Consider a country that has a deposit insurance strategy to protect banks against bank runs. The insurance premium is 1414 cents for every $100$⁢100 in bank deposits for banks where liabilities account for less than 50%50% of assets and 7070 cents for every $100$⁢100 in bank deposits for banks where liabilities account for more than 50%50% of assets. Suppose that an individual wants to make a deposit and is thinking about two options: Bank A that has $3,500$⁢3,500 billion of assets and $1,925$⁢1,925 billion of liabilities and Bank B that has $3,200$⁢3,200 billion of assets and $1,440$⁢1,440 billion of liabilities. If the planned total insurance premium that the chosen bank will pay for the deposit is $35$⁢35, what was the deposit amount? Round to the nearest whole number if necessary.

We are tasked with finding the deposit amount given the insurance premiums for two banks and the total premium of $35. The premium rate is determined by the proportion of liabilities to assets for each bank.

### Step 1: Calculate the liability-to-asset ratios for Bank A and Bank B.

For **Bank A**:
- Assets = $3,500 billion
- Liabilities = $1,925 billion

Liability-to-asset ratio for Bank A:
\[
\text{Ratio} = \frac{1,925}{3,500} \approx 0.55
\]

Since the liability-to-asset ratio for Bank A is greater than 50%, the premium rate for Bank A is **70 cents per $100** of deposit.

For **Bank B**:
- Assets = $3,200 billion
- Liabilities = $1,440 billion

Liability-to-asset ratio for Bank B:
\[
\text{Ratio} = \frac{1,440}{3,200} = 0.45
\]

Since the liability-to-asset ratio for Bank B is less than 50%, the premium rate for Bank B is **14 cents per $100** of deposit.

### Step 2: Analyze the premium formulas.

Let $ x $ be the deposit amount in dollars.

- For **Bank A**: The premium is $ 0.70 \times \frac{x}{100} = \frac{0.7x}{100} $.
- For **Bank B**: The premium is $ 0.14 \times \frac{x}{100} = \frac{0.14x}{100} $.

### Step 3: Find the correct bank and deposit amount.

We know that the total insurance premium paid is $35, so we now calculate the deposit amount for each case:

#### Case 1: Bank A (Premium rate: 70 cents per $100)
\[
\frac{0.7x}{100} = 35
\]
Solving for $ x $:
\[
0.7x = 35 \times 100
\]
\[
0.7x = 3500
\]
\[
x = \frac{3500}{0.7} = 5000
\]

The deposit amount would be **$5,000**.

#### Case 2: Bank B (Premium rate: 14 cents per $100)
\[
\frac{0.14x}{100} = 35
\]
Solving for $ x $:
\[
0.14x = 35 \times 100
\]
\[
0.14x = 3500
\]
\[
x = \frac{3500}{0.14} = 25,000
\]

The deposit amount would be **$25,000**.

### Step 4: Conclusion

Since the problem specifies the total insurance premium as $35, the calculated deposit amounts are $5,000 for Bank A and $25,000 for Bank B. Either bank could be the choice, depending on which premium rate applies. Therefore, the two possible deposit amounts are:

- **$5,000** for Bank A.
- **$25,000** for Bank B.

### Questions and Further Exploration:
1. How does the premium rate affect the deposit decision between Bank A and Bank B?
2. What would happen to the premium if the liabilities of Bank A were reduced?
3. How does the liability-to-asset ratio influence the financial stability of a bank?
4. What are the benefits of deposit insurance for consumers in this context?
5. How would the premium change for a $10,000 deposit in each bank?

### Tip:
Always verify the liability-to-asset ratio to understand which premium rate applies when comparing banks.

Consider a country that has a deposit insurance strategy to protect banks against bank runs. The insurance premium is 1414 cents for every $100 in bank deposits for banks where liabilities account for less than 50% of assets and 70 cents for every $100 in bank deposits for banks where liabilities account for more than 50% of assets. Suppose that an individual wants to make a deposit and is thinking about two options: Bank A that has $3,500 billion of assets and $1,925 billion of liabilities and Bank B that has $3,200 billion of assets and $1,440 billion of liabilities. If the planned total insurance premium that the chosen bank will pay for the deposit is $35, what was the deposit amount?

Learn how to calculate the deposit insurance premium for Bank A and Bank B using their liability-to-asset ratios. The solution covers ratios, algebra, and step-by-step calculations to find the deposit amount given a premium of $35. Suitable for Grades 9-12.

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