Problem 15 - Elementary Algebra Within Reach

Question

Find a ratio that compares the relative sizes of the quantities. (Use the same units of measurement for both quantities.) \(\$ 40\) to \(\$ 60\)

Step-by-Step Solution

Answer

The ratio between \$40 and \$60 is 2:3.

Step 1: Understand the Problem

We are required to find a ratio of two dollar amounts, \$40 and \$60. The ratio gives us the relative size of these two numbers.

Step 2: Calculate the ratio

A ratio is obtained by dividing the first quantity by the second quantity. Hence, the ratio of \$40 to \$60 can be calculated as: \(\frac{40}{60}\).

Step 3: Simplify the ratio

Now, we simplify the calculated ratio to its smallest form by dividing both numbers by their greatest common divisor, which is 20, we get the new ratio as \(\frac{2}{3}\). So, \$40 is to \$60 as 2 is to 3.

Key Concepts

Dollar Amounts

Working with dollar amounts is a common mathematical activity, especially when dealing with real-life situations like comparing prices or budgeting. When you're asked to find the ratio of two dollar amounts, like \(\\(40\) and \(\\)60\), it means you're comparing the relative size of these amounts.

• The first step is ensuring both amounts are in the same currency, which they usually are, such as dollars in this case.
• The purpose here is to express how much larger or smaller one amount is compared to the other.

This doesn't require converting to another unit, just keeping them consistent allows for straightforward comparison. In the exercise, we compare \(\\(40\) and \(\\)60\) to find their ratio.

Relative Sizes

Understanding relative sizes is pivotal in many aspects of mathematics and practical applications. It's all about expressing one quantity in relation to another. When you hear the term "relative size," think about how one value compares to another.

• In the context of a ratio, the first number in the ratio tells us how many parts of the first quantity there are compared to the parts of the second quantity represented by the second number in the ratio.
• For example, the ratio of \(\\(40\) to \(\\)60\) can be calculated as \(\frac{40}{60}\).
• This means for every \(40\) units of the first quantity, there are \(60\) units of the second.

A ratio thus provides an exact measure of how one size stands in relation to another, which aids in making informed decisions or comparisons.

Simplifying Ratios

Simplifying ratios is an essential skill that makes them easier to understand and work with. To simplify a ratio, you convert it into its most reduced form by finding and using the greatest common divisor (GCD). Here is how it works:

• First, you identify the GCD of the two numbers in the ratio; for \(40\) and \(60\), the GCD is \(20\).
• You then divide both parts of the ratio by this number, so \(\frac{40}{60}\) becomes \(\frac{40 \div 20}{60 \div 20}\).
• This simplifies to \(\frac{2}{3}\), meaning \(40\) to \(60\) is the same as \(2\) to \(3\).

Having the ratio in simplified form makes it easier to interpret the relationship between the two values and can be particularly handy in more complex calculations.