Problem 21
Question
In \(2008,\) the euro varied in value from \(\$ 1.24 /\) euro to \(\$ 1.60 /\) euro. Express the variation in the dollar cost, \(c,\) of a 140 euro hotel room during 2008 as an inequality.
Step-by-Step Solution
Verified Answer
Answer: 173.60 ≤ c ≤ 224
1Step 1: Find the minimum dollar cost
The given information states that the minimum value of the euro was \(\$ 1.24 /\) euro in 2008. To find the minimum dollar cost, we multiply this exchange rate by the cost of the hotel room in euros:
Minimum dollar cost = 140 euros × \(\$ 1.24 /\) euro
2Step 2: Calculate the minimum dollar cost
Now, we will perform the multiplication:
Minimum dollar cost = 140 × 1.24 = 173.60
3Step 3: Find the maximum dollar cost
The given information states that the maximum value of the euro was \(\$ 1.60 /\) euro in 2008. To find the maximum dollar cost, we multiply this exchange rate by the cost of the hotel room in euros:
Maximum dollar cost = 140 euros × \(\$ 1.60 /\) euro
4Step 4: Calculate the maximum dollar cost
Now, we will perform the multiplication:
Maximum dollar cost = 140 × 1.60 = 224
5Step 5: Write the inequality
Now that we have the minimum and maximum dollar costs, we will write the inequality that represents the variation in the dollar cost, c, of a 140 euro hotel room during 2008:
173.60 ≤ c ≤ 224
Key Concepts
Understanding Exchange RatesIntroduction to Algebraic ExpressionsMultiplication in Currency Conversion
Understanding Exchange Rates
When you travel or purchase items in a different country, you often deal with exchange rates. Exchange rates tell us how much one unit of a currency is worth in another currency. For example, if the exchange rate is \( \\(1.24 / \text{euro} \), it means you need \(1.24\) US dollars to buy one euro.
Exchange rates fluctuate due to factors like supply and demand, interest rates, and economic stability. This fluctuation is what we're exploring in our exercise, where the euro varied between \(\\)1.24\) and \(\$1.60\) against the US dollar in 2008.
In our example, you calculate the cost in dollars for a transaction made in euros by using the exchange rate. For the hotel room exercise, to find how much \(140\) euros would cost in dollars, we multiply the euro amount by the exchange rates given, which are \(1.24\) and \(1.60\). This determines the possible range of costs in US dollars.
Exchange rates fluctuate due to factors like supply and demand, interest rates, and economic stability. This fluctuation is what we're exploring in our exercise, where the euro varied between \(\\)1.24\) and \(\$1.60\) against the US dollar in 2008.
In our example, you calculate the cost in dollars for a transaction made in euros by using the exchange rate. For the hotel room exercise, to find how much \(140\) euros would cost in dollars, we multiply the euro amount by the exchange rates given, which are \(1.24\) and \(1.60\). This determines the possible range of costs in US dollars.
Introduction to Algebraic Expressions
Algebraic expressions are mathematical phrases involving numbers, variables, and operators (like addition and multiplication). They help us express mathematical ideas concisely.
In the exercise, we express the dollar cost of a hotel room in terms of euros and the exchange rates. We start with the base expression that shows the calculation for different scenarios:
This is how we can communicate the variation in cost as an inequality: \(173.60 \leq c \leq 224\). So, no matter the exact exchange rate at the moment of booking, the cost \(c\) will fall within this range.
In the exercise, we express the dollar cost of a hotel room in terms of euros and the exchange rates. We start with the base expression that shows the calculation for different scenarios:
- For the minimum cost: \(140 \times 1.24\)
- For the maximum cost: \(140 \times 1.60\)
This is how we can communicate the variation in cost as an inequality: \(173.60 \leq c \leq 224\). So, no matter the exact exchange rate at the moment of booking, the cost \(c\) will fall within this range.
Multiplication in Currency Conversion
Multiplication is a simple, yet essential, operation we often use in various applications from shopping to budgeting.
In the context of converting currency, multiplication is used to determine how much money you'd need in one currency to have the same value in another currency.
In our exercise, we multiply the hotel room cost (\(140\) euros) by the given exchange rates (\(1.24\) and \(1.60\)) to find out how many US dollars would be required, reflecting different valuations of the euro throughout 2008.
Thus, the operation looks like this:
In the context of converting currency, multiplication is used to determine how much money you'd need in one currency to have the same value in another currency.
In our exercise, we multiply the hotel room cost (\(140\) euros) by the given exchange rates (\(1.24\) and \(1.60\)) to find out how many US dollars would be required, reflecting different valuations of the euro throughout 2008.
Thus, the operation looks like this:
- \(140 \times 1.24 = 173.60\), representing the minimum dollar cost
- \(140 \times 1.60 = 224\), representing the maximum dollar cost
Other exercises in this chapter
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