Scenario 5 : Bundling Insurance John is considering bundling his auto and homeowners insurance, which will save him $$ 10 \% $$ on the total combined cost. 1. Calculate the combined cost of his auto and homeowners insurance without the umbrella policy or the discount. 2. Apply the $$ 10 \% $$ discount to find the new total cost. 3. What would John's final cost be if he combined the bundled policy with the umbrella policy? Remember, the umbrella policy costs $$ \$ 300 $$ per year. If John pays his premium every 6 months (twice a year), he gets an additional $$ 5 \% $$ discount on whole policy (Auto, Homeowners, and Umbrella). Calculate his premium payment each 6 mon using the discount.

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To help John evaluate his insurance bundling options, we'll break down each part of the scenario step by step. Since the specific costs of John's auto and homeowners insurance aren't provided, we'll use variables to represent these amounts. You can substitute the actual values once they're known.

Let's define:
- **$$ A $$** = Annual cost of Auto Insurance
- **$$ H $$** = Annual cost of Homeowners Insurance
- **Umbrella Policy Cost** = \$300 per year

### 1. Calculate the Combined Cost Without Discount and Without Umbrella Policy

**Combined Cost = Auto Insurance + Homeowners Insurance**

$$
\text{Combined Cost} = A + H
$$

*Example:*  
If John's Auto Insurance costs \$1,000 annually and his Homeowners Insurance costs \$1,200 annually:
$$
\text{Combined Cost} = \$1{,}000 + \$1{,}200 = \$2{,}200
$$

### 2. Apply the 10% Discount to the Combined Cost

**Discounted Combined Cost = Combined Cost × (1 - Discount Percentage)**

$$
\text{Discounted Combined Cost} = (A + H) \times 0.90
$$

*Using the example numbers:*
$$
\text{Discounted Combined Cost} = \$2{,}200 \times 0.90 = \$1{,}980
$$

### 3. Add the Umbrella Policy to the Bundled Policy

**Final Annual Cost with Umbrella Policy = Discounted Combined Cost + Umbrella Policy Cost**

$$
\text{Final Annual Cost} = (A + H) \times 0.90 + \$300
$$

*With example values:*
$$
\text{Final Annual Cost} = \$1{,}980 + \$300 = \$2{,}280
$$

### 4. Calculate the Premium Payment Every 6 Months with an Additional 5% Discount

When paying semi-annually, John receives an additional 5% discount on the entire policy (Auto, Homeowners, and Umbrella).

**Total After Additional Discount = Final Annual Cost × (1 - Additional Discount)**

$$
\text{Total After Additional Discount} = \$2{,}280 \times 0.95 = \$2{,}166
$$

**Premium Payment Every 6 Months = Total After Additional Discount / 2**

$$
\text{6-Month Premium} = \$2{,}166 \div 2 = \$1{,}083
$$

### Summary with Example Numbers:

1. **Combined Cost Without Discounts:**
   $$
   \$1{,}000 \, (\text{Auto}) + \$1{,}200 \, (\text{Homeowners}) = \$2{,}200
   $$

2. **After 10% Bundle Discount:**
   $$
   \$2{,}200 \times 0.90 = \$1{,}980
   $$

3. **Adding Umbrella Policy:**
   $$
   \$1{,}980 + \$300 = \$2{,}280
   $$

4. **After Additional 5% Semi-Annual Discount:**
   $$
   \$2{,}280 \times 0.95 = \$2{,}166 \
   \text{6-Month Premium} = \$2{,}166 \div 2 = \$1{,}083
   $$

### Final Notes:

- **Without Specific Costs:** If you provide the actual annual costs for John's auto and homeowners insurance, you can substitute them into the formulas above to get precise numbers.
  
- **Discount Applications:** Ensure that the discounts are applied in the correct order:
  1. Bundle the Auto and Homeowners insurance to get a 10% discount.
  2. Add the Umbrella policy.
  3. Apply the additional 5% discount if paying semi-annually.

Feel free to provide the specific insurance costs if you need a more detailed calculation!
John's final cost after bundling and discounts is \$2,166 per year, with a semi-annual premium payment of \$1,083.

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