To solve this problem, we need to first calculate the work rates of men and boys, then combine their rates to find the number of days it would take for 5 men and 8 boys to complete the same job.

Step 1: Find the work rate of 1 man.

If 10 men can complete the job in 15 days, the work done by 10 men in one day is:

$\frac{1}{15} \, \text{(job per day)}.$

Thus, the work rate of 1 man is:

$\frac{1}{15} \div 10 = \frac{1}{150} \, \text{(job per day)}.$

So, 1 man can complete $\frac{1}{150}$ of the job per day.

Step 2: Find the work rate of 1 boy.

Similarly, if 15 boys can complete the job in 16 days, the work done by 15 boys in one day is:

$\frac{1}{16} \, \text{(job per day)}.$

Thus, the work rate of 1 boy is:

$\frac{1}{16} \div 15 = \frac{1}{240} \, \text{(job per day)}.$

So, 1 boy can complete $\frac{1}{240}$ of the job per day.

Step 3: Find the combined work rate of 5 men and 8 boys.

Now, we calculate the combined work rate of 5 men and 8 boys.

The work rate of 5 men is:

$5 \times \frac{1}{150} = \frac{5}{150} = \frac{1}{30} \, \text{(job per day)}.$

The work rate of 8 boys is:

$8 \times \frac{1}{240} = \frac{8}{240} = \frac{1}{30} \, \text{(job per day)}.$

So, the combined work rate of 5 men and 8 boys is:

$\frac{1}{30} + \frac{1}{30} = \frac{2}{30} = \frac{1}{15} \, \text{(job per day)}.$

Step 4: Calculate the number of days to complete the job.

Since 5 men and 8 boys can complete $\frac{1}{15}$ of the job in one day, they will complete the entire job in:

$15 \, \text{days}.$

Therefore, 5 men and 8 boys can complete the job in 15 days.

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Would you like more details or have any questions? Here are some related problems to explore:

1. How long will it take 3 men and 6 boys to complete the same job?
2. What if 12 boys and 4 men work together? How many days would it take?
3. If only 8 boys are available, how long will it take them to finish the job?
4. How many men are needed to complete the job in 10 days?
5. If the work rate of a man doubles, how does that affect the time required for 5 men to finish the job?

Tip: When combining rates, always sum the individual contributions to get the overall rate before calculating the total time.