Problem 57 - Algebra 1: Concepts and Skills

Question

SELLING CARDS Your community center is selling cards. Your goal is to sell 100 dollars worth of cards. Each box sells for 3 dollars. Using mental math, solve the inequality \(3 b \geq 100\) to determine at least how many boxes you must sell to meet your goal.

Step-by-Step Solution

Answer

You have to sell a minimum of 34 boxes in order to reach or exceed your sales goal of 100 dollars.

Step 1: Understanding the Problem

We are given that each box sells for 3 dollars and the goal is to sell at least 100 dollars worth of boxes. We need to find the least number of boxes 'b' which will meet this condition.

Step 2: Setting up the Inequality

Because each box sells for 3 dollars and we want to reach 100 dollars, we can form the inequality \(3 b \geq 100\). Here, 'b' is the number of boxes that needs to be sold.

Step 3: Solving the Inequality

To solve for 'b', opposite operation is performed for both sides of the inequality. In this case, the opposite operation will be division by 3. So, we are going to divide both sides of the inequality by 3, giving \(b \geq \frac{100}{3}\).

Step 4: Calculating the Number of Boxes

Calculating this division gives \(b \geq 33.33...\). As the number of boxes must be a whole number, we should round it up to the nearest whole number. Thus, 'b' should be at least 34.

Key Concepts

Mental Math

Mental math is a powerful tool that allows you to solve simple calculations in your head without the need for paper, pencil, or a calculator. It involves recognizing patterns and using logic to quickly arrive at an answer. For instance, when determining how many boxes of cards need to be sold to reach a goal of $100, you can use mental math techniques to estimate.
To split 100 evenly by 3, you'd first estimate how many times 3 goes into 100 by rounding numbers up or down. Realizing that 3 times 30 equals 90—close to but less than 100—helps you understand that you need to aim higher. By continually adjusting these estimates in your head, you can mentally determine that you need more than 33 and move to the next number.

• Break down large numbers into manageable parts.
• Use rounding to simplify calculations.
• Refine your answer using logical adjustments.

Mental math sharpens your number sense and can speed up solving problems like inequalities, where quick approximations are needed.

Solving Inequalities

Solving inequalities involves finding the range or set of values that make the inequality true. Unlike equations, inequalities show that one side is either greater or smaller than the other, thus leading to multiple possible solutions.
In the case provided, the inequality is written as \(3b \geq 100\). The goal is to find the minimum number of boxes \(b\) that will meet or exceed $100. Solving inequalities typically involves similar steps to solving equations, but it's important to pay attention to the direction of the inequality sign.

• Set up your inequality based on the conditions given.
• Simplify and isolate the variable using algebraic operations.
• Always remember: making the variable subject changes the inequality direction only if you multiply or divide by a negative number. Otherwise, it remains unchanged.

The result here, \(b \geq 33.33\ldots\), is interpreted as needing to sell at least 34 boxes (since you can't sell a fraction of a box). This demonstrates the practical side of inequalities in determining minimal requirements.

Division in Algebra

Division in algebra is a key operation that's often used to isolate variables within equations or inequalities. It's the process of determining how many times one number fits into another. In solving the inequality \(3b \geq 100\), division was employed to isolate \(b\) by dividing both sides by 3.
The step \(b \geq \frac{100}{3}\) relies on understanding that division simplifies the inequality to find the solution for \(b\). This algebraic division must be executed with precision, especially when solutions need to be whole numbers.

• Ensure both sides of an inequality or equation undergo the same operation.
• Check calculations with division to guarantee accuracy, especially when dealing with decimals.
• Remember to interpret fractional or non-integer results by rounding appropriately, where applicable, to suit context requirements (like selling whole items).

Effective use of division in algebra allows tackling real-world problems efficiently and finding workable solutions by interpreting numerical results meaningfully.