Problem 123 - College Algebra

Question

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per text. Plan \(\mathrm{B}\) has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per text. How many text messages in a month make plan A the better deal?

Step-by-Step Solution

Answer

Plan A becomes the better deal when more than 300 text messages are sent in a month.

Step 1: Understand the problem

The problem provides two plans of monthly texting charges, Plan A and Plan B. For Plan A, there is a monthly fee of $15 with a charge of $0.08 per text. For Plan B, the monthly fee is $3 and the charge per text is $0.12. The task is to determine the number of text messages in a month that would make Plan A the better (cheaper) deal.

Step 2: Formulate equations

Convert the plans into linear equations. For Plan A, the monthly cost (C_A) is described by \(C_A = 15 + 0.08t\) where \(t\) is the number of text messages. For Plan B, the monthly cost (C_B) is described by \(C_B = 3 + 0.12t\).

Step 3: Create inequality

Plan A is a better deal if the cost of Plan A is less than the cost of Plan B. This can be expressed with the inequality \(15 + 0.08t < 3 + 0.12t\).

Step 4: Rearrange and solve inequality

Subtract \(15\) and \(0.08t\) from both sides to simplify the equation: \(0 < -12 + 0.04t\). Further simplify this to \(12 < 0.04t\). Divide both sides by \(0.04\) to get that \(300 < t\).

Step 5: Interpret the solution

The solution to the inequality, \(300 < t\), implies that if more than 300 texts per month are sent, Plan A becomes the cheaper or 'better' plan.

Key Concepts

Word Problems

Word problems involve real-life scenarios that require mathematical reasoning to solve. They require careful translation of everyday language into mathematical expressions. This can sometimes be intimidating, but by following systematic steps, you can solve them efficiently.
To tackle a word problem like this one, you need to identify the relevant information and what exactly is being asked. Here, the problem involves comparing two different text messaging plans to determine which is more cost-effective based on the number of texts sent. This involves understanding costs associated with each plan and direct comparisons linked to a variable, like the number of texts, which often leads to forming equations or inequalities.

Text Messaging Plans

Text messaging plans usually come with a combination of a fixed monthly fee and a variable charge per text message. Plan A and Plan B in this exercise illustrate different pricing structures:

• Plan A charges a higher monthly fee ($15) but has a lower cost per text ($0.08).
• Plan B offers a lower monthly fee ($3) but a higher cost per text ($0.12).

The cost structures indicate that the choice between plans depends on how many texts you send. As such, a modeling equation helps determine the tipping point where one plan becomes more advantageous than the other. By using variables, like 't' for the number of texts, you can express the total monthly cost for each plan mathematically and explore their relationships.

Cost Comparison

Cost comparison involves determining which option provides better economic value based on specific criteria. In this case, the inequality strategy is used to make cost comparisons between Plan A and Plan B. The approach involves setting up both cost models as equations, and then creating an inequality to find out when Plan A becomes cheaper:

The inequality formed was: \(15 + 0.08t < 3 + 0.12t\)
By simplifying, you solve for the number of texts needed to make Plan A the better option. Steps include combining like terms and isolating variables until you find that sending more than 300 texts makes Plan A more economical.
This process illustrates a critical idea in cost comparison: focusing on the point where different pricing options intersect to determine the best deal depending on usage. Understanding this concept is particularly vital in everyday decisions involving cost savings and budget optimization.