Sure, let's work through this step by step.
First, we need to understand the relationship between the number of labourers, the number of hours they work per day, the number of days they work, and the size of the field they dig.
1. We start by calculating the amount of work done per labourer per hour in the first scenario.
In the first scenario:
- 60 labourers
- working 8 hours a day
- for 12 days
- dig a field of 25 ropanees
Total labour hours in the first scenario:
[tex]\( \text{Total labour hours} = 60 \times 8 \times 12 = 5760 \)[/tex]
Work done per labour hour in the first scenario:
[tex]\( \text{Work per labour hour} = \frac{25 \text{ ropanees}}{5760 \text{ labour hours}} = 0.00434 \text{ ropanees per labour hour} \)[/tex]
2. Now, we consider the second scenario and use it to find out how many hours are needed per day.
In the second scenario:
- 64 labourers
- working some number of hours a day (let's call this number [tex]\( x \)[/tex])
- for 18 days
- to dig a field of 30 ropanees
Total labour hours in the second scenario ([tex]\( H_2 \)[/tex]):
[tex]\( \text{Total labour hours in this scenario} = 64 \times x \times 18 \)[/tex]
Work needed per labour hour in the second scenario ([tex]\( W_2 \)[/tex]):
[tex]\( \text{Work per labour hour in second scenario} = \frac{30 \text{ ropanees}}{64 \times 18 \times x} \)[/tex]
3. We know that the work done per labour hour in both scenarios should be the same for equivalence.
Therefore,
[tex]\( \frac{30}{64 \times 18 \times x} = 0.00434 \)[/tex]
4. Solving for [tex]\( x \)[/tex]:
[tex]\( 30 = 0.00434 \times 64 \times 18 \times x \)[/tex]
[tex]\( 30 = 4.99488 \times x \)[/tex]
[tex]\( x = \frac{30}{4.99488} \)[/tex]
[tex]\( x \approx 6 \)[/tex]
5. Therefore, 64 labourers need to work 6 hours a day to dig a field of 30 ropanees in 18 days.
