John, a United States resident, is on vacation in Spain and uses his credit card to purchase a souvenir for 184 euros. The bank that issues the credit card converts the purchase price at the foreign exchange rate for that day, and an additional fee of `6%` of the converted cost is applied before the bank posts the charge. If the bank posts a charge of `$212` to John's account, what exchange rate, in Euros per one U.S dollar, did the bank use?

Answer

To find the exchange rate in Euros per one U.S. dollar that the bank used, we can follow these steps:

### Step 1: Define the variables
Let:
- \( P \) = Purchase price in euros = 184 euros
- \( R \) = Exchange rate in euros per U.S. dollar (what we need to find)
- \( C \) = Converted cost in dollars
- \( F \) = Final charge posted to John's account = $212

### Step 2: Set up the equation for the converted cost
The bank converts the purchase price using the exchange rate:
\[ C = P \times R \]
Substituting the value of \( P \):
\[ C = 184 \times R \]

### Step 3: Calculate the final charge including the fee
The bank adds a 6% fee to the converted cost:
\[ F = C + 0.06 \times C = C \times \left(1 + 0.06\right) = C \times 1.06 \]
Substituting the expression for \( C \):
\[ F = (184 \times R) \times 1.06 \]

### Step 4: Set the equation equal to the final charge
Now we can set this equal to the final charge:
\[ 212 = (184 \times R) \times 1.06 \]

### Step 5: Solve for \( R \)
First, simplify the equation:
\[ 212 = 184 \times R \times 1.06 \]
Now divide both sides by 1.06:
\[ \frac{212}{1.06} = 184 \times R \]
Calculating the left side:
\[ 212 \div 1.06 \approx 200 \]
So we have:
\[ 200 = 184 \times R \]
Now, divide both sides by 184 to isolate \( R \):
\[ R = \frac{200}{184} \]

### Step 6: Simplify \( R \)
Now simplify the fraction:
\[ R = \frac{200 \div 8}{184 \div 8} = \frac{25}{23} \]
Calculating the decimal form:
\[ R \approx 1.087 \]

### Step 7: Find the exchange rate in euros per U.S. dollar
The exchange rate \( R \) is in dollars per euro, but we need euros per dollar. Therefore, we take the inverse:
\[ \text{Exchange rate (euros per dollar)} = \frac{1}{R} = \frac{1}{1.087} \approx 0.92 \]

Thus, the exchange rate used by the bank is approximately **0.92 euros per one U.S. dollar**.

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