Problem 4 - Algebra 2

Question

On January \(1,1999,\) the euro became legal tender in 11 participating countries in Europe. Based on the exchange rate on one particular day, the linear function \(d(x)=0.8881 x\) could be used to convert \(x\) euros to U.S. dollars. On that day, what was the value in euros of 500 U.S. dollars?

Step-by-Step Solution

Answer

The value in euros is approximately 563.19.

Step 1: Understand the Linear Function

The function given is \( d(x) = 0.8881x \), where \( x \) represents the amount in euros, and \( d(x) \) represents the amount in U.S. dollars. Thus, it converts euros to dollars.

Step 2: Convert the Function to Solve for Euros

To find the value of euros from U.S. dollars, we need to manipulate the function. Rearrange \( d(x) = 0.8881x \) to solve for \( x \): \( x = \frac{d(x)}{0.8881} \).

Step 3: Substitute the Dollar Amount into the Formula

We are given \( d(x) = 500 \) U.S. dollars. Substitute this value into the rearranged formula: \( x = \frac{500}{0.8881} \).

Step 4: Perform the Calculation

Calculate the division to find \( x \): \( x = \frac{500}{0.8881} \approx 563.19 \). Therefore, the value in euros is approximately 563.19.

Key Concepts

Currency Conversion

Currency conversion is a process that allows you to determine the equivalent value of one currency in the terms of another. This is essential when dealing with international transactions, travel, or global financial analyses. Each currency pair has an exchange rate, which is the key factor in this conversion. For example, euros to U.S. dollars might have an exchange rate of 0.8881, which means 1 euro can be exchanged for 0.8881 dollars on a specific day. It's important to understand two main points:

• Exchange rates can fluctuate, meaning the equivalent amount you receive can change from day to day.
• Often, financial institutions add a small fee or conversion rate difference to the market rate, resulting in slightly different conversion results depending on where you do the exchange.

These conversion rates and calculations allow businesses and individuals to understand and manage their finances more effectively when dealing with multiple currencies.

Algebraic Manipulation

Algebraic manipulation involves rearranging mathematical expressions to isolate variables and solve equations. This skill is particularly useful in currency conversion when we need to reverse a calculation, such as determining euros from a given amount of dollars.Starting with the linear function given for conversion, such as \( d(x) = 0.8881x \), you can solve for \( x \) by applying algebraic manipulation.Here's how you do it step by step:

• Identify the equation: \( d(x) = 0.8881x \).
• Rearrange the formula to solve for \( x \): This involves dividing both sides of the equation by the coefficient of \( x \), which is 0.8881.
• This gives you the new equation: \( x = \frac{d(x)}{0.8881} \).

By systematically manipulating equations, you can find unknown variables, which is a crucial skill not only in currency conversion but also in broader mathematical and real-life problem-solving scenarios.

Euro to Dollar Exchange

Exchanging euros for U.S. dollars or vice versa involves understanding how the specific exchange rates determine the equivalent value you receive in return. Linear functions simplify this conversion process, allowing quick calculations.For instance, if the linear function is \( d(x) = 0.8881x \), it indicates the relationship between the euro (\( x \)) and the dollar (\( d(x) \)). When you have a certain number of dollars, such as 500, and want to know how many euros that equates to, algebraic manipulation comes into play.Following these steps will help you convert dollars back to euros:

• Start with the conversion formula: \( d(x) = 0.8881x \).
• To find euros, rearrange the formula to solve for \( x \): \( x = \frac{d(x)}{0.8881} \).
• Substitute \( d(x) = 500 \) into the equation, and calculate: \( x = \frac{500}{0.8881} \approx 563.19 \).

This result implies that 500 U.S. dollars is about 563.19 euros at that specific exchange rate. Such calculations are pivotal in determining how currency values relate during international trading or travel.