Problem 312 - Elementary Algebra

Question

In the following exercises, solve. Marcela is registering for her college classes, which cost \(\$ 105\) per unit. How many units can she take to have a maximum cost of \(\$ 1,365 ?\)

Step-by-Step Solution

Answer

Marcela can take up to 13 units.

Step 1: Understand the Relationship

The problem describes a linear relationship between the number of units Marcela wants to register for and the total cost. Specifically, the total cost is the product of the number of units and the cost per unit.

Step 2: Set Up the Inequality

Let the number of units be represented by the variable . According to the problem, the cost per unit is \( \$105 \) and the total cost should not exceed \( \$1,365 \). Therefore, we set up the inequality: \times 105 ≤ 1365.

Step 3: Solve for the Number of Units

To find the maximum number of units (), we need to solve the inequality: \times 105 ≤ 1365. Dividing both sides of the inequality by 105, we get ≤ \( \frac{1365}{105} \).

Step 4: Simplify the Fraction

Now, simplify \( \frac{1365}{105} \). By dividing the numerator and the denominator by 105, we find that \( \frac{1365}{105} = 13 \).

Step 5: Conclusion

Since must be a whole number of units, the maximum number of units Marcela can take without exceeding the budget is = 13.

Key Concepts

linear relationship

In the given exercise, we deal with a **linear relationship** between the cost of college classes and the number of units Marcela wants to take. A linear relationship is one where the change in one variable is directly proportional to the change in another variable. This means if you know the relationship's formula, you can easily predict one variable based on the other.

For Marcela's situation, the cost (C) is directly proportional to the number of units (u) she registers for, at a rate of $105 per unit. This relationship can be expressed as:

\(\text{Cost} (C) = \text{number of units} (u) \times \text{cost per unit}\rm{(105)}\)

Understanding this concept helps you easily set up and solve related problems by identifying the pattern and using the given formula. So, recognizing this linear relationship is the foundation for proceeding further in solving the problem.

inequalities

Inequalities are mathematical expressions that show the relationship between two values where they are not necessarily equal. They use symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

In this exercise, we deal with an inequality to ensure Marcela's total cost does not exceed a certain limit. Mathematically, we set this up as:

\(( \text{number of units} \times 105 ) \text{≤} 1365 \)

Solving this inequality involves isolating the variable (number of units) on one side. Here's how you break it down:

1. Write down the inequality: \( 105u \text{≤} 1365\)
2. Divide both sides by 105 to isolate u: \(u \text{≤} \frac{1365}{105}\)
3. Simplify the fraction: \(u \text{≤} 13\)

This final inequality, \(\text{u} \text{≤} 13\)\, tells us that Marcela can register for up to 13 units without exceeding her budget. Inequalities are powerful tools in math for expressing limits and ranges within specific constraints.

cost calculation

Cost calculation is essential in budgeting and expense management. Here, we calculate how much Marcela can spend without exceeding a set amount. The cost per unit is multiplied by the number of units to get the total cost.

Given Marcela's maximum budget of \$1365\, and the cost per unit being \$105\, her total expense for \(u\) units is expressed as:

\( \text{Total Cost} = u \times 105 \)

To stay within her budget, this total cost must not exceed \$1365\. This sets up the inequality \( u \times 105 \text{≤} 1365 \)
, as previously discussed.

To solve for the number of units, you follow these steps:

• Divide both sides of the inequality by 105: \(\frac{1365}{105} = 13\)
• Marcela can afford a maximum of 13 units, as confirmed by simplifying and solving the inequality.

By understanding cost calculations, Marcela can effectively plan her expenses and make financial decisions to meet her educational goals without overspending.