Question

A takes 49 days more than C and 45 days morethan B to complete a work. C takes as much timeas A and B together to complete the work. In howmany days A and B together will complete thework?​

Answer 1

Let time taken by A, B and C to complete the work separately be x, y and z days respectively.

According to the question:

1. A takes 49 days more than C

$\Rightarrow x = z + 49 ...... (1)$

2. A takes 49 days more than B

$\Rightarrow x = y + 45 ...... (2)$

3. C takes as much time as A and B together to complete the work.

As per the formula:

$\Rightarrow z = \dfrac{xy}{(x+y)} ...... (3)$

Putting value of z in equation (1):

$\Rightarrow x = \dfrac{xy}{(x+y)} + 49$

Taking LCM and solving:

$\Rightarrow x\times (x+y) = xy + 49 \times (x+y)\\\Rightarrow x^{2} + xy = xy + 49x + 49y\\\Rightarrow x^{2} - 49x - 49y = 0$

Putting $y = (x - 45)$ from equation (2):

$\Rightarrow x^{2} - 49x - 49(x - 45) = 0\\\Rightarrow x^{2} - 49x - 49x + 49\times45 = 0\\\Rightarrow x^{2} - 98x + 2205 = 0$

Solving the quadratic equation:

$\Rightarrow x^{2} - 35x - 63x + 2205 = 0\\\Rightarrow (x-35)(x-63)=0\\\Rightarrow x = 35, 63$

Using $x = 35$:

As per equation (1), $z = -14 \text { days }$

Negative number of days are not possible.

So, x = 35 is not possible.

Using $x = 63$:

$\Rightarrow z = 14 days.$

So, number of days taken by A and B together to complete the work is 14 days.