#### Answer

Using Hamilton's method, each clinic is apportioned the following number of doctors:
Clinic A is apportioned 6 doctors.
Clinic B is apportioned 8 doctors.
Clinic C is apportioned 12 doctors.
Clinic D is apportioned 14 doctors.

#### Work Step by Step

We can find the total patient load.
total load = 275 + 392 + 611 + 724 = 2002
We can find the standard divisor.
$standard~divisor = \frac{total~load}{number~of~ doctors}$
$standard~divisor = \frac{2002}{40}$
$standard~divisor = 50.05$
The standard divisor is 50.05
We can find the standard quota for each clinic.
Clinic A:
$standard~quota = \frac{patient~load}{standard~divisor}$
$standard~quota = \frac{275}{50.05}$
$standard~quota = 5.49$
Clinic B:
$standard~quota = \frac{patient~load}{standard~divisor}$
$standard~quota = \frac{392}{50.05}$
$standard~quota = 7.83$
Clinic C:
$standard~quota = \frac{patient~load}{standard~divisor}$
$standard~quota = \frac{611}{50.05}$
$standard~quota = 12.21$
Clinic D:
$standard~quota = \frac{patient~load}{standard~divisor}$
$standard~quota = \frac{724}{50.05}$
$standard~quota = 14.47$
Hamilton's method is an apportionment method that involves rounding each standard quota down to the nearest whole number. Surplus doctors are given, one at a time, to the clinics with the largest decimal parts in their standard quotas until there are no more surplus doctors.
Initially, each clinic is apportioned its lower quota.
Clinic A is apportioned 5 doctors.
Clinic B is apportioned 7 doctors.
Clinic C is apportioned 12 doctors.
Clinic D is apportioned 14 doctors.
The total number of doctors which have been apportioned is 5 + 7 + 12 + 14 = 38 doctors
Since there is a total of 40 doctors, there are two surplus doctors. The first doctor is given to Clinic B because it has the largest decimal part (0.83) in its standard quota. The second doctor is given to Clinic A because it has the second largest decimal part (0.49) in its standard quota.
Using Hamilton's method, each clinic is apportioned the following number of doctors:
Clinic A is apportioned 5 + 1 = 6 doctors.
Clinic B is apportioned 7 + 1 = 8 doctors.
Clinic C is apportioned 12 doctors.
Clinic D is apportioned 14 doctors.