1.2: How Can I Use The Features Of This Book To My Advantage?

This textbook was written for students like you, designed to address common student needs in mathematics.

I Don’t Know How to Solve a Math Problem

Do you find it difficult to start a math problem? What do you do first? What is the next step? The reason this is challenging for some students is that they have never been shown problem-solving techniques. This textbook will use a step by step approach to solving problems, as illustrated in the example below.  Successfully working through a problem involves a careful approach and this model provides a structured problem-solving approach that will aid you not only in mathematics but in any problem-solving situation you may encounter.

Example 1.2.1

When Sandra borrowed [latex]\$7,100[/latex] from Sanchez, she agreed to reimburse him [latex]\$8,615.19[/latex] three years from now including interest compounded quarterly. What interest rate is being charged?

Solution

Step 1: What are we looking for?

Find the nominal quarterly compounded rate of interest ([latex]IY[/latex]).

Step 2: What do we already know from the question?

The present value, future value, term, and compounding are known, as illustrated in the timeline above (Figure 1.2.1).

[latex]\begin{eqnarray*}PV&=&\$7,100\\FV&=&\$8,615.19\\\text{C/Y}\;&=&\;\text{quarterly}\;=\;4\\\text{Term}\;&=&\;3\;\text{years}\end{eqnarray*}[/latex]

Step 3: Make substitutions using the information known above.

Calculate [latex]n[/latex] using Formula 6.2a[latex]n=\text{C/Y}\times\text{Number of Years}[/latex].

[latex]\begin{eqnarray*}n&=&C/Y\times(\text{Number of Years})\\n&=&4\times3\\n&=&12\end{eqnarray*}[/latex]

Using the [latex]n[/latex] value, we substitute it into the Compound Interest for Single Payments Formula. Rearrange and solve for the Periodic Rate value, [latex]i[/latex].

[latex]\begin{eqnarray*}FV&=&PV{(1+i)}^n\\\$8,615.19&=&\$7,100{(1+i)}^{12}\;\\1.213407&=&{(1+i)}^{12}\\1.213407^\frac1{12}&=&1+i\\1.01624996&=&1+i\\i&=&0.01624996\end{eqnarray*}[/latex]

Substitute [latex]i[/latex] value into Formula 6.1a[latex]\begin{align*}i=\frac{\text{I/Y}}{\text{C/Y}}\end{align*}[/latex] and solve for [latex]\text{I/Y}[/latex].

[latex]\begin{eqnarray*}i&=&\frac{\text{I/Y}}{\text{C/Y}}\\0.01624996&=&\frac{I/Y}4\\I/Y&=&0.06499985\;=\;0.065\;\text{or}\;6.5\%\end{eqnarray*}[/latex]

Note that this amount would be compounded quarterly.

Table 1.2.1

n	I/Y*	PV	PMT	FV	P/Y	C/Y
[latex]12[/latex]	[latex]6.499985[/latex]*	[latex]-7,100[/latex]	[latex]0[/latex]	[latex]8,615.19[/latex]	[latex]4[/latex]	[latex]4[/latex]

_{*Answer provided by calculations above}

Step 4: Provide the information in a worded statement. 

Sanchez is charging an interest rate of [latex]6.5\%[/latex] compounded quarterly on the loan to Sandra.

Once you repeatedly practice these steps and ingrain this practice in your thinking process, you’ll have a structured approach to solving not just math problems but any life problem.  As you keep practicing, you’ll find that this becomes a way of thinking. You can never skip one of the steps, but you won’t have to write them all out anymore because you’ll execute some steps with rapid-fire thought! 

I Can’t Find What I Want When I Need It

When practicing your business math through homework and assignments, you will need to go back to look up various formulas or techniques or to review concepts. The challenge of most textbooks is in finding what you need when you need it buried in the vast pages of the textbook. Browse the table of contents of this book and notice the functional layout. This textbook incorporates a handbook design that aids in finding information. When you need to look something up, you will notice every part of every section in every chapter is designed and laid out using an identical structure.

While not every part is required in every section, the order is always as follows:

Are There Any Steps to Help Me Remember?

Students have always asked if they need to follow a logical sequence of steps to solve a problem. The answer, of course, is YES! A section called “HOW TO” helps you with this by providing step-by-step procedures, techniques, and formulas you need to solve the problem.

Can This Book Help Me Understand the Formulas Better?

You can’t avoid formulas—this is math, after all. What helps, though, is to understand what the formulas do and how each component comes together to produce the solution. If you can understand, you do not need to memorize! This book lays formulas out visually, clearly labelling all symbols. In addition, each component is fully explained (see below). You are not required to cross-reference paragraph-style explanations to formulas, as the visual layout makes it all clear. The goal is for you to understand what exactly the formula does and how, so that you understand enough to recognize when solutions make sense or might be wrong.

[latex]\boxed{4.1\text{a}}[/latex] Single Discount

[latex]{\color{red}{N}}\;\text{is the Net Price:}[/latex] The net price is the price of the product after the discount is removed from the list price. It is a dollar amount representing what price remains after you have applied the discount.

[latex]{\color{green}{L}}\;\text{is List Price:}[/latex] The list price is the normal or regular dollar price of the product before any discounts. It is the Manufacturer’s Suggested Retail Price (MSRP – a price for a product that has been published or advertised in some way), or any dollar amount before you remove the discount.

[latex]{\color{blue}{d}}\;\text{is the Discount Rate:}[/latex] The discount rate represents the percentage (in decimal format) of the list price that is deducted.

Additionally, you will not find any of those archaic algebraic symbols in this book. Representative symbols make the algebra easier to remember and understand. Those ancient mathematicians just loved to speak Greek (literally!), but this text speaks modern-day English and understands that we have technology here in the twenty-first century! We’ll use symbols like [latex]N[/latex] for net price, [latex]L[/latex] for list price, [latex]P[/latex] for profit, and [latex]E[/latex] for expenses, letters that actually remind you of the variable they stand for!

Does Anyone Actually Do Any of This in the Real World?

As you read through the guided examples and do your homework, pause to consider how the question applies to you. Does it demonstrate a business application you can use to be a smarter business professional? How can you extend the question to similar situations? This textbook shows you examples from both business and personal situations that you either have already encountered or probably will encounter in the future. (Remember: Math is all around you!)

• Business

Why do business programs require a business math course in the first term or two? How is it relevant to your future? This textbook provides numerous examples of applications in marketing, finance, economics, accounting, and more. Once you complete this textbook, you should see clearly how business mathematics is relevant and important to your chosen future career.

• Personal

You will see companies and products from your everyday life. Real-world situations show you how to put a few extra dollars in your pocket by making smart mathematical choices. What if you could save [latex]\$100[/latex] per month simply by making smarter choices? Over [latex]40[/latex] years at a very conservative rate of interest, that is approximately [latex]\$100,000[/latex] in your pocket. You will also find life lessons. Are you ready to start planning your RRSP? Would you like to know the basics on how to do this? If you are thinking of buying a home, do you know how the numbers work on your mortgage? To learn how to buy a car at the lowest cost, read on!

Are There Any Secrets?

There are never any secrets. However, I will point out some tricks of the trade and commonly made mistakes throughout this book.

Paths To Success

This feature is designed to make you feel like part of the “in” crowd by letting you take advantage of shortcuts or tricks of the trade, such as some easy way to remember a particular technique or formula. It helps you see how some of the math could be made a little simpler or performed in a different way.

Is There Any Way to Know That I’m “Getting It”?

The second skill is much harder to acquire than the first. A feature called “Try It” poses scenarios and questions for which no calculations are required. You need to visualize and conceptualize various mathematical theorems. Ultimately, you perform a self-test to see if you are “getting it.”

In Today’s World, Who Does This By Hand Anymore?

That is a perfectly valid question. You are right: Most people do not do this work by hand and instead use mathematical tools such as calculators, spreadsheet software, and apps. However, as the saying goes, you must learn to walk before you can run. The learning of mathematics requires pencil and paper first. Once you master these, you can then employ technology to speed up the process. This basic version textbook incorporates one technology to aid you in your application of business mathematics:

• The Calculator

Integrated throughout this book are instructions for one of the financial calculators most widely used in Canada—the Texas Instruments BAII Plus. Calculator functions with step-by-step button sequences are presented with guided examples illustrating how you solve the math problem not only by hand but also using the financial calculator.

Is There Any Way I Can Assess My Ability?

At the end of almost every section of this book you will find approximately 18 practice questions covering the concepts specifically introduced in that section, with final solutions listed to all questions at the end of this textbook. These questions are divided into three categories to help develop your mathematical skills and assess your abilities:

Mechanics

This group of questions focuses on fundamental mathematical skills and formula usage. They are your first taste of using the section formulas and working with the mathematical concepts at an introductory level.

Do not stop here, though! You do not yet have enough skills to meet the basic requirements of most business math courses. Once you have mastered the Mechanics section, you must progress to the next section, where you learn to apply your knowledge.

Applications

These questions are more typical of business math course expectations (check with your instructor). The key difference from the Mechanics section is that you must now execute your problem-solving skills. These questions require you to determine the unknown variable and figure out how the various pieces of the puzzle come together in the solution.

Challenge, Critical Thinking, and Other Applications

These questions put your knowledge and understanding to the test. The difficulty bar is set high, as these questions commonly require multistep solutions with advanced problem-solving techniques. As well, success might require you to integrate formulas, concepts, or procedures in novel ways. You will be challenged!

THE FOLLOWING LATEX CODE IS FOR FORMULA TOOLTIP ACCESSIBILITY. NEITHER THE CODE NOR THIS MESSAGE WILL DISPLAY IN BROWSER.[latex]n=\text{C/Y}\times\text{Number of Years}[/latex][latex]\begin{align*}i=\frac{\text{I/Y}}{\text{C/Y}}\end{align*}[/latex]

Attribution

“Chapter 1: Succeeding in Business Mathematics” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.