Problem 56 - Prealgebra

Question

If 1 cup of regular coffee contains 105 milligrams of caffeine, how much caffeine is contained in 3.5 cups of coffee?

Step-by-Step Solution

Answer

3.5 cups of coffee contain 367.5 milligrams of caffeine.

Step 1: Identify the known values

We know that 1 cup of coffee contains 105 milligrams of caffeine. This is our unit rate of caffeine per cup.

Step 2: Set up the calculation

We want to find out how much caffeine is in 3.5 cups. To do this, we'll multiply the unit rate of caffeine by the number of cups: \[\text{Caffeine in 3.5 cups} = 105 \times 3.5\]

Step 3: Perform the multiplication

Calculate the multiplication:\[105 \times 3.5 = 367.5\]This means 3.5 cups of coffee contain 367.5 milligrams of caffeine.

Key Concepts

Understanding Unit Rate

In many real-life situations, you often need to compare quantities, and a unit rate is a way to make these comparisons straightforward. A unit rate tells you how much of a certain quantity corresponds to a single unit of something else. For example, in the context of the caffeine content of coffee, a unit rate describes how many milligrams of caffeine there are in one cup of coffee.

• The unit rate helps you establish a baseline for computation, making it essential for scaling to larger or smaller quantities.
• Unit rates are particularly useful in finding out the cost per item, speed per hour, or—as in our exercise here—the amount of caffeine per cup.

Understanding the unit rate serves as a foundational step in tackling a variety of problems, enabling clear comprehension and effective calculation.

Caffeine Calculation

Calculating caffeine content in multiple cups of coffee is a simple yet important application of multiplication in mathematics. You begin with a known quantity—105 milligrams per cup—and use multiplication to find the caffeine content in a desired number of cups.

• First, identify the unit rate, which in this case, is 105 milligrams per cup.
• Next, multiply the unit rate by the number of cups you have to find the total caffeine content. For 3.5 cups, the calculation is:

\[ \text{Caffeine in 3.5 cups} = 105 \times 3.5 \]
This results in a total of \[ 367.5 \text{ milligrams} \] of caffeine across 3.5 cups. This calculation is straightforward but illustrates how unit rates act as a building block for understanding and solving practical problems.

Effective Problem Solving Techniques

Solving the caffeine content problem involves a structured approach to problem solving. This not only helps get to the correct answer but also enhances your understanding of the process.

• **Identify Known Values:** Always start by clearly defining what you already know—this includes both the unit rates and the quantity you're solving for.
• **Set Up the Calculation:** Clearly arrange what needs to be calculated. This involves formulating a strategy which, in our example, is multiplying the unit rate by the number of cups.
• **Execute the Calculation:** Carefully perform the necessary arithmetic operation. Double-check your work as needed.

Adopting these systematic approaches in problem solving can significantly increase efficiency and accuracy. Breaking down the problem into these manageable steps helps build confidence and competence, especially when faced with new challenges.