P takes 6 days less than Q to finish the work individually. If P and Q working together complete the work in 4 days, then how many days are required by Q to complete the work alone ?a)7 daysb)10 daysc)5 daysd)12 dayse)None of theseCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

A takes 6 days less than B to do a work. If both A and B working together can do it in 4 days, how many days will B take to finish it?

Let the time taken by B to complete the work be x days.

in these type of questions , we should convert the work done into 1 day.

So , According to the question

1/x + 1/x-6 = 1/4

=> x-6+x / x(x-6) = 1/4

=> 2x-6 / x^2-6x = 1/4

=> 8x-24 = x^2-6x

=> x^2-14x+24 = 0

=> (x^2-12x)+(-2x+24) = 0

=> (x-12) (x-2) = 0

=> x = 12 , 2

if the time taken by B to complete the work is 2 days then the time taken by A to complete the work will be -4 days which is not possible.

=> time taken by B to complete the work is 12 days

Answer

Let's assume that Q takes x days to complete the work alone.

Given that P takes 6 days less than Q to finish the work individually, we can say that P takes (x - 6) days to complete the work alone.

Now, let's calculate the individual work rates of P and Q.

The work rate of P is 1/(x - 6) (since P completes the work in (x - 6) days) and the work rate of Q is 1/x (since Q completes the work in x days).

When P and Q work together, their work rates are additive.

So, the work rate when P and Q work together is (1/(x - 6)) + (1/x) = 1/4 (since they complete the work in 4 days when working together).

To solve this equation, we can find a common denominator by multiplying both sides by (x(x - 6)).

(x(x - 6))/(x - 6) + (x(x - 6))/x = (x(x - 6))/4

Simplifying this equation, we get:

x + (x - 6) = (x(x - 6))/4

2x - 6 = (x^2 - 6x)/4

Multiplying both sides by 4 to eliminate the denominator, we get:

8x - 24 = x^2 - 6x

Rearranging this equation, we get:

x^2 - 14x + 24 = 0

This is a quadratic equation. We can either factorize it or use the quadratic formula to find the values of x.

By solving this equation, we find that the values of x are 12 and 2.

Since Q cannot take 2 days (as P takes 6 days less than Q), the value of x is 12.

Therefore, Q takes 12 days to complete the work alone.

Hence, the correct answer is option D) 12 days.

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