Exercise : 4


1. A alone would take 8 days more to complete the job than if both A and B would together. If B worked alone, he took 412 days more to complete the job than A and B worked together. What time would they take if both A and B worked together?
(a) 7 days
(b) 5 days
(c) 4 days
(d) 6 days
(e) None of these
Ans.d

Let if both A and B work together, they take x days.

∴ (A + B)’s 1 days’s work = 1x th work

A’s 1 day’s work = 1(x + 8) th work.

B’s 1 day’s work = 1(x + 9/2) th work.

Now, \(\frac{1}{x + 8} + \frac{2}{2x + 9} = \frac{1}{x}\)

⇒ x(2x + 9 + 2x + 16) = (x + 8)(2x + 9)

⇒ 4x2 + 25x = 2x2 + 25x + 72

⇒ x2 = 36 ⇒ x = 6 days

2. 10 men and 15 women together can complete a work in 6 days. It takes 100 days for one man alone to complete the same work. How many days will be required for one woman alone to complete the same work?
(a) 90
(b) 125
(c) 145
(d) 225
(e) None of these
Ans.d

1 man’s 1 day’s work = 1100

(10 men + 15 women)’s 1 day’s work = 16

15 women’s 1 day’s work

= (16 - 10100) = (16 - 110) = 115

∴ 1 woman’s 1 day’s work = 1225

∴ 1 woman alone can complete the work in 225 days.

3. A contractor undertook to do a piece of work in 9 days. He employed certain number of laboures but 6 of them were absent from the very first day and the rest could finish the work in only 15 days. Find the number of men originally employed .
(a) 15
(b) 6
(c) 13
(d) 9
(e) None of these
Ans.a

Let the number of men originally employed be x.

9x = 15(x – 6)

or x = 15

4. After working for 8 days, Anil finds that only 13 of the work has been done. He employs Rakesh who is 60 % efficient as Anil. How many more days will Anil take to complete the job?
(a) 15 days
(b) 12 days
(c) 10 days
(d) 8 days
(e) None of these
Ans.c

In 8 days, Anil does = 13 rd work

∴ in 1 day, he does = 124 th work.

∴ Rakesh’s one day’s work

= 60% of 124 = 140 th work.

Remaining work = 1 - 13 = 23

(Anil and Rakesh)’s one day’s work

= 124 + 140 = 115 th work.

Now, 115 th work is done by them in one day.

23 rd work is done by them in 15 × 23 = 10 days

5. A sum of 25 was paid for a work which A can do in 32 days, B in 20 days, B and C in 12 days and D in 24 days.How much did C receive if all the four work together ?
(a) 143
(b) 163
(c) 153
(d) 173
(e) None of these
Ans.b

A's one day's work = 132

B's one day's work = 120

(B + C)'s one day's work = 112

∴ C's one day's work = 112 - 120 = 130

D's one day's work = 124

∴ (A + B + C + D)'s one day's work

\(\frac{1}{32} + \frac{1}{20} + \frac{1}{30} + \frac{1}{24} = \frac{75 + 120 + 80 + 100}{2400} \)

= 3752400 = 1596 = 532

∴ Out of 532 of work done

130 of the work is done by C.

⇒ Out of 25 paid for the work, C will receive

\(\frac{\frac{1}{30}}{\frac{5}{32}} \times 25\), i.e, 130 × 325 × 25, i.e, 163

6. A and B can do a job in 15 days and 10 days, respectively. They began the work together but A leaves after some days and B finished the remaining job in 5 days. After how many days did A leave?
(a) 2 days
(b) 3 days
(c) 1 day
(d) 4 days
(e) None of these
Ans.b

A’s one day’s work = 115 th work.

B’s one day’s work = 110 th work.

(A + B)’s one day’s work = 115 + 110 = 16 th work.

Let A left after x days.

∴ (A + B)’s x days’ work = x6 th work.

Remaining work = 1 - x6 = (6 - x)6 th work.

Now, in 5 days, work done by B = (6 - x)6 th work.

∴ in 1 day work done by B = (6 - x)30 th work.

and (6 - x)30 = 110

∴ x = 3 days

7. Mr. Suresh is on tour and he has 360 for his expenses. If he exceeds his tour by 4 days he must cut down daily expenses by 3. The number of days of Mr. Suresh’s tour programme is :
(a) 20 days
(b) 24 days
(c) 40 days
(d) 42 days
(e) None of these
Ans.a

Let Suresh undertakes a tour of x days.

Then, expenses for each day = 360x

now, \(\frac{360}{x + 4} = \frac{360}{x} - 3\)

or \(360\left(\frac{1}{x} - \frac{1}{x + 4} \right) = 3\)

or x2 + 4x - 480 = 0 or x = – 24 or x = 20

Since, x ≠ -24 we have x = 20

8. A can knit a pair of socks in 3 days. B can knit the same thing in 6 days. If they are knitting together, in how many days will they knit two pairs of socks?
(a) 4 days
(b) 2 days
(c) 412 days
(d) 3 days
(e) None of these
Ans.a

A’s one day’s work = 13 rd work.

B’s one day’s work = 16 rd work.

(A + B)’s one day’s work = 13 + 16 = 12 nd work.

∴ A and B together can complete the work (knit a pair of socks) in 2 days.

∴ They together knit two pair of socks in 4 days.

9. A can do a job in 3 days less time than B. A works at it alone for 4 days and then B takes over and completes it. If altogether 14 days were required to finish the job, how many days would each of them take alone to finish it?
(a) 17 days, 20 days
(b) 12 days, 15 days
(c) 13 days, 16 days
(d) 14 days, 11 days
(e) None of these
Ans.b

Let B can finish the work in x days.

Then A can finish the work in (x – 3) days.

B’s one day’s work = 1xth work.

A’s one day’s work = 1(x - 3)th work.

A’s 4 days’ work = 4(x - 3)th work.

Remaining work = 1 - 4(x - 3) = \(\frac{x - 7}{x - 3}\)

The remaining work done by B in 14 – 4 = 10 days.

Now, in 10 days, work done by B = \(\frac{x - 7}{x - 3}\) th work.

∴ in 1 day, work done by B = \(\frac{1}{10}\left(\frac{x - 7}{x - 3} \right)\) th work.

and \(\frac{1}{10}\left(\frac{x - 7}{x - 3} \right) = \frac{1}{x}\)

⇒ x = 15 days

∴ B → 15 days and A → 12 days

10. Two workers A and B working together completed a job in 5 days. If A worked twice as efficiently as he actually did and B worked 13 as efficiently as he actually did, the work would have completed in 3 days. Find the time for A to complete the job alone.
(a) 614 days
(b) 534 days
(c) 5 days
(d) 3 days
(e) None of these
Ans.a

(A + B)’s one day’s work 15th work

Let A can do job in x days.

Then,A’s one day’s work = 1x th work.

and B’s one day’s work = 15 - 1x = \(\frac{x - 5}{5x}\)th work.

Now, (2A)'s work + (13) B's work = 13 rd work.

\(\frac{2}{x} + \frac{1}{3}\left(\frac{x - 5}{5x} \right) = \frac{1}{3}\) ⇒ x = 254 = 614 days