Solution:

Let's assume that the sum of 16 positive integers is X.

We know that the average of 16 positive integers is P.

Therefore, P = X/16

Now, we increase each integer by 50%. This means that each integer becomes 1.5 times its original value.

So, the new sum of 16 integers will be 1.5X.

The new average will be (1.5X)/16.

Calculating the new average

To find out by how much the new average has increased, we need to compare it with the old average.

New average = (1.5X)/16

Old average = X/16 = P

Let's substitute the value of P in the above equation.

Old average = X/16 = (2P - P)/16 = (P + P)/16 = 2P/16

Now, we can compare the new average with the old average.

New average = (1.5X)/16

Old average = 2P/16

We can simplify the above expressions by dividing both the numerator and denominator by 2.

New average = (0.75X)/8

Old average = P/8

Calculating the increase in average

To find out by how much the new average has increased, we need to find the difference between the new average and the old average.

Increase in average = New average - Old average

Increase in average = (0.75X)/8 - P/8

Increase in average = (0.75X - P)/8

Therefore, the new average is increased by (0.75X - P)/8.

Conclusion

In this way, we have found out that the new average is increased by (0.75X - P)/8 when each integer is increased by 50%.