A piece of work was to be completed in 40 days, a number of men employed upon it did only half the work in 24 days, 16 more men were then set on and the work was completed in the specified time, how many men were employed at first?

5 men + 3 boys can reap 23 hectares in 4 days ......(i)
3 men + 2 boys can reap 7 hectares in 2 days ......(ii)

? From (i),
14( 5 men + 3 boys) can reap 23 x 14 hectares in 4 days
Now, from (ii)
? 23 (3 men + 2 boys) can reap 7 x 2 x 23 hectares in 4 days

? 14(5 men + 3 boys) = 23 (3 men + 2 boys)
? 70 men + 42 boys = 69 men + 46 boys
? 1 men = 4 boys
Now, 5 men + 3 boys = 23 boys

? 23 boys can reap 23 hectare 4 days
? 1 boy can reap 1 hectares in 4 days
? 4 boys can reap 1 hectare in 1 day
? 4 x 45 boys can reap 45 hectaes in 1 day
? 4 x 45 / 6 boys can reap 45 hectares in 6 days
? 30 boys can reap 45 hectares in 6 days

But 30 boys = 28 boys + 2 boys = 7 men + 2 boys
Hence, 2 boys can assist 7 men for the work.

? 14 person complete in 16 days = 1 work
? 8 person complete in 12 days = 1 x (8/14) x (12/16) = 3/7
? Remaining work = 1 - 3/7 = 4/7
and total number of person = 8 + 8 = 16
? 14 person do 1 work in 16 days
? 16 person do 4/7 work in 16 x (14/16) x (4/7) = 8 days

Work of (A + B + C) for 1 hour = 1/16 + 1/20 + 1/24 = 37/240
? Work of (A + B + C) for 4 hours = 4 x 37/240 = 37/60
? Remaining work = 1 - 37/60 = 23/60
? Work of (B + C) for 1 hour = 1/20 + 1/24 = 11/120

? 11/120 work is done by (B + C) in 1 hour
? 23/60 work is done by (B + C) = 120/11 x 23/60 = 46/11 hours = 4 hours 11 min.
? The type at which the report was typed = 01 : 00 + 04 : 11 = 05 : 11 p.m.

24 men complete the work in 16 days
? Work that 16 men complete in 12 days = (16/24) x (12/ 16) = 1/2 part of work in 12 days.

32 women complete the work in 24 days
? Work that 16 women complete in 14 days = (16/32) x (14/24) = 7/24 part of work in (12 + 2) = 14 days.

So, the remaining part of the work which done by (sixteen men + sixteen women) and required additional no. of men in 2 days = 1 - (1/2 + 7/24) = 1/2 - 7/24 = 5/24 (part)

Now, in 2 days 5/24 part part of the work is done by (24 x 16)/ 2 x (5/ 24) = 40 men
? Required additional no. of men = 40 - 16 =24.

(B + C)'s 2 day's work = 2 x (1/10 + 1/15) = 1/3
Remaining work = (1 - 1/3) = 2/3
? 1/9 work is done by A in 1 day
? 2/3 work is done by A in (9 x 2) / 3 = 6 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

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

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

(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 the number of men originally employed = x
So, (x - 6) men could finish the work in 15 days and x men could finish the work in 9 days.
? 9x = 15(x - 6)
? x = 15