P can complete a work in 12 days working 8 hours a day. Q can complete the same work in 8 days working 10 hours a day. If both P and Q work together, working 8 hours a day, in how many days can they complete the work?

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

Then, 4x + 6y =	1	and 3x + 7y =	1	.
	8		10

Solving the two equations, we get: x =	11	, y =	1
	400		400

∴ 1 woman's 1 day's work =	1	.
	400

⟹ 10 women's 1 day's work =	❨	1	x 10	❩	=	1	.
		400				40

Hence, 10 women will complete the work in 40 days.

2(A + B + C)'s 1 day's work =	❨	1	+	1	+	1	❩	=	15	=	1	.
		30		24		20			120		8

Therefore, (A + B + C)'s 1 day's work =	1	=	1	.
	2 x 8		16

Work done by A, B, C in 10 days =	10	=	5	.
	16		8

Remaining work =	❨	1 -	5	❩	=	3	.
			8			8

A's 1 day's work =	❨	1	-	1	❩	=	1	.
		16		24			48

Now,	1	work is done by A in 1 day.
	48

So,	3	work will be done by A in	❨	48 x	3	❩	= 18 days.
	8				8

A's 2 day's work =	❨	1	x 2	❩	=	1	.
		20				10

(A + B + C)'s 1 day's work =	❨	1	+	1	+	1	❩	=	6	=	1	.
		20		30		60			60		10

Work done in 3 days =	❨	1	+	1	❩	=	1	.
		10		10			5

Now,	1	work is done in 3 days.
	5

∴ Whole work will be done in (3 x 5) = 15 days.

(A + B)'s 1 day's work =	❨	1	+	1	❩	=	1	.
		15		10			6

Work done by A and B in 2 days =	❨	1	x 2	❩	=	1	.
		6				3

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

Now,	1	work is done by A in 1 day.
	15

∴	2	work will be done by a in	❨	15 x	2	❩	= 10 days.
	3				3

Hence, the total time taken = (10 + 2) = 12 days.

(A + B + C)'s 1 day's work =	1	,
	4

A's 1 day's work =	1	,
	16

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

∴ C's 1 day's work =	1	-	❨	1	+	1	❩	=	❨	1	-	7	❩	=	5	.
	4			16		12				4		48			48

So, C alone can do the work in	48	= 9	3	days.
	5		5

Number of pages typed by Ravi in 1 hour =	32	=	16	.
	6		3

Number of pages typed by Kumar in 1 hour =	40	= 8.
	5

Number of pages typed by both in 1 hour =	❨	16	+ 8	❩	=	40	.
		3				3

∴ Time taken by both to type 110 pages =	❨	110 x	3	❩	hours
			40

= 8	1	hours (or) 8 hours 15 minutes.
	4

(A + B)'s 1 day's work =	1
	10

C's 1 day's work =	1
	50

(A + B + C)'s 1 day's work =	❨	1	+	1	❩	=	6	=	3	. .... (i)
		10		50			50		25

A's 1 day's work = (B + C)'s 1 day's work .... (ii)

From (i) and (ii), we get: 2 x (A's 1 day's work) =	3
	25

⟹ A's 1 day's work =	3	.
	50

∴ B's 1 day's work	❨	1	-	3	❩	=	2	=	1	.
		10		50			50		25

So, B alone could do the work in 25 days.

(20 x 16) women can complete the work in 1 day.

∴ 1 woman's 1 day's work =	1	.
	320

(16 x 15) men can complete the work in 1 day.

∴ 1 man's 1 day's work =	1
	240

So, required ratio	= 1 : 1 240 320
	= 1 : 1 3 4
	= 4 : 3 (cross multiplied)

Work done by X in 4 days =	❨	1	x 4	❩	=	1	.
		20				5

Remaining work =	❨	1 -	1	❩	=	4	.
			5			5

(X + Y)'s 1 day's work =	❨	1	+	1	❩	=	8	=	2	.
		20		12			60		15

Now,	2	work is done by X and Y in 1 day.
	15

So,	4	work will be done by X and Y in	❨	15	x	4	❩	= 6 days.
	5			2		5

Hence, total time taken = (6 + 4) days = 10 days.

A's 1 day's work =	1	;
	15

B's 1 day's work =	1	;
	20

(A + B)'s 1 day's work =	❨	1	+	1	❩	=	7	.
		15		20			60

(A + B)'s 4 day's work =	❨	7	x 4	❩	=	7	.
		60				15

Therefore, Remaining work =	❨	1 -	7	❩	=	8	.
			15			15