Problem 46 - Elementary Algebra

Question

The classic "Shirley Temple" cocktail requires 1 part cherry syrup to 4 parts lemon-lime soda. How much cherry syrup is needed to mix the cocktail given a 12 -ounce can of lemon-lime soda?

Step-by-Step Solution

Answer

You need 3 ounces of cherry syrup.

Step 1: Understand the Ratio

The recipe describes a ratio of 1 part cherry syrup to 4 parts lemon-lime soda. This means for every 1 unit of cherry syrup, you need 4 units of soda.

Step 2: Determine the Total Parts

In this ratio, there are a total of 1 + 4 = 5 parts. This means you are dividing the total mixture into 5 equal parts.

Step 3: Calculate One Part

Since you have 12 ounces of lemon-lime soda representing 4 parts, one part is equal to \( \frac{12}{4} = 3 \) ounces.

Step 4: Calculate Cherry Syrup Amount

According to the ratio, you need 1 part cherry syrup. Since one part is 3 ounces (as calculated), you need 3 ounces of cherry syrup.

Key Concepts

Unit Conversion

Unit conversion is the process of changing a measurement from one unit to another. It's crucial in many real-world applications, especially in recipes where different ingredients may be measured in different units. For our cocktail problem, we had every ingredient measured in ounces, making it unnecessary to convert units. However, understanding unit conversion can be beneficial for tackling similar problems.

• Suppose the lemon-lime soda was measured in milliliters instead of ounces, you’d need to convert it to ounces.
• Knowing conversion factors can help, for example, 1 ounce is approximately 29.57 milliliters.

Learning to convert between various units allows you to work with compatible measurements, ensuring accuracy in recipes and scientific calculations alike.

Proportions

Proportions are mathematical expressions that state two ratios are equal. They're crucial for solving problems where two quantities vary together in a fixed ratio. In our cocktail scenario, we used proportions to maintain the correct balance between cherry syrup and lemon-lime soda. In the recipe:

• Understanding the given ratio of 1:4 tells us how much of each ingredient to use relative to each other.
• By acknowledging that 12 ounces of soda make up '4 parts', we managed to evaluate '1 part' for cherry syrup.

By keeping the proportion constant, the taste and quality remain unaffected by the quantity change. This concept is widely applicable in cooking, chemistry, and even in mixing paint colors!

Problem Solving

Problem solving is the art of finding solutions to complex questions or situations. It's a skill that you hone by familiarizing yourself with problem-solving strategies like breaking down the problem. For our cocktail recipe, we adopted a step-by-step approach to reach a solution. To solve the problem, we:

• Identified the ratio needed for cherry syrup and soda.
• Calculated the total number of parts in the ratio.
• Determined how much one part was by dividing the total soda amount.
• Finally, calculated the cherry syrup requirement using our understanding of the ratio.

Each step served a purpose and was built on the previous one, illustrating the importance of a structured approach. Problem solving is about understanding the given data, employing logical reasoning, and drawing fact-based conclusions. This structured methodology is useful beyond classroom exercises, aiding in decision-making and strategic planning in everyday situations.