A and B together can do a piece of work in 30 days. A having worked for 16 days, B finishes the remaining work alone in 44 days. In how many days shall B finish the whole work alone?

B's 10 day's work =	❨	1	x 10	❩	=	2	.
		15				3

Remaining work =	❨	1 -	2	❩	=	1	.
			3			3

Now,	1	work is done by A in 1 day.
	18

∴	1	work is done by A in	❨	18 x	1	❩	= 6 days.
	3				3

Ratio of times taken by Sakshi and Tanya = 125 : 100 = 5 : 4.

Suppose Tanya takes x days to do the work.

5 : 4 :: 20 : x    ⟹ x =	❨	4 x 20	❩
		5

⟹ x = 16 days.

Hence, Tanya takes 16 days to complete the work.

Let 1 man's 1 day's work = x and 1 boy's 1 day's work = y.

Then, 6x + 8y =	1	and 26x + 48y =	1	.
	10		2

Solving these two equations, we get : x =	1	and y =	1	.
	100		200

(15 men + 20 boy)'s 1 day's work =	❨	15	+	20	❩	=	1	.
		100		200			4

∴ 15 men and 20 boys can do the work in 4 days.

Suppose A, B and C take x,	x	and	x	days respectively to finish the work.
	2		3

Then,	❨	1	+	2	+	3	❩	=	1
		x		x		x			2

⟹	6	=	1
	x		2

⟹ x = 12.

So, B takes (12/2) = 6 days to finish the work.

Ratio of rates of working of A and B = 2 : 1.

So, ratio of times taken = 1 : 2.

B's 1 day's work =	1	.
	12

∴ A's 1 day's work =	1	; (2 times of B's work)
	6

(A + B)'s 1 day's work =	❨	1	+	1	❩	=	3	=	1	.
		6		12			12		4

So, A and B together can finish the work in 4 days.

Formula: If A can do a piece of work in n days, then A's 1 day's work =	1	.
	n

(A + B + C)'s 1 day's work =	❨	1	+	1	+	1	❩	=	7	.
		24		6		12			24

Formula: If A's 1 day's work =	1	,	then A can finish the work in n days.
	n

So, all the three together will complete the job in	❨	24	❩ days	=	3	3	days.
		7				7

(B + C)'s 1 day's work =	❨	1	+	1	❩	=	7	.
		9		12			36

Work done by B and C in 3 days =	❨	7	x 3	❩	=	7	.
		36				12

Remaining work =	❨	1 -	7	❩	=	5	.
			12			12

Now,	1	work is done by A in 1 day.
	24

So,	5	work is done by A in	❨	24 x	5	❩	= 10 days.
	12				12

A's 1 hour's work =	1	;
	4

(B + C)'s 1 hour's work =	1	;
	3

(A + C)'s 1 hour's work =	1	.
	2

(A + B + C)'s 1 hour's work =	❨	1	+	1	❩	=	7	.
		4		3			12

B's 1 hour's work =	❨	7	-	1	❩	=	1	.
		12		2			12

∴ B alone will take 12 hours to do the work.

Whole work is done by A in	❨	20 x	5	❩	= 25 days.
			4

Now,	❨	1 -	4	❩	i.e.,	1	work is done by A and B in 3 days.
			5			5

Whole work will be done by A and B in (3 x 5) = 15 days.

A's 1 day's work =	1	, (A + B)'s 1 day's work =	1	.
	25		15

∴ B's 1 day's work =	❨	1	-	1	❩	=	4	=	2	.
		15		25			150		75

So, B alone would do the work in	75	= 37	1	days.
	2		2

Ratio of times taken by A and B = 100 : 130 = 10 : 13.

Suppose B takes x days to do the work.

Then, 10 : 13 :: 23 : x     ⟹     x =	❨	23 x 13	❩	⟹     x =	299	.
		10			10

A's 1 day's work =	1	;
	23

B's 1 day's work =	10	.
	299

(A + B)'s 1 day's work =	❨	1	+	10	❩	=	23	=	1	.
		23		299			299		13

Therefore, A and B together can complete the work in 13 days.