Problem 58 - Prealgebra

Question

Cooking. A cake recipe calls for \(\frac{1}{2}\) cup of sugar. If the only measuring cup available is a \(\frac{1}{8}\) cup, how many of these will have to be filled with sugar to make a total of \(\frac{1}{2}\) cup of sugar?

Step-by-Step Solution

Answer

You need 4 of the \( \frac{1}{8} \) cup measures.

Step 1: Understand the Requirement

We need to determine how many \( \frac{1}{8} \) cup measures are needed to equal \( \frac{1}{2} \) cup of sugar.

Step 2: Set Up the Equation

Let \( x \) be the number of \( \frac{1}{8} \) cups needed. We can set up the equation as: \( x \times \frac{1}{8} = \frac{1}{2} \).

Step 3: Solve for x

To solve for \( x \), multiply both sides of the equation by 8: \( x = 8 \times \frac{1}{2} = 4 \).

Step 4: Conclusion

We need 4 of the \( \frac{1}{8} \) cup measurements to equal \( \frac{1}{2} \) cup.

Key Concepts

Understanding Ratio and Proportion

Ratio and proportion are important concepts that help understand relationships between quantities. When dealing with fractions, a ratio signifies how one value compares to another. For example, if a recipe requires a certain proportion of sugar, knowing how to maintain that ratio with different measuring tools is key.
In our task, the ratio of the needed sugar is given as \( \frac{1}{2} \) cup, compared to what the available measuring cup can hold, which is \( \frac{1}{8} \) cup. When expressed as a proportion, we compare these values to determine how many smaller measuring cups will equate to the larger requirement.

To find the total number of cups needed, establish the proportion: \(\frac{x}{8} = \frac{1}{2}\). Solving this helps find how many times you need the smaller cup to reach the necessary sugar amount.

Setting Up Equations

Equations act as powerful tools for solving mathematical problems efficiently. When we are given conditions like in the cake recipe, transforming these conditions into an equation can provide a clear path to finding a solution.

In our exercise, the equation is formulated as \( x \times \frac{1}{8} = \frac{1}{2} \). Here, \( x \) represents the number of small cups needed to achieve the total amount of sugar required. This showcases the application of equations in expressing real-life scenarios in a mathematical format.

• The left side of the equation shows the total sugar using small cups.
• The right side represents the sugar needed by the actual recipe.

By setting up this equation, we can solve for \( x \), simplifying the problem-solving process.

Effective Problem Solving Strategies

Problem-solving involves a mix of strategy and analysis, especially when faced with practical tasks like cooking. A good problem-solving approach will enhance accuracy and understanding of the task at hand.

1. **Understand the Problem**: Clearly define what is required. In this scenario, understanding that you need \( \frac{1}{2} \) cup of sugar and you have a \( \frac{1}{8} \) cup as a measuring tool.
2. **Translate into Mathematical Terms**: Convert the problem into a mathematical representation, setting up an equation: \( x \times \frac{1}{8} = \frac{1}{2} \).
3. **Solve the Equation**: Use basic algebraic techniques to solve the equation. Multiply both sides by 8 to isolate \( x \), resulting in \( x = 4 \).

By following these simple steps, one can solve the task effectively, regardless of the complexity or context.