Problem 41 - Algebra 1: Concepts and Skills

Question

You want to paint a piece of pottery. The total price is the cost of the piece plus an hourly painting rate. Studio A sells a vase for \(\$12\) and lets you paint for \(\$7\) an hour. Studio B sells a similar vase for \(\$15\) and lets you paint for \(\$4\) an hour. Which equation would you use to compare the total price at each studio? A. \(7 x-12=4 x-15\) B. \(12+7 x=15+4 x\)

Step-by-Step Solution

Answer

The correct equation to compare the total price at each studio is: \(12 + 7x = 15 + 4x\), which is Option B.

Step 1: Identifying the costs for the studios

First, we must recognize how the total cost is calculated in each studio. Studio A sells a vase for \(\$12\) and charges \(\$7\) for each hour of painting. Therefore, the total cost for Studio A can be represented as \(12 + 7x\), where \(x\) signifies the number of hours spent painting. Similarly, Studio B sells their vase for \(\$15\) and charges \(\$4\) for each hour of painting. So, the cost for Studio B is represented as \(15 + 4x\).

Step 2: Comparing the cost equations

The next step is to compare the cost equations presented in the exercise options. Option A, \(7x - 12 = 4x - 15\), doesn't accurately reflect the total cost at each studio, as it suggests subtracting the cost of the vase, which is wrong. Option B, \(12 + 7x = 15 + 4x\), correctly represents the total cost for each studio. It adds the cost of each vase to the respective hourly painting fee.

Key Concepts

Cost Comparison

When deciding between two services, understanding cost comparison is crucial. In this scenario, you want to compare the total cost of painting a vase at two different studios. Each studio has a different initial cost for the vase and different hourly rates for painting. By comparing these costs, you can make informed decisions based on your time and budget. To compare, list down each expense:

• Studio A: Vase cost is \( \\(12 \) and \( \\)7 \) per hour of painting.
• Studio B: Vase costs \( \\(15 \) and \( \\)4 \) per hour of painting.

These costs are then used to form equations that include both the initial vase cost and the variable time cost. For the best price, calculate the total expense at various hours and find the breakeven point where the costs equate. This helps you determine the most economical choice based on how long you plan to paint.

Hourly Rate

The hourly rate is the fee charged for each hour of service. It's a common way to calculate costs for time-based activities. In this exercise, both studios charge different hourly rates for painting:

• Studio A charges \( \\(7 \) per hour.
• Studio B charges \( \\)4 \) per hour.

Understanding and comparing these rates is key, especially when an activity's duration can vary. The hourly rate not only impacts the total cost but also how you budget your time. For instance, if you know you'll need more than a few hours, the studio with the lower hourly rate might be the better option. However, this decision should take into account the initial costs as well.

Linear Equations

Linear equations are an essential concept in algebra. They are used for calculating and predicting costs in real-life situations when variables are involved. In the pottery painting example, linear equations allow us to represent the total cost at each studio with the equation: \[12 + 7x = 15 + 4x\]Here, \(x\) represents the painting hours. Each side of the equation models the total cost for one studio. Linear equations like this are particularly useful because they provide a clear way to see how both constants (vase cost) and variables (hourly cost) interact. Solving such an equation will show you when both studios charge the same total price, known as the breakeven point. In everyday situations, they help you determine the best financial choice by solving for \(x\), which in this case identifies how long you can paint before prices at both studios match.