Question

Shwetha takes 6 days less than the number of days taken by Ankitha to complete a piece of work. If both Shwetha and Ankitha together can complete the same work in $4$ days. In how many days will Ankitha alone complete the work?

Answer 1

Hint: Assume the number of days Ankitha takes to complete the work be any variable. Then using given conditions find the number of days Shwetha takes to complete the same work in assumed variables. Then calculate the number of days Shwetha and Ankitha will take to complete the same work together using a work formula and equate it to the given value. Solve the quadratic equation and find the number of days Ankitha will take to complete the work alone.

Formula used: If A can do a piece of work in $p$ days and B can do it in $q$ days, then A and B can together can complete the same in $\dfrac{{pq}}{{p + q}}$ days.
For factorising an algebraic expression of the type $a{x^2} + bx + c$, we find two factors p and q such that $ac = pq$ and $p + q = b$

Complete step-by-step solution:
Assume the number of days Ankitha takes to complete the work be any variable.
Let the number of days Ankitha takes to complete the work be $x$ days.
Now we have to find the number of days Shwetha takes to complete the same work.
It is given that Shwetha takes $6$ days less than the number of days taken by Ankitha to complete a piece of work.
So, the number of days Shwetha takes to complete the same work is $x - 6$ days.
It is also given that both Shwetha and Ankitha together can complete the same work in $4$ days.
We can also calculate the number of days Shwetha and Ankitha will take to complete the same work together using a work formula and equate it to the given value.
Now we first find the number of days Shwetha and Ankitha will take to complete the same work together using a work formula.
As the number of days Ankitha takes to complete the work is $x$ days and the number of days Shwetha takes to complete the same work is $x - 6$ days.
So, putting $p = x$ and $q = x - 6$ in $\dfrac{{pq}}{{p + q}}$, we get
$ \Rightarrow \dfrac{{x\left( {x - 6} \right)}}{{x + x - 6}}$
On adding the denominator term and we get
$ \Rightarrow \dfrac{{x\left( {x - 6} \right)}}{{2x - 6}}$
So, the number of days Shwetha and Ankitha will take to complete the same work together is $\dfrac{{x\left( {x - 6} \right)}}{{2x - 6}}$ days.
It is given that both Shwetha and Ankitha together can complete the same work in $4$ days.
So, Equating $\dfrac{{x\left( {x - 6} \right)}}{{2x - 6}}$ to $4$ and solve the quadratic equation.
$ \Rightarrow \dfrac{{x\left( {x - 6} \right)}}{{2x - 6}} = 4$
Cross multiplying the equation, we get
$ \Rightarrow x\left( {x - 6} \right) = 4\left( {2x - 6} \right)$
On multiply the term, we get
$ \Rightarrow {x^2} - 6x = 8x - 24$
Taking all the term as LHS and adding the same term we get,
$ \Rightarrow {x^2} - 14x + 24 = 0$
Find the product of the first and last constant term of the expression.
$ \Rightarrow 24 \times 1 = 24$
Find the factors of $24$ in such a way that addition or subtraction of those factors is the middle term.
So we can write it as, $ - 2 \times - 12 = 24$and $ - 2 - 12 = - 14$
Splitting middle term in these factors, we get
$ \Rightarrow {x^2} - 2x - 12x + 24 = 0$
Taking $x$ common in $\left( {{x^2} - 2x} \right)$ and $ - 12$ common in $\left( { - 12x + 24} \right)$, we get
$ \Rightarrow x\left( {x - 2} \right) - 12\left( {x - 2} \right) = 0$
Take $\left( {x - 2} \right)$ common in above equation, we get
$ \Rightarrow \left( {x - 2} \right)\left( {x - 12} \right) = 0$
This implies, $x - 2 = 0$
$ \Rightarrow x = 2$
Also, $x - 12 = 0$
$ \Rightarrow x = 12$
If we put $x = 2$ in the number of days Shwetha takes to complete the same work, we will get a negative value and the number of days can’t be negative.
Therefore, the number of days Ankitha takes to complete the work is $12$ days.
So, putting $x = 12$ in the number of days Shwetha takes to complete the same work.
$ \Rightarrow 12 - 6 = 6$
Therefore , the number of days Shwetha takes to complete the work is $6$ days.

Thus, Ankitha alone can complete the work in $12$ days.

Note: If two persons A and B can individually do some work in $p$ and $q$ days respectively, we can find out how much work can be done by them together in one day. Since A can do ${\left( {\dfrac{1}{p}} \right)^{{\text{th}}}}$ part of the work in one day and B can do ${\left( {\dfrac{1}{q}} \right)^{{\text{th}}}}$ part of the work in one day, the two of them together do ${\left( {\dfrac{1}{p} + \dfrac{1}{q}} \right)^{{\text{th}}}}$ part of the work in one day. From this we can find out the number of days that they take to complete the work.