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Chapter 4: Problem 56

When hired at a new job selling electronics, you are given two pay options: Option A: Base salary of $$\$ 14,000$$ a year with a commission of \(10 \%\) of your sales Option B: Base salary of $$\$ 19,000$$ a year with a commission of \(4 \%\) of your sales How much electronics would you need to sell for option A to produce a larger income?

Short Answer

Expert verified
You need to sell more than $83,333.33 in electronics for Option A to provide a larger income.

Step by step solution

01

Understand the Pay Options

There are two pay options: - **Option A**: Base salary = $14,000, Commission = 10% - **Option B**: Base salary = $19,000, Commission = 4% Our goal is to find the sales amount for which Option A produces a larger income than Option B.
02

Set Up Income Equations

For Option A, the income can be expressed as:\[ \text{Income from A} = 14000 + 0.10X \]For Option B, the income is:\[ \text{Income from B} = 19000 + 0.04X \]where \(X\) is the amount of electronics sold in dollars.
03

Set the Inequality

To find when Option A produces a larger income than Option B, set the inequality:\[ 14000 + 0.10X > 19000 + 0.04X \]
04

Solve the Inequality for X

Rearrange the inequality:1. Subtract \(14000\) from both sides: \[ 0.10X > 5000 + 0.04X \]2. Subtract \(0.04X\) from both sides:\[ 0.06X > 5000 \]3. Divide both sides by \(0.06\):\[ X > \frac{5000}{0.06} \]\[ X > 83333.33 \]Thus, more than $83,333.33 of electronics need to be sold for Option A to be more beneficial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Commission Calculations
When deciding between two pay structures that include commission, it's essential to understand how commission works. A commission is a percentage of sales that an employee earns in addition to a base salary. To calculate the commission:
  • Identify the commission rate, which is the percentage of sales given as a bonus.
  • Apply this rate to the sales amount. For example, if you sell items worth $10,000 with a 10% commission rate, your commission is $1,000.
In this problem, Option A provides a 10% commission rate while Option B gives 4%. This means for Option A: - If you sell $1 worth of electronics, you earn $0.10 as commission. - For Option B, selling the same amount earns you $0.04. Commissions can significantly impact overall income, depending on the rate and total sales made. In this scenario, a higher commission rate only makes Option A lucrative if sales are beyond a certain threshold.
Income Comparison
When faced with multiple job offers, comparing the income structures is vital to make an informed decision. This involves adding both the base salary and the potential commission to determine total earnings for each option.For this exercise:
  • Option A starts with a base salary of \(14,000, plus 10% of sales
  • Option B offers a baseline of \)19,000, plus 4% of sales
The core question is to identify at what sales point Option A results in higher income than Option B. You need to set up income equations for each option:- Income from A becomes \(14,000 + 0.10X\)- Income from B is \(19,000 + 0.04X\)Here, \(X\) stands for the dollar amount of electronics sold. By comprehending these equations, you can visualize at what level of sales each option becomes financially advantageous.
Linear Inequalities
Linear inequalities are a fundamental concept in algebra used to compare quantities and find conditions where one is greater or less than another. In this task, we used a linear inequality to determine at what point Option A yields more income than Option B.To set up the inequality:- Start with the income equations: \(14,000 + 0.10X > 19,000 + 0.04X\)- The goal is to isolate \(X\) to find the amount of sales necessary.Break down the steps:
  • Subtract 14,000 from both sides: \(0.10X > 5,000 + 0.04X\)
  • Eliminate the 0.04X on the right: \(0.06X > 5,000\)
  • To isolate \(X\), divide both sides by 0.06, resulting in \(X > 83,333.33\)
This final result tells you more than $83,333.33 in sales is required for Option A to surpass Option B in income. Here, solving linear inequalities provided a clear pathway to the solution.

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Most popular questions from this chapter

A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of $$\$ 10$$ and then a certain amount of money pei megabyte (MB) of data used on the phone. If a customer uses \(20 \mathrm{MB}\), the monthly cost will be $$\$ 11.20 .$$ If the customer uses \(130 \mathrm{MB}\), the monthly cost will be $$\$ 17.80$$. (a) Find a linear equation for the monthly cost of the data plan as a function of \(x,\) the number of \(\mathrm{MB}\) used. (b) Interpret the slope and \(y\) -intercept of the equation. c) Use your equation to find the total monthly cost if \(250 \mathrm{MB}\) are used.

In \(2004,\) a school population was 1001. By 2008 the population had grown to 1697 . Assume the population is changing linearly. (a) How much did the population grow between the year 2004 and \(2008 ?\) (b) How long did it take the population to grow from 1001 students to 1697 students? (c) What is the average population growth per year? (d) What was the population in the year \(2000 ?\) (e) Find an equation for the population, \(P\), of the school \(t\) year after 2000 . (f) Using your equation, predict the population of the school in \(2011 .\)

For the following exercises, consider this scenario: A town has an initial population of 75,000 . It grows at a constant rate of 2,500 per year for 5 years. If the function \(P\) is graphed, find and interpret the \(x\) - and \(y\) -intercepts.

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For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ h(x)=3 x-5 $$
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