A can do a piece of work in 10 days. B can do it in 24 days. If C also works with them then it take only 6 days to complete the whole work. In how many days C alone can complete the whole work?

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

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.

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

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.