Problem 48 - Prealgebra

Question

It takes Carolyn two hours to complete a cross-stitch pattern. Carolyn can spend no more than fourteen hours cross-stitching. Write an inequality that represents this situation and use it to determine whether Carolyn can complete 8 cross-stitch patterns.

Step-by-Step Solution

Answer

Carolyn cannot complete 8 patterns; she can complete at most 7.

Step 1: Understand the Situation

Carolyn can complete one cross-stitch pattern in two hours. She has a maximum of fourteen hours available for cross-stitching.

Step 2: Formulate the Inequality

Let the variable \( x \) represent the number of cross-stitch patterns Carolyn wants to complete. Since each pattern takes 2 hours, the total time she spends is \( 2x \). We know that she can spend no more than 14 hours. Therefore, the inequality is given by:\[ 2x \leq 14 \]

Step 3: Solve the Inequality

To find out how many patterns Carolyn can complete, solve the inequality:1. Divide both sides of the inequality by 2. \[ x \leq \frac{14}{2} \]2. Simplify: \[ x \leq 7 \]This means Carolyn can complete at most 7 cross-stitch patterns.

Step 4: Check the Question

The question asks if Carolyn can complete 8 cross-stitch patterns. From our solution, we found that Carolyn can complete at most 7 patterns.

Key Concepts

Understanding Cross-Stitch as a Hobby

Cross-stitch is a beautiful form of embroidery that involves creating stitched designs on fabric. This craft is loved by many due to its simplicity and flexibility. You can create anything from intricate patterns to simple designs by following a grid.
Cross-stitch is not just a hobby; it's a relaxing activity that helps improve focus and reduce stress.

• Each cross-stitch pattern includes a color chart that guides you on where to place the stitches using threads.
• Patterns can range from small motifs to large landscapes, and each has a different completion time.

For individuals like Carolyn, cross-stitching provides a creative outlet to express skills and artistic vision. Partaking in this hobby requires patience and careful attention to detail. This makes effective time management skills crucial for completing patterns efficiently.

The Importance of Time Management

Time management plays a pivotal role in achieving goals, especially when tackling hobbies like cross-stitching. This skill helps prioritize tasks and ensures that you meet your set objectives without feeling overwhelmed.
Carolyn's example highlights the need for a plan to balance cross-stitching with other daily activities. Here are a few tips to effectively manage your time:

• Set daily goals: Define how many hours or patterns you aim to complete each day.
• Use a planner or calendar: This helps in organizing daily tasks and allocating specific times for cross-stitching.
• Break tasks: Divide your stitching work into smaller tasks to avoid burnout and maintain motivation.

By efficiently planning her time, Carolyn can enjoy her cross-stitching journey without compromising on any other commitments.

Solving Inequalities in Time Allocation

Inequalities are mathematical expressions used to compare quantities and establish limits. They help solve real-world problems where resources or time are limited. In Carolyn's case, the inequality helps determine how many cross-stitch patterns can be completed within her time constraint.
To break this down:

• Let the variable \( x \) represent the number of patterns Carolyn can complete.
• Since each pattern takes 2 hours, the total time spent stitching is \( 2x \).
• The constraint is that she can't exceed 14 hours, forming the inequality: \( 2x \leq 14 \).

Solving this gives \( x \leq 7 \), meaning Carolyn can complete up to 7 patterns but not 8. Understanding this inequality allows for a proper allocation of time, ensuring she stays within her limits while maximizing productivity.