Problem 56 - Elementary Algebra Within Reach

Question

Writing In your own words, explain what the following proportion represents. $$ \frac{1 \text { gallon of milk }}{\$ 3.89}=\frac{3 \text { gallons of milk }}{\$ 11.67} $$

Step-by-Step Solution

Answer

The proportion represents the equivalent cost of milk per gallon. It means that if one gallon costs \$3.89, then three gallons should cost \$11.67, maintaining the same rate of cost per gallon.

Step 1: Identify the Proportion

Proportions are mathematical statements that two ratios are equal, in this case, the ratio is the price of milk to the amount purchased. In this notation, \(\frac{1 gallon of milk}{\$ 3.89}\) stands for how much one gallon of milk costs, and \(\frac{3 gallons of milk}{\$ 11.67}\) states the price of three gallons of milk.

Step 2: Interpret the Proportion

This proportionality establishes that one gallon of milk costs \$3.89, and this price scales up proportionately such that three gallons of milk cost \$11.67. This proportion thus represents the equality of ratios, ensuring that the cost per gallon stays the same for different quantities purchased.

Key Concepts

Ratios

Ratios are a fundamental concept in mathematics that compare two quantities, showing how much of one thing exists in relation to another. In our everyday life, we use ratios to simplify comparisons and to understand relationships between different items. For instance, in the given exercise, the ratio \( \frac{1 \text{ gallon of milk}}{\\( 3.89} \) indicates that for every one gallon of milk, it costs \\)3.89. This makes ratios incredibly useful when trying to determine proportional relationships or when making comparisons in a consistent format.

When dealing with ratios, it's important to maintain the same units for each part being compared. This ensures that you are making an accurate and meaningful comparison. Additionally, ratios can be simplified much like fractions, providing a clearer understanding of the relationship between the quantities involved.

To master ratios, practice is key. Try considering various scenarios where you can identify and create ratios to see how they aid in making comparisons.

Price Comparison

Price comparison is the process of evaluating the cost of similar items to determine the best value for money. It is a practical application of ratios where prices are compared to the quantities received. In the given exercise, the comparison is made between the cost of one gallon of milk and the cost of three gallons. By establishing a ratio, it becomes easier to see that although the quantity increases, the price per gallon remains the same, demonstrating uniformity in pricing.

When engaging in price comparisons, consider the following:

• Compare consistently – ensure that quantities and units are the same to make valid comparisons.
• Look for proportional relationships – this can indicate whether you're getting a fair deal irrespective of the quantity purchased.

Comparison shopping isn't just for the consumer—it's also beneficial in business and economics, where understanding pricing structures is crucial.

Equality of Ratios

The equality of ratios forms the basis of proportions in mathematics. When two ratios are equal, it implies that there's a consistent relationship between the quantities involved. In the exercise provided, the two ratios: \( \frac{1 \text{ gallon of milk}}{\\( 3.89} = \frac{3 \text{ gallons of milk}}{\\) 11.67} \) demonstrate this equality.

This means that despite an increase in the quantity of milk, the price per gallon doesn't change; it's as if prices were scaled proportionately. Therefore, understanding this equality helps in verifying consistent pricing across different purchases or verifying consistent measurements in recipes, projects, and more.

Noting these equal ratios can also aid in solving more complex mathematical problems where proportionality is key, such as in solving geometry problems or converting units.

Mathematical Statements

Mathematical statements like proportions allow us to express complex relationships succinctly and clearly. They are used to represent ideas and relationships in an abstract way, using symbols and numbers to convey accurate information.

In the context of the exercise, the mathematical statement \( \frac{1 \text{ gallon of milk}}{\\( 3.89} = \frac{3 \text{ gallons of milk}}{\\) 11.67} \) communicates the concept of consistent pricing between differing quantities. This form of representation is powerful, as it permits a straightforward comparison and analysis.

To better understand and use mathematical statements, it's important to:

• Familiarize yourself with mathematical symbols and their meanings.
• Practice translating word problems into mathematical statements to enhance comprehension.
• Understand the logic behind each statement, ensuring clarity in problem-solving processes.

These statements form the backbone of mathematical problem-solving, helping students and professionals alike to see and solve real-world problems with precision.