A contractor undertook to do a piece of work in 9 days. He employed a certain number of laborious but 6 of them being absent from the very first day, the rest could finish the work in 15 days. Find the number of men originally employed.

(A + B +)'s 2 h work = 1/8 + 1/12 = 5/24
(A + B)'s 8 h work = (5 x 8)/(24 x 2) = 5/6
Work done by A in 9th h = 1/8
Total work done upto 9th h = 5/6 + 1/8 = 23/24
Remaining wok = 1/24
B's 1 h work = 1/12
B can do 1/24 of the work = (1/24) x 12 = 1/2 h
So, both can finish the job in 9¹/₂ h.

Let us assume that each soldiers eats one unit of food per day. Thus, total units of food at the beginning will be 1000 x 30 = 30000. After 10 days the 100 soldiers would have eaten 1000 x 10 = 10000 units of food. Thus, food left after 10 days equals 20000 units. Now, there are a total of 2000 soldiers who eat one unit of food everyday.

Now, 30 x 1000 = 1000 x 10 + 2000 x D (here, D is number of days food will last)
? 30000 - 10000 = 2000 x D
? D = 20000/2000 = 10 days

Using formula M₁D₁T₁W₁ = M₂D₂T₂W₂
M₁ = 15, M₂ = 10, D₁ = 16, D₂ = ?
T₁ = 5, T₂ = 8, W₁ = 15, W₂ = 12
Now, 15 x 16 x 5 x 15 = 10 x D₂ x 8 x 12
D₂ = (15 x 16 x 5 x 15)/(10 x 8 x 12)
= (15 x 15)/12 = 75/4 = 18³/₄ days

Efficiency of A = 100/10 = 10%
Efficiency of B = 100/24 = 4.16%
Efficiency of (A+B+C) = 100/6 = 16.66%

Efficiency of C = (16.66) - (10 + 4.16) = 2.5%
Number of days required by C alone to finish the work = 100/2.5 = 40 days

Let N men were employed at first
? N men do 1/2 of the work in 24 days
? 1 man do the whole work in 24 x 2 x N = 48N days

Now, from the question,
(N + 16) men do the remaining work (1 - 1/2) = 1/2 in (40 - 24) = 16 days

? 1 man do the whole work in 16 x 2 x (N + 16) days
? 48N = 32(N + 16)
? N = 32 men

Work done by C in one min = (1/40 + 1/60) - 1/30 = 1/120
Hence, C can empty the tank in 120 min or 2 h.

The weight of baggage deliver in 1 min by 1st belt = 3/5 tonne
The weight of baggage deliver in 1 min by 2nd belt = 1/2 tonne
The weight of baggage deliver in 1 min by both belt = 3/5 + 1/2 = 11/10 tonne
So, the two belts delivers 1 tonne in 10/11 min.
Hence, required time to deliver 33 tonne = (10/11) x 33 = 30 min

Let Rohit, Harsh and Sanjeev can type x, y and z pages respectively in 1 h.
Therefore, they together can type 4(x + y + z) pages in 4 h
? 4(x + y + z) = 216
? x + y + z = 54 .....(i)
Also, z - y = y - x
i.e., 2y = x + z ......(ii)

From Eqs. (i) and (ii), we get
3y = 54
? y = 18

From Eq. (ii), x + z = 36 ....(iv)
From Eqs. (iii) and (iv),
we get x = 15 and z = 21

(A + B)'s one day work = 1/12
(B + C)'s one day work = 1/16
Now, from the question,
A's 5 days work + B's 7 days work + C's 13 days work = 1
? A's 5 days work + B's 5 days work + B's 2 days work + C's 2 days work + C's 11 days work = 1
(A + B)'s 5 days work + (B + C)'s 2 days work + C's 11 days work = 1
? 5/12 + 2/16 + C's 11 days work = 1
? C's 11 days work = 1 - (5/12 + 2/16) = 11/24
? C's 1 day work = (11/24) x 11 = 1/24
Hence, C can do this work in 24 days.

Fraction of work complete by (A and B) + Fraction of work complete by (B and C) + Fraction of work completed by C = 1
As B did the work for 8 days we can say B did the work for 3 days with A and B did the work for 5 days with C. So, A and B together did the work for 3 days, B and C together did the work for 5 days, C alone did the work for 5 days.
? 3/12 + 5/15 + 5/x = 1 or 5/x = 5/12
i.e., x = 12 or C alone can complete the work in 12 days