maths: time and work

1.	A can do a work in 15 days and B in 20 days. If they work on it together for 4 days, then the fraction of the work that is left is :

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

2.	A can lay railway track between two given stations in 16 days and B can do the same job in 12 days. With help of C, they did the job in 4 days only. Then, C alone can do the job in:

(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

3.	A, B and C can do a piece of work in 20, 30 and 60 days respectively. In how many days can A do the work if he is assisted by B and C on every third day?

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.

4.	A is thrice as good as workman as B and therefore is able to finish a job in 60 days less than B. Working together, they can do it in:

Ratio of times taken by A and B = 1 : 3.

The time difference is (3 - 1) 2 days while B take 3 days and A takes 1 day.

If difference of time is 2 days, B takes 3 days.

If difference of time is 60 days, B takes		3	x 60		= 90 days.
		2

So, A takes 30 days to do the work.

A's 1 day's work =	1
	30

B's 1 day's work =	1
	90

(A + B)'s 1 day's work =		1	+	1		=	4	=	2
		30		90			90		45

A and B together can do the work in	45	= 22	1	days.
	2		2

5.	A alone can do a piece of work in 6 days and B alone in 8 days. A and B undertook to do it for Rs. 3200. With the help of C, they completed the work in 3 days. How much is to be paid to C?

C's 1 day's work =	1	-		1	+	1		=	1	-	7	=	1	.
	3			6		8			3		24		24

A's wages : B's wages : C's wages =	1	:	1	:	1	= 4 : 3 : 1.
	6		8		24

C's share (for 3 days) = Rs.		3 x	1	x 3200		= Rs. 400.
			24

6.	If 6 men and 8 boys can do a piece of work in 10 days while 26 men and 48 boys can do the same in 2 days, the time taken by 15 men and 20 boys in doing the same type of work will be:

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.

7.	A can do a piece of work in 4 hours; B and C together can do it in 3 hours, while A and C together can do it in 2 hours. How long will B alone take to do it?

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.

8.	A can do a certain work in the same time in which B and C together can do it. If A and B together could do it in 10 days and C alone in 50 days, then B alone could do it in:

(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.

9.	A does 80% of a work in 20 days. He then calls in B and they together finish the remaining work in 3 days. How long B alone would take to do the whole 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

10.

A machine P can print one lakh books in 8 hours, machine Q can print the same number of books in 10 hours while machine R can print them in 12 hours. All the machines are started at 9 A.M. while machine P is closed at 11 A.M. and the remaining two machines complete work. Approximately at what time will the work (to print one lakh books) be finished ?

(P + Q + R)'s 1 hour's work =		1	+	1	+	1		=	37	.
		8		10		12			120

Work done by P, Q and R in 2 hours =		37	x 2		=	37	.
		120				60

Remaining work =		1 -	37		=	23	.
			60			60

(Q + R)'s 1 hour's work =		1	+	1		=	11	.
		10		12			60

Now,	11	work is done by Q and R in 1 hour.
	60

So,	23	work will be done by Q and R in		60	x	23		=	23	hours  2 hours.
	60			11		60			11

So, the work will be finished approximately 2 hours after 11 A.M., i.e., around 1 P.M.

11.	A can finish a work in 18 days and B can do the same work in 15 days. B worked for 10 days and left the job. In how many days, A alone can finish the remaining work?

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

12.	4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 10 women complete it?

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.

13.	A and B can together finish a work 30 days. They worked together for 20 days and then B left. After another 20 days, A finished the remaining work. In how many days A alone can finish the work?

(A + B)'s 20 day's work =		1	x 20		=	2	.
		30				3

Remaining work =		1 -	2		=	1	.
			3			3

Now,	1	work is done by A in 20 days.
	3

Therefore, the whole work will be done by A in (20 x 3) = 60 days.

14.	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?

P can complete the work in (12 x 8) hrs. = 96 hrs.

Q can complete the work in (8 x 10) hrs. = 80 hrs.

P's1 hour's work =	1	and Q's 1 hour's work =	1	.
	96		80

(P + Q)'s 1 hour's work =		1	+	1		=	11	.
		96		80			480

So, both P and Q will finish the work in		480		hrs.
		11

Number of days of 8 hours each =		480	x	1		=	60	days = 5	5	days.
		11		8			11		11

15.	10 women can complete a work in 7 days and 10 children take 14 days to complete the work. How many days will 5 women and 10 children take to complete the work?

1 woman's 1 day's work =	1
	70

1 child's 1 day's work =	1
	140

(5 women + 10 children)'s day's work =		5	+	10		=		1	+	1		=	1
		70		140				14		14			7

 5 women and 10 children will complete the work in 7 days.