example of indirect variation is speed and time. The problems prices and area are a direct variation problem.
