Question

A takes 5 days less than B to complete the work. Both complete the work in 6 days. In how many days A complete the work?
A) 10
B) 15
C) 20
D) 30

Answer 1

Hint: Here, we will first calculate A’s one day work and B’s one day work, and equate them with one day work when both A and B do the same work together.

Complete step by step answer
Given, A takes 5 days less than B to complete the work.
Let B take x days to complete the work, then according to the question, A takes (x – 5) days to complete the same work.
Also given that both A and B together complete the work in 6 days.
Work done in one day = Reciprocal of days taken to complete the total work
B’s one day work = $\dfrac{1}{x}$
A’s one day work = $\dfrac{1}{{x - 5}}$
(A + B)’s one day work = $\dfrac{1}{6}$
According to the question,
$\dfrac{1}{x} + \dfrac{1}{{x - 5}} = \dfrac{1}{6}$
Solving equation,
$\dfrac{{x - 5 + x}}{{x(x - 5)}} = \dfrac{1}{6}$
$\dfrac{{2x - 5}}{{{x^2} - 5x}} = \dfrac{1}{6}$
On cross-multiplying, we get
$12x - 30 = {x^2} - 5x$
Rearranging the equation
${x^2} - 17x + 30 = 0$
Now, this is the quadratic equation, so solving the equation by factorization method.
${x^2} - 15x - 2x + 30 = 0$
$x(x - 15) - 2(x - 15) = 0$
$(x - 15)(x - 2) = 0$
Either $(x - 15) = 0$ or $(x - 2) = 0$
For $x - 15 = 0$
$x = 15$
And
For $x - 2 = 0$
$x = 2$
Here $x = 2$ is not possible because $\left( {x - 5 = 2 - 5 = - 3} \right)$days cannot be negative.
So, $x = 15$ is correct.
Therefore, A complete the work in $\left( {x - 5} \right)$days i.e. $15 - 5 = 10$days

Note: In this type of question, always calculate and compare one day's work by taking reciprocal of the total time taken to do the work. As if a particular work is done in x days and the work done in one day will be $\dfrac{1}{x}$ part of the total work. Never calculate by considering the number of days to complete the total work. And also after finding the unknown value, put that value in the equation formed as it must satisfy the equation.