Question

A takes 5 hours more time than that taken by B to complete a work. If working together, they can complete a work in 6 hours, then the number of hours that A takes to complete the work individually is
(a) 15
(b) 12
(c) 10
(d) 9

Answer 1

Hint:
Here, we need to find the number of hours taken by A to complete the work individually. Using the given information, we will find the amount of work done by A and B individually in 1 hour. Then, we will find the amount of work done by them together in 1 hour by adding the work done by them in 1 hour individually. Then, using the given information, we will find another equation for the amount of work done by A and B together in 1 hour. Finally, we will equate the two equations, and simplify to find the value of \[x\], and hence the number of hours taken by A to complete the work individually.

Complete step by step solution:
Let the number of hours taken by A and B to complete the work individually be \[x\] and \[y\] hours respectively.
A takes 5 more hours than B to complete the work.
Therefore, we get
\[ \Rightarrow x = y + 5\]
Rewriting the equation, we get
\[ \Rightarrow y = x - 5\]
Now, we know that A takes \[x\] hours to complete the work and B takes \[y\] hours to complete the work.
Therefore, we get
Amount of work done by A in 1 hour individually \[ = \dfrac{1}{x}\]
Amount of work done by B in 1 hour individually \[ = \dfrac{1}{y}\]
The amount of work done by A and B together in 1 hour is the sum of the amount of work done by A and B individually in 1 hour.
Therefore, we get
Amount of work done by A and B together in 1 hour \[ = \dfrac{1}{x} + \dfrac{1}{y}\]………\[\left( 1 \right)\]
Now, it is given that A and B together can complete the work in 6 hours.
Dividing by 6, we get
Amount of work done by A and B together in 1 hour \[ = \dfrac{1}{6}\]………\[\left( 2 \right)\]
From equation \[\left( 1 \right)\] and equation \[\left( 2 \right)\], we can observe that
\[ \Rightarrow \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{6}\]
We will simplify this equation to find the required values.
Substituting \[y = x - 5\] in the equation, we get
\[ \Rightarrow \dfrac{1}{x} + \dfrac{1}{{x - 5}} = \dfrac{1}{6}\]
Taking the L.C.M., we get
\[\begin{array}{l} \Rightarrow \dfrac{{x - 5 + x}}{{x\left( {x - 5} \right)}} = \dfrac{1}{6}\\ \Rightarrow \dfrac{{2x - 5}}{{x\left( {x - 5} \right)}} = \dfrac{1}{6}\end{array}\]
Rewriting the equation, we get
\[ \Rightarrow 6\left( {2x - 5} \right) = x\left( {x - 5} \right)\]
Multiplying the terms of the equation, we get
\[\begin{array}{l} \Rightarrow 12x - 30 = {x^2} - 5x\\ \Rightarrow {x^2} - 5x - 12x + 30 = 0\\ \Rightarrow {x^2} - 17x + 30 = 0\end{array}\]
We can observe that this is a quadratic equation.
We will factorise the quadratic equation to find the value of \[x\].
Factoring the quadratic equation by splitting the middle term, we get
\[\begin{array}{l} \Rightarrow {x^2} - 15x - 2x + 30 = 0\\ \Rightarrow x\left( {x - 15} \right) - 2\left( {x - 15} \right) = 0\\ \Rightarrow \left( {x - 15} \right)\left( {x - 2} \right) = 0\end{array}\]
Therefore, we get
\[ \Rightarrow x - 15 = 0\] or \[x - 2 = 0\]
Simplifying the equations, we get
\[ \Rightarrow x = 15\] or \[x = 2\]
If \[x = 2\], then the number of hours taken by B to complete the work are
\[y = x - 5 = 2 - 5 = - 3\]
The number of hours cannot be negative. Therefore, \[x\] cannot be equal to 2.
Thus, we get
\[ \Rightarrow x = 15\]
Therefore, we get the number of hours taken by A to complete the work individually as 15 hours.

Thus, the correct option is option (a).

Note:
Here, the work done when completed is taken as 1. This is why we divided 1 by the number of hours to complete the work to get the amount of work done in 1 hour.
Also, a common mistake is to leave the answer as \[x = 15\] or \[x = 2\]. We need to remember to reject the value of \[x\] which is not possible.