Answer :
Certainly! Let's work through the problem step by step.
### Step 1: Determine the total amount of work
First, we'll calculate the total units of work necessary to complete the job based on the given information. We know that 45 men and 60 boys can complete the work in 20 days.
So, the total work done can be represented as:
[tex]\[ \text{Total Work} = (\text{Number of men} \times \text{Number of days}) + (\text{Number of boys} \times \text{Number of days}) \][/tex]
[tex]\[ = (45 \times 20) + (60 \times 20) \][/tex]
[tex]\[ = 900 + 1200 \][/tex]
[tex]\[ = 2100 \text{ units of work} \][/tex]
### Step 2: Determine the work done per day by 15 men and 20 boys
The next step involves determining how much work 15 men and 20 boys can accomplish in one day. For this, we can break down how much work one man and one boy can do per day based on the initial scenario.
Given that the total work done by 45 men and 60 boys in 20 days is 2100 units:
[tex]\[ \text{Work done by 45 men in 20 days} = 45 \times 20 = 900 \][/tex]
[tex]\[ \text{Work done by 60 boys in 20 days} = 60 \times 20 = 1200 \][/tex]
Now, we find the amount of work done per day by all the men together and all the boys together:
[tex]\[ \text{Total work per day by men} = \frac{900 \text{ units}}{20 \text{ days}} = 45 \text{ units/day} \][/tex]
[tex]\[ \text{Total work per day by boys} = \frac{1200 \text{ units}}{20 \text{ days}} = 60 \text{ units/day} \][/tex]
Then, work done by one man per day is:
[tex]\[ \text{Work done by one man per day} = \frac{45 \text{ units/day}}{45} = 1 \text{ unit/day} \][/tex]
Similarly, work done by one boy per day is:
[tex]\[ \text{Work done by one boy per day} = \frac{60 \text{ units/day}}{60} = 1 \text{ unit/day} \][/tex]
Thus, 15 men and 20 boys can do:
[tex]\[ \text{Work done per day by 15 men} = 15 \times 1 = 15 \text{ units/day} \][/tex]
[tex]\[ \text{Work done per day by 20 boys} = 20 \times 1 = 20 \text{ units/day} \][/tex]
Combining these:
[tex]\[ \text{Total work per day by 15 men and 20 boys} = 15 + 20 = 35 \text{ units/day} \][/tex]
### Step 3: Calculate the number of days needed to complete the work
Finally, we determine how many days it would take for 15 men and 20 boys to complete the 2100 units of work:
[tex]\[ \text{Total days required} = \frac{\text{Total work}}{\text{Work per day by 15 men and 20 boys}} \][/tex]
[tex]\[ = \frac{2100 \text{ units}}{35 \text{ units/day}} \][/tex]
[tex]\[ = 30 \text{ days} \][/tex]
### Conclusion
Therefore, it will take 15 men and 20 boys 30 days to complete the same piece of work.
### Step 1: Determine the total amount of work
First, we'll calculate the total units of work necessary to complete the job based on the given information. We know that 45 men and 60 boys can complete the work in 20 days.
So, the total work done can be represented as:
[tex]\[ \text{Total Work} = (\text{Number of men} \times \text{Number of days}) + (\text{Number of boys} \times \text{Number of days}) \][/tex]
[tex]\[ = (45 \times 20) + (60 \times 20) \][/tex]
[tex]\[ = 900 + 1200 \][/tex]
[tex]\[ = 2100 \text{ units of work} \][/tex]
### Step 2: Determine the work done per day by 15 men and 20 boys
The next step involves determining how much work 15 men and 20 boys can accomplish in one day. For this, we can break down how much work one man and one boy can do per day based on the initial scenario.
Given that the total work done by 45 men and 60 boys in 20 days is 2100 units:
[tex]\[ \text{Work done by 45 men in 20 days} = 45 \times 20 = 900 \][/tex]
[tex]\[ \text{Work done by 60 boys in 20 days} = 60 \times 20 = 1200 \][/tex]
Now, we find the amount of work done per day by all the men together and all the boys together:
[tex]\[ \text{Total work per day by men} = \frac{900 \text{ units}}{20 \text{ days}} = 45 \text{ units/day} \][/tex]
[tex]\[ \text{Total work per day by boys} = \frac{1200 \text{ units}}{20 \text{ days}} = 60 \text{ units/day} \][/tex]
Then, work done by one man per day is:
[tex]\[ \text{Work done by one man per day} = \frac{45 \text{ units/day}}{45} = 1 \text{ unit/day} \][/tex]
Similarly, work done by one boy per day is:
[tex]\[ \text{Work done by one boy per day} = \frac{60 \text{ units/day}}{60} = 1 \text{ unit/day} \][/tex]
Thus, 15 men and 20 boys can do:
[tex]\[ \text{Work done per day by 15 men} = 15 \times 1 = 15 \text{ units/day} \][/tex]
[tex]\[ \text{Work done per day by 20 boys} = 20 \times 1 = 20 \text{ units/day} \][/tex]
Combining these:
[tex]\[ \text{Total work per day by 15 men and 20 boys} = 15 + 20 = 35 \text{ units/day} \][/tex]
### Step 3: Calculate the number of days needed to complete the work
Finally, we determine how many days it would take for 15 men and 20 boys to complete the 2100 units of work:
[tex]\[ \text{Total days required} = \frac{\text{Total work}}{\text{Work per day by 15 men and 20 boys}} \][/tex]
[tex]\[ = \frac{2100 \text{ units}}{35 \text{ units/day}} \][/tex]
[tex]\[ = 30 \text{ days} \][/tex]
### Conclusion
Therefore, it will take 15 men and 20 boys 30 days to complete the same piece of work.
