Question

A takes 6 days less than the time taken by B to finish a piece of work. If both A and B together can finish it in 4 days, find the time taken by B to finish the work.​

Answer 1

$\huge\underline\mathbb {SOLUTION:-}$

$\mathsf {Let\:B\:takes\:x\:days\:to\:finish\:the\:work,}$

$\mathsf {Then,\:A\:alone\:can\:finish\:it\:in\:( x - 6)\:days}$

$\underline\mathsf \red{According\:to\:question,}$

$\mathsf {\frac{1}{x} + \frac{1}{x - 6} = \frac{1}{4} }$

$\implies \mathsf {\frac{x - 6 + x}{x^2 - 6x} = \frac{1}{4} }$

$\implies \mathsf {\frac{2x - 6}{x^2 - 6x} = \frac{1}{4} }$

$\implies \mathsf {x^2 - 6x = 8x - 24}$

$\implies \mathsf {x^2 - 14x + 24 = 0}$

$\implies \mathsf {x^2 - 12x - 2x + 24 = 0}$

$\implies \mathsf {x(x - 12) -2 (x - 12) = 0}$

$\implies \mathsf {(x - 12)\:(x - 2) = 0}$

$\implies \mathsf {x = 12\:or\:x = 2}$

$\mathsf {x = 2\:(Neglect)}$

$\mathsf \blue {So,\:B\:takes\:x = 12\:days.}$

Answer 2

Given:-

If A and B work together they can finish the work in 4 day

● Let the work of B is y

● Work of A = y-6

To find:-

How many time taken by B to finish the work

A.T.Q

So first find How much work they do in 1 day

1 day work of A =1/y-6

1 day Work of B= 1/y

If they work together they finish 1/4 of work in 1 day

Now the solution of your question

$\sf\implies \dfrac{1}{y}+\dfrac{1}{y-6}=\dfrac{1}{4}\\ \\ \sf\implies \dfrac{y-6+y}{y(y-6)}=\dfrac{1}{4}\\ \\ \sf\implies \dfrac{2y-6}{y{}^{2}-6y}=\dfrac{1}{4}\\ \\ \sf\implies \:\:\: cross\: multiplication\\ \\ \sf\implies 4(2y-6)= y{}^{2}-6y\\ \\ \sf\implies 8y-24=y{}^{2}-6y\\ \\ \sf\implies 0=y{}^{2}-6y-8y+24\\ \\ \sf\:\:\:\:\:We\: write\:it\:as\\ \\ \sf\implies y{}^{2}-6y-8y-24=0\\ \\ \sf\implies y{}^{2}-14y+24=0\\ \\ \sf\implies y{}^{2}-(12+2)y+24=0 \\ \\ \sf\implies y{}^{2}-12y-2y+24=0\\ \\ \sf\implies y(y-12)-2(y-12)=0\\ \\ \sf\implies (y-12)(y-2)=0\\ \\ \sf\:\:\:Vlaue\:of\:y =12\:\:and\:\:2$

Now find the work of A

• Put y= 2

Work of A =y-6

• => 2-6= (-4)

(work can't be negative)

So B complete his work In 12 day

and A =12-6= 6days

$\boxed{\sf{\dag{\blue{B\:take \:12days\:to\: complete\:his\:work}}}}$