Problem 69 - Intermediate Algebra

Question

Use a proportion to solve each problem. Caffeine. Many convenience stores sell super-size 44 -ounce soft drinks in refillable cups. For each of the products listed in the table below, find the amount of caffeine contained in one of the large cups. Round to the nearest milligram. $$ \begin{array}{|l|c|c|} \hline \text { Soft drink } & \begin{array}{c} \text { Milligrams of } \\ \text { caffeine in } \\ \text { 12 ounces } \end{array} & \begin{array}{c} \text { Milligrams of } \\ \text { caffeine in } \\ \text { 44 ounces } \end{array} \\ \hline \text { Mountain Dew } & 55 & ? \\ \hline \text { Pepsi } & 38 & ? \\ \hline \text { Coca-Cola Classic } & 34 & ? \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Mountain Dew: 202 mg, Pepsi: 139 mg, Coca-Cola Classic: 125 mg for 44 ounces.

Step 1: Understanding Proportions

We are given a table with caffeine content for 12-ounce servings of various soft drinks. The goal is to find the caffeine content for a 44-ounce serving using a proportion. For each soft drink, we will set up a proportion comparing the known caffeine content to the desired quantity.

Step 2: Set Up Proportion for Mountain Dew

For Mountain Dew, the proportion is set as follows: \( \frac{55}{12} = \frac{x}{44} \), where 55 is the caffeine content for 12 ounces, and \( x \) is the caffeine content for 44 ounces.

Step 3: Solve the Proportion for Mountain Dew

To find \( x \), cross-multiply: \( 55 \times 44 = 12 \times x \). This simplifies to \( 2420 = 12x \). Divide both sides by 12 to solve for \( x \): \( x = \frac{2420}{12} \approx 201.67 \). Rounding to the nearest milligram gives approximately 202 mg for 44 ounces of Mountain Dew.

Step 4: Set Up Proportion for Pepsi

Similarly, for Pepsi, we set up the proportion \( \frac{38}{12} = \frac{x}{44} \), where 38 is the caffeine content for 12 ounces, and \( x \) is for 44 ounces.

Step 5: Solve the Proportion for Pepsi

Cross-multiply: \( 38 \times 44 = 12 \times x \). This simplifies to \( 1672 = 12x \). Divide both sides by 12, resulting in \( x = \frac{1672}{12} \approx 139.33 \). Rounding to the nearest milligram gives approximately 139 mg for 44 ounces of Pepsi.

Step 6: Set Up Proportion for Coca-Cola Classic

For Coca-Cola Classic, the proportion is \( \frac{34}{12} = \frac{x}{44} \), where 34 mg is the caffeine content for 12 ounces, and \( x \) is for the 44 ounces.

Step 7: Solve the Proportion for Coca-Cola Classic

Cross-multiply: \( 34 \times 44 = 12 \times x \). This simplifies to \( 1496 = 12x \). Divide both sides by 12 to solve for \( x \): \( x = \frac{1496}{12} \approx 124.67 \). Rounding to the nearest milligram gives approximately 125 mg for 44 ounces of Coca-Cola Classic.

Key Concepts

Solving Proportions

Proportions are a powerful mathematical tool that allow us to find an unknown value when we have a set of equivalent ratios.
In the context of the caffeine content exercise, using proportions helps us scale up the known caffeine content from a smaller amount to a larger amount.
This is done by setting two ratios equal to each other. To solve a proportion problem, follow these simple steps:

• Identify the known quantities. In our exercise, these are the caffeine amounts for 12-ounce servings.
• Set up the proportion equation, where the first ratio includes the known caffeine content and the other represents the unknown caffeine amount for 44 ounces.
• Cross-multiply the terms and solve for the unknown variable by dividing both sides of the equation.
• Round your answer to the nearest milligram as needed.

Using these steps, you can effectively determine the desired caffeine content from a larger serving size.

Caffeine Content

Caffeine is a stimulant found in various beverages, and understanding its content in drinks is crucial for making informed choices.
The exercise addresses caffeine content in popular soft drinks such as Mountain Dew, Pepsi, and Coca-Cola Classic, offering insight into the amount of caffeine in a standard refillable cup size (44 ounces).
Manufacturers usually provide caffeine quantities for a typical serving size, like 12 ounces, which is standard for cans and smaller bottles. However, larger containers, like the 44-ounce cups discussed, require a scaled understanding of caffeine content. For our example:

• Mountain Dew has 55 mg in 12 ounces, which gives an approximate caffeine content of 202 mg per 44 ounces after calculations.
• Pepsi, containing 38 mg in 12 ounces, translates to about 139 mg per 44 ounces.
• Coca-Cola Classic, with 34 mg per 12 ounces, ends up with about 125 mg per 44 ounces.

Knowing caffeine content can assist you to manage daily caffeine intake and understand the effect of such beverages on your energy levels.

Algebraic Methods

Algebraic methods enable us to solve mathematical problems by employing operations to isolate variables and find their values.
In solving proportions—an essential skill for algebra—using cross-multiplication simplifies the process.
Through cross-multiplication, you multiply diagonally across the equal sign to find a product that helps in solving for the unknown.Here's a brief explanation of the cross-multiplication used in the exercise:

• Write the proportion, for example, for Mountain Dew: \( \frac{55}{12} = \frac{x}{44} \).
• Cross-multiply: multiply 55 by 44 and equate it to the product of 12 and \( x \).
• Solve the resulting equation \( 2420 = 12x \) by dividing both sides by 12 to derive \( x \). This results in \( x \approx 201.67 \), which rounds to 202.

Algebraic methods like these offer a structured approach for tackling various quantitative problems beyond caffeine content, enhancing general problem-solving skills.