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Hamilton's method, also known as the method of largest remainders, is used to apportion seats based on population. Here’s how we can apply it to this problem step by step:

### Step 1: Calculate the standard divisor.
- First, sum the populations of all the counties:
  - Adams: 161,000
  - Grant: 121,000
  - Colton: 215,000
  - Davis: 336,000
  - Hayes: 95,000

\[
\text{Total Population} = 161,000 + 121,000 + 215,000 + 336,000 + 95,000 = 928,000
\]

The legislature has 48 seats, so the **standard divisor** is the total population divided by the total number of seats:

\[
\text{Standard Divisor} = \frac{928,000}{48} \approx 19,333.33
\]

### Step 2: Calculate the standard quota for each county.
The standard quota is found by dividing each county's population by the standard divisor:

- Adams: \(\frac{161,000}{19,333.33} \approx 8.33\)
- Grant: \(\frac{121,000}{19,333.33} \approx 6.26\)
- Colton: \(\frac{215,000}{19,333.33} \approx 11.12\)
- Davis: \(\frac{336,000}{19,333.33} \approx 17.38\)
- Hayes: \(\frac{95,000}{19,333.33} \approx 4.91\)

### Step 3: Assign the whole number part of the quotas.
- Adams: 8
- Grant: 6
- Colton: 11
- Davis: 17
- Hayes: 4

Total seats assigned so far: \(8 + 6 + 11 + 17 + 4 = 46\).

### Step 4: Distribute the remaining seats based on the fractional parts.
We need to assign 2 more seats (since 48 seats are available, but only 46 have been assigned so far). We give these to the counties with the largest fractional parts.

- Adams: 0.33
- Grant: 0.26
- Colton: 0.12
- Davis: 0.38
- Hayes: 0.91

Hayes and Davis have the largest fractional parts (0.91 and 0.38), so they each receive one additional seat.

### Final Apportionment:
- **Adams**: 8 seats
- **Grant**: 6 seats
- **Colton**: 11 seats
- **Davis**: 18 seats
- **Hayes**: 5 seats

Would you like any details on this method or any further explanation?

Here are some related questions for further understanding:
1. What is the importance of the standard divisor in Hamilton's method?
2. How do we determine which counties receive the remaining seats?
3. Can Hamilton's method result in more than one solution in certain cases?
4. How does Hamilton's method compare to other apportionment methods like Jefferson's?
5. What happens if the fractional parts are the same for two counties?

**Tip:** Hamilton’s method always ensures that no more seats are distributed than available by rounding down quotas first and then distributing based on largest fractional parts.

The legislature in a state has 48 seats. Apportion these seats to the five counties below using Hamilton's method.

Apportion 48 legislative seats among five counties using Hamilton's method. This detailed solution breaks down the process step-by-step, explaining key concepts in apportionment, including the standard divisor and quota calculations.

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