Question

A can do $${\displaystyle \frac{2}{3}}$$ of a certain work in $$16$$ days and B can do $${\displaystyle \frac{1}{4}}$$ of the same work in $$3$$ days. In how many days can both finish the work working together?

Answer 1

Hint: Let us first take the whole work as $$1$$ unit.
We need to find first the work done by $$A$$ and $$B$$ in $$1$$ day. Then we can calculate the work done by $$A$$ and $$B$$ in $$1$$ day.
Then applying the unitary method on the work done by $$A$$ and $$B$$ in $$1$$ day, we can find out the number of days they both need to finish the work, working together.
(The unitary method is a technique for solving a problem by first finding the value of a single unit and then finding the necessary value by multiplying the single unit value.)

Complete step by step answer:
It is given that, $$A$$ can do $${\displaystyle \frac{2}{3}}$$ of a certain work in $$16$$ days and $$B$$ can do $${\displaystyle \frac{1}{4}}$$ of the same work in $$3$$ days.
We need to find out the number of days they both need to finish the work, working together.
Let us first take the whole work as a $$1$$ unit.
Now, $$A$$ can be done in $$16$$ days $${\displaystyle \frac{2}{3}}$$ units of a certain work.
In $$1$$ day $$A$$ can do:
$$\Rightarrow {\displaystyle \frac{{\displaystyle \frac{2}{3}}}{16}}={\displaystyle \frac{2}{3}}\times {\displaystyle \frac{1}{16}}={\displaystyle \frac{1}{24}}$$ units of work.
$$B$$ can do in $$3$$ days, $${\displaystyle \frac{1}{4}}$$ units of the work.
In $$1$$ day $$B$$ can do:
$$\Rightarrow {\displaystyle \frac{{\displaystyle \frac{1}{4}}}{3}}={\displaystyle \frac{1}{4}}\times {\displaystyle \frac{1}{3}}={\displaystyle \frac{1}{12}}$$ units of the work.
Now, $$A$$ and $B$ together can do in $$1$$ day:
$$\Rightarrow {\displaystyle \frac{1}{24}}+{\displaystyle \frac{1}{12}}={\displaystyle \frac{1+2}{24}}={\displaystyle \frac{3}{24}}={\displaystyle \frac{1}{8}}$$ unit of work.
Thus $$A$$ and $$B$$ together can do $${\displaystyle \frac{1}{8}}$$ unit of work in $$1$$ day.
$$A$$ and $$B$$ together can do the whole work that is, $$1$$unit of work in $${\displaystyle \frac{1}{{\displaystyle \frac{1}{8}}}}=1\times {\displaystyle \frac{8}{1}}=8$$ days.

Hence, $$A$$ and $$B$$ together can finish the work in $$8$$ days.

Note:
We have solved the problem with the unitary method.
Now we need to know what is the unitary method?
In short, this method is used to find the value of a unit from the value of a multiple, and hence the value of a multiple.