Problem 175 - Introductory Chemistry Atoms First

Question

If 1 U.S. dollar is worth \(1.54\) Canadian dollars, how many U.S. dollars are needed to purchase an item that costs 350 Canadian dollars?

Step-by-Step Solution

Answer

Approximately 227.27 U.S. dollars are needed to purchase an item that costs 350 Canadian dollars. However, since we generally deal with whole currency amounts, you'll need 228 U.S. dollars to be sure you have enough to pay the equivalent of 350 Canadian dollars.

Step 1: Determine the conversion rate between currencies

We are given the exchange rate as 1 U.S. dollar being equal to 1.54 Canadian dollars. We can represent this as: 1 USD = 1.54 CAD

Step 2: Set up a proportion to find the amount of U.S. dollars needed

We want to know the amount in U.S. dollars (USD) needed to purchase an item costing 350 Canadian dollars (CAD). We can set up a proportion to find the value of X (amount in USD) needed to buy the item: \(\frac{1 USD}{1.54 CAD}\) \(=\) \(\frac{X USD}{350 CAD}\)

Step 3: Solve for X by cross-multiplying

To find the value of X, we'll cross-multiply the proportion and then divide by the remaining factor: \((1 USD)(350 CAD) = (1.54 CAD)(X USD)\)

Step 4: Simplify and solve for X

Now we can simplify the equation and solve for X: \(350 USD = 1.54X\) To find the value of X, divide both sides of the equation by 1.54: \(X = \frac{350}{1.54}\)

Step 5: Calculate the value of X

Now, calculate the value of X as: \(X ≈ 227.27\)

Step 6: Interpret the result

It means that approximately 227.27 U.S. dollars are needed to purchase an item that costs 350 Canadian dollars. However, since we generally deal with whole currency amounts, you'll need 228 U.S. dollars to be sure you have enough to pay the equivalent of 350 Canadian dollars.

Key Concepts

Exchange Rate

An exchange rate is a value that represents how much one currency can be exchanged for another. It fluctuates daily with the international foreign exchange markets, where currency values are continuously traded.

For example, if the exchange rate between U.S. dollars (USD) and Canadian dollars (CAD) is 1 USD to 1.54 CAD, it means that for each U.S. dollar you exchange, you'll receive 1.54 Canadian dollars in return. Understanding exchange rates is crucial for international transactions and travel, as it allows you to calculate how much of one currency you need to have to obtain a desired amount in another currency.

Proportion

Proportion in mathematics is a statement that two ratios or fractions are equal. It's often used to solve problems involving scaling or conversion from one thing to another, such as currency. Proportions can help you determine unknown values based on known values.

When you set up a proportion for a currency conversion, you are essentially saying that the relationship between one currency and another will be the same when you scale it up to the amount you're interested in exchanging. It's a way of predicting the equivalent value in a different currency.

Cross-Multiplication

Cross-multiplication is a technique used to solve proportions. It involves multiplying diagonally across the equal sign in a proportion equation, essentially multiplying the numerator of one fraction by the denominator of the other and then comparing the result to the product of the remaining numerator and denominator.

This method allows you to isolate the variable (usually represented as 'X') and thus solve for the unknown number. Cross-multiply has its name because if you draw the multiplication as lines across the proportion, they form a cross.

Unit Conversion

Unit conversion is the process of converting a measurement from one unit to another within the same measurement system or between different systems. In terms of currency, unit conversion helps you change an amount of money in one currency (like Canadian dollars) to the equivalent amount in another currency (such as U.S. dollars).

To perform a unit conversion, you need a conversion factor, which in the case of currency is the exchange rate. Unit conversion follows the principle that the amount before and after the conversion should represent the same inherent value, even though numbers and currency may differ.