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<!DOCTYPE html>
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<title>Chapter 1 Preliminaries | Business Mathematics</title>
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<li><a href="./">Business Mathematics</a></li>
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<li><a href="index.html#section" id="toc-section"></a></li>
<li class="part"><span><b>I Essentials</b></span></li>
<li class="chapter" data-level="1" data-path="preliminaries.html"><a href="preliminaries.html"><i class="fa fa-check"></i><b>1</b> Preliminaries</a>
<ul>
<li class="chapter" data-level="" data-path="preliminaries.html"><a href="preliminaries.html#order-of-operations"><i class="fa fa-check"></i>1.2 Order of Operations</a></li>
<li class="chapter" data-level="" data-path="preliminaries.html"><a href="preliminaries.html#rational-numbers"><i class="fa fa-check"></i>1.3 Rational Numbers</a></li>
<li class="chapter" data-level="" data-path="preliminaries.html"><a href="preliminaries.html#expanding-and-factoring"><i class="fa fa-check"></i>1.4 Expanding and Factoring</a></li>
<li class="chapter" data-level="" data-path="preliminaries.html"><a href="preliminaries.html#solving-systems-of-equations"><i class="fa fa-check"></i>1.5 Solving Systems of Equations</a></li>
<li class="chapter" data-level="" data-path="preliminaries.html"><a href="preliminaries.html#ratios-proportions-and-percentages"><i class="fa fa-check"></i>1.6 Ratios, Proportions and Percentages</a></li>
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<li class="chapter" data-level="" data-path=""><a href="#part-economics"><i class="fa fa-check"></i>(PART) ECONOMICS</a>
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<li class="chapter" data-level="" data-path="preliminaries.html"><a href="preliminaries.html#supply-and-demand-economics"><i class="fa fa-check"></i>2.2 Supply and Demand Economics</a></li>
<li class="chapter" data-level="" data-path="preliminaries.html"><a href="preliminaries.html#linear-programming"><i class="fa fa-check"></i>2.3 Linear Programming</a></li>
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<li class="chapter" data-level="2" data-path="business-economics.html"><a href="business-economics.html"><i class="fa fa-check"></i><b>2</b> Business Economics</a>
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<li class="chapter" data-level="" data-path="business-economics.html"><a href="business-economics.html#introduction-to-calculus"><i class="fa fa-check"></i>3.1 Introduction to Calculus</a></li>
<li class="chapter" data-level="" data-path="business-economics.html"><a href="business-economics.html#merchandizing"><i class="fa fa-check"></i>3.2 Merchandizing</a></li>
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<li class="chapter" data-level="" data-path=""><a href="#part-simple-and-compound-interest"><i class="fa fa-check"></i>(PART) Simple and Compound Interest</a>
<ul>
<li class="chapter" data-level="" data-path="business-economics.html"><a href="business-economics.html#simple-interest"><i class="fa fa-check"></i>4.2 Simple Interest</a></li>
<li class="chapter" data-level="" data-path="business-economics.html"><a href="business-economics.html#present-and-future-value"><i class="fa fa-check"></i>4.4 Present and Future Value</a></li>
<li class="chapter" data-level="" data-path="business-economics.html"><a href="business-economics.html#rate-of-interest-and-term-of-investments"><i class="fa fa-check"></i>4.5 Rate of Interest and Term of Investments</a></li>
<li class="chapter" data-level="" data-path="business-economics.html"><a href="business-economics.html#equivalent-payments"><i class="fa fa-check"></i>4.6 Equivalent Payments</a></li>
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<li class="chapter" data-level="3" data-path="compound-interest.html"><a href="compound-interest.html"><i class="fa fa-check"></i><b>3</b> Compound Interest</a>
<ul>
<li class="chapter" data-level="" data-path="compound-interest.html"><a href="compound-interest.html#present-and-future-values"><i class="fa fa-check"></i>5.2 Present and Future Values</a></li>
<li class="chapter" data-level="" data-path="compound-interest.html"><a href="compound-interest.html#calculating-the-periodic-interest-rate-and-number-of-compounding-periods"><i class="fa fa-check"></i>5.3 Calculating the Periodic Interest Rate and Number of Compounding Periods</a></li>
<li class="chapter" data-level="" data-path="compound-interest.html"><a href="compound-interest.html#equivalent-payments-and-payment-streams"><i class="fa fa-check"></i>5.4 Equivalent Payments and Payment Streams</a></li>
<li class="chapter" data-level="" data-path="compound-interest.html"><a href="compound-interest.html#changing-interest-rates-1"><i class="fa fa-check"></i>5.6 Changing Interest Rates</a></li>
<li class="chapter" data-level="" data-path="compound-interest.html"><a href="compound-interest.html#continuously-compounded-interest"><i class="fa fa-check"></i>5.7 Continuously Compounded Interest</a></li>
</ul></li>
<li class="part"><span><b>II Annuities</b></span></li>
<li class="chapter" data-level="4" data-path="annuity-basics.html"><a href="annuity-basics.html"><i class="fa fa-check"></i><b>4</b> Annuity Basics</a>
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<li class="chapter" data-level="" data-path="annuity-basics.html"><a href="annuity-basics.html#effective-and-equivalent-interest-rates"><i class="fa fa-check"></i>6.1 Effective and Equivalent Interest Rates</a></li>
<li class="chapter" data-level="" data-path="annuity-basics.html"><a href="annuity-basics.html#annuities"><i class="fa fa-check"></i>6.2 Annuities</a></li>
<li class="chapter" data-level="" data-path="annuity-basics.html"><a href="annuity-basics.html#calculating-the-periodic-payment-and-number-of-payments"><i class="fa fa-check"></i>6.3 Calculating the Periodic Payment and Number of Payments</a></li>
<li class="chapter" data-level="" data-path="annuity-basics.html"><a href="annuity-basics.html#ordinary-general-annuities"><i class="fa fa-check"></i>6.4 Ordinary General Annuities</a></li>
<li class="chapter" data-level="" data-path="annuity-basics.html"><a href="annuity-basics.html#using-the-ti-baii-plus-calculator-for-annuities"><i class="fa fa-check"></i>6.5 Using the TI BAII plus Calculator for Annuities</a></li>
</ul></li>
<li class="chapter" data-level="5" data-path="more-on-annuities.html"><a href="more-on-annuities.html"><i class="fa fa-check"></i><b>5</b> More on Annuities</a>
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<li class="chapter" data-level="" data-path="more-on-annuities.html"><a href="more-on-annuities.html#annuities-due"><i class="fa fa-check"></i>7.1 Annuities Due</a></li>
<li class="chapter" data-level="" data-path="more-on-annuities.html"><a href="more-on-annuities.html#deferred-annuities"><i class="fa fa-check"></i>7.2 Deferred Annuities</a></li>
<li class="chapter" data-level="" data-path="more-on-annuities.html"><a href="more-on-annuities.html#perpetuities"><i class="fa fa-check"></i>7.3 Perpetuities</a></li>
<li class="chapter" data-level="" data-path="more-on-annuities.html"><a href="more-on-annuities.html#constant-growth-annuities"><i class="fa fa-check"></i>7.4 Constant Growth Annuities</a></li>
<li class="chapter" data-level="" data-path="more-on-annuities.html"><a href="more-on-annuities.html#special-case-annuities"><i class="fa fa-check"></i>7.5 Special Case Annuities</a></li>
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<li class="chapter" data-level="" data-path=""><a href="#part-investments"><i class="fa fa-check"></i>(PART) Investments</a>
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<li class="chapter" data-level="" data-path="more-on-annuities.html"><a href="more-on-annuities.html#mortgage-fundamentals"><i class="fa fa-check"></i>8.1 Mortgage Fundamentals</a></li>
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<div id="preliminaries" class="section level1 hasAnchor" number="1">
<h1><span class="header-section-number">Chapter 1</span> Preliminaries<a href="preliminaries.html#preliminaries" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<div id="order-of-operations" class="section level2 unnumbered hasAnchor">
<h2>1.2 Order of Operations<a href="preliminaries.html#order-of-operations" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<ul>
<li>Simplify <span class="math inline">\(\$250 ( 1 + 0.065 × 172/365)\)</span>
<ul>
<li>$257.66 <a href="https://www.youtube.com/watch?v=bpgDtSvmLt0&list=PLhMAuJ_imnt6A_N3MEIxSXKEwY4RChEWX&index=1">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li>Simplify <span class="math inline">\(\$1775 ( 1 + 0.6/2)^3( 1 + 7.84/4)^2\)</span>
<ul>
<li>$34,167.39 <a href="https://www.youtube.com/watch?v=dPtWFsmhBI0&list=PLhMAuJ_imnt6A_N3MEIxSXKEwY4RChEWX&index=2">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li>Simplify <span class="math inline">\(\$1500/(1+ 0.08/12)^4\)</span>
<ul>
<li>$1460.66 <a href="https://www.youtube.com/watch?v=nAlCJJ7F3nQ&list=PLhMAuJ_imnt6A_N3MEIxSXKEwY4RChEWX&index=3">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li>A hockey team says that its season ticket sales represent 72% of
the stadium seating capacity of 18,500 seats. Rounded to the nearest 100,
how many seats were not sold to season ticket holders?
<ul>
<li>5,200 <a href="https://www.youtube.com/watch?v=r-YA7n_3RWY&list=PLhMAuJ_imnt6A_N3MEIxSXKEwY4RChEWX&index=4">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li>Billy is a Realtor. He receives 44% of the 7% commission that the real estate
agency charges on sales. If his sales for the past year were $14,500,000, what was the dollar
value of his commission?
<ul>
<li>$446,600 <a href="https://www.youtube.com/watch?v=qoo0mt3FgQ8&list=PLhMAuJ_imnt6A_N3MEIxSXKEwY4RChEWX&index=5">Click here for full solution</a>
<p>
</p></li>
</ul></li>
</ul>
</div>
<div id="rational-numbers" class="section level2 unnumbered hasAnchor">
<h2>1.3 Rational Numbers<a href="preliminaries.html#rational-numbers" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Simplify the following:</p>
<ul>
<li><span class="math inline">\(3/4 - x/4 - 2x^2 - 0.5 x^2 − 10/4x + 3/2\)</span>
<ul>
<li><span class="math inline">\(-2.5x^2 - (5/4)x + 7/4\)</span> <a href="https://www.youtube.com/watch?v=tIk7EkumoAw&list=PLhMAuJ_imnt6VyuU4Pf_T29UDSy3eJrzk">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li><span class="math inline">\(y(1 − 0.08 * 200/365) + 2y/(1 + 0.08 \times 60/365)\)</span>
<ul>
<li><span class="math inline">\(2.9292y\)</span> <a href="https://www.youtube.com/watch?v=u3EWjZLyUvU&list=PLhMAuJ_imnt6VyuU4Pf_T29UDSy3eJrzk&index=2">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li><span class="math inline">\(x/(1 + 0.04/2)^2 − 3x (1 + 0.04/2)^3\)</span>
<ul>
<li><span class="math inline">\(-2.070224x\)</span> <a href="https://www.youtube.com/watch?v=aev4KtTo99c&list=PLhMAuJ_imnt6VyuU4Pf_T29UDSy3eJrzk&index=3">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li><span class="math inline">\(500((1 + 0.04/2)^25 - 1)/(0.04/2)\)</span>
<ul>
<li><span class="math inline">\(13,775\)</span> <a href="https://www.youtube.com/watch?v=uTtT9hn4rmA&list=PLhMAuJ_imnt6VyuU4Pf_T29UDSy3eJrzk&index=5">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li><span class="math inline">\(500/(0.06/4) + 500\)</span>
<ul>
<li><span class="math inline">\(33,833.33\)</span> <a href="https://www.youtube.com/watch?v=FLYaSFsqH5A&list=PLhMAuJ_imnt6VyuU4Pf_T29UDSy3eJrzk&index=5">Click here for full solution</a>
<p>
</p></li>
</ul></li>
</ul>
</div>
<div id="expanding-and-factoring" class="section level2 unnumbered hasAnchor">
<h2>1.4 Expanding and Factoring<a href="preliminaries.html#expanding-and-factoring" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Simplify the following:</p>
<ul>
<li><span class="math inline">\(12x^2y + (−5x^2y) − (−7x^2y)\)</span>
<ul>
<li><span class="math inline">\(14x^2y\)</span> <a href="https://www.youtube.com/watch?v=PMlvcOMKILc">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li><span class="math inline">\((8m^3 − 3m − 4m^2 + 14) − (2m^3 − 6 + 8m − 14m^2)\)</span>
<ul>
<li><span class="math inline">\(6m^3 + 10m^2 - 11m + 20\)</span> <a href="https://www.youtube.com/watch?v=poKHlgg34pU&list=PLhMAuJ_imnt7b2MKNxawcVvV2EiNB2VxQ&index=2">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li><span class="math inline">\(10x− [8 − 4(3x − 4)]\)</span>
<ul>
<li><span class="math inline">\(22x - 24\)</span> <a href="https://www.youtube.com/watch?v=LOTdpEzEVrk&list=PLhMAuJ_imnt7b2MKNxawcVvV2EiNB2VxQ&index=3">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li><span class="math inline">\(−2xy(6x^2 − 2xy − 5y^2)\)</span>
<ul>
<li><span class="math inline">\(-12x^3y + 4x^2y^2 + 10xy^3\)</span> <a href="https://www.youtube.com/watch?v=QCpCmtvZrr8&list=PLhMAuJ_imnt7b2MKNxawcVvV2EiNB2VxQ&index=4">Click here for full solution</a>
<p>
</p></li>
</ul></li>
<li><span class="math inline">\((4a^2b^7 - 8a^5b^3)/(2a^2b^2)\)</span>
<ul>
<li><span class="math inline">\(2b^5 - 4a^3b\)</span> <a href="https://www.youtube.com/watch?v=Zb8b77JlH_o&list=PLhMAuJ_imnt7b2MKNxawcVvV2EiNB2VxQ&index=5">Click here for full solution</a>
<p>
</p></li>
</ul></li>
</ul>
</div>
<div id="solving-systems-of-equations" class="section level2 unnumbered hasAnchor">
<h2>1.5 Solving Systems of Equations<a href="preliminaries.html#solving-systems-of-equations" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Solve the following systems of equations by any method:</p>
<ul>
<li><span class="math inline">\(9x-6y = 18\)</span> and <span class="math inline">\(2y = 2x - 2\)</span>
<ul>
<li><span class="math inline">\(x = 4, y = 3\)</span>
<p>
</p></li>
</ul></li>
<li><span class="math inline">\(2x - 2y = 2\)</span> and <span class="math inline">\(6x + 8y = 40\)</span>
<ul>
<li><span class="math inline">\(x = 24/7, y = 17/7\)</span>
<p>
</p></li>
</ul></li>
<li><span class="math inline">\(x - y = 20\)</span> and <span class="math inline">\(4/3x + 5/2y = 0\)</span>
<ul>
<li><span class="math inline">\(x = -9.443, y = -29.443\)</span>
<p>
</p></li>
</ul></li>
<li>A car rental company charges an hourly rate plus a per kilometre charge. Jack paid $60 for a two-hour rental during which he drove 50 km. Jules paid $145 for five hours and 100 km driven. What rate is charged per hour and per kilometre?
<ul>
<li>$25 per hour and $0.20 per kilometre
<p>
</p></li>
</ul></li>
<li>Hank bought 8 steaks and 30 potatoes for $75. Kevin purchased 11 steaks and 20 potatoes for $75. What were the prices of the steak and potatoes.
<ul>
<li>Steak: $4.41 and potatoes: $0.53
<p>
</p></li>
</ul></li>
<li>A company pays monthly salaries of $5500 to each production worker and $4600 to each assembly worker. At the beginning of COVID, the firm decides to reduce its total monthly manufacturing payroll from $390,000 to $300,000 by laying off 18% of its production workers and 22% of its assembly workers. How many layoffs will there be from each of the assembly and production divisions?
<ul>
<li>50 production and 23 assembly
<p>
</p></li>
</ul></li>
</ul>
</div>
<div id="ratios-proportions-and-percentages" class="section level2 unnumbered hasAnchor">
<h2>1.6 Ratios, Proportions and Percentages<a href="preliminaries.html#ratios-proportions-and-percentages" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<ul>
<li>Solve the ratio <span class="math inline">\(165 : 245 = 65 : x\)</span>
<ul>
<li><span class="math inline">\(x = 96.33\)</span>
<p>
</p></li>
</ul></li>
<li>Solve the ratio <span class="math inline">\(1500: 2500 : 750 = x : 750 : y\)</span>
<ul>
<li><span class="math inline">\(x = 450\)</span> and <span class="math inline">\(y = 1250\)</span>
<p>
</p></li>
</ul></li>
<li>Jim, Karen, and Sean started their partnership with a total investment of
$245,000 contributed in the ratio of 4 : 6 : 3. If each partner contributes another $30,000, what will be the ratio of their total contributions?
<ul>
<li>Approximately <span class="math inline">\(1.26 : 1.69 : 1.00\)</span>
<p>
</p></li>
</ul></li>
<li>The CPI was at 120 when the union signed their contract 5 years ago, the members of a union were earning $45.55 per hour. Now the current CPI at 135 and they want a new hourly
rate that will restore their former purchasing power. What hourly rate are they seeking?
<ul>
<li>$51.42
<p>
</p></li>
</ul></li>
<li>I live in a 4-plex. We need a new roof with costs totaling $35,000 and these costs must be divided between owners A, B, C, and D based on their square-footage 1300, 1400, 1750 and 2100 square feet, respectively. How much does each owner pay?
<ul>
<li>Owner A: $6,947.50, owner B: $7,476.00, owner C: $9,352.00, and owner D: $11,225.00.
<p>
</p></li>
</ul></li>
<li>Jenny, Kathy, and Fred share the costs of their coffee fund in the ratio <span class="math inline">\(3/2 : 5/4 : 4/3\)</span>. How
will costs of $50 be allocated among them?
<ul>
<li>Jenny: $11.72, Kathy: $14.66, Fred: $19.62.
<p>
</p></li>
</ul></li>
<li>Lisa receives a monthly base salary of $2500. On the first $15,000 of sales above
her monthly quota of $25,000, she is paid a commission of 6%. On any additional sales, the
commission rate is 10%. What were her total earnings for a month in which she had sales amounting to $45,000?
<ul>
<li>$3,900
<p>
</p></li>
</ul></li>
<li>The price tags in Mel’s Flower Shop include the 8.25% tax. How much tax will she report for a plant sold at $49.55?
<ul>
<li>$4.08
<p>
</p></li>
</ul></li>
<li>Okanagan Food-Mart usually spends half as much on radio advertising as on newspaper
advertising, and 70% as much on television advertising as on radio advertising. If next year’s
total advertising budget is $225,000, how much should be allocated to each form of advertising?
<ul>
<li>Newspaper advertising: $121,622.16, Radio advertising: $60,811.08, Television advertising: $42,467.76</li>
</ul></li>
</ul>
<p>
</p>
<ul>
<li>The government forecasts expenditure of $2.85 billion on education next year represents 21% of the provincial budget. Rounded to the nearest million dollars, what is the province’s total budget for the
next year?
<ul>
<li>$13,570,000,000
<p>
</p></li>
</ul></li>
</ul>
</div>
</div>
<div id="supply-and-demand-economics" class="section level2 unnumbered hasAnchor">
<h2>2.2 Supply and Demand Economics<a href="preliminaries.html#supply-and-demand-economics" class="anchor-section" aria-label="Anchor link to header"></a></h2>
</div>
<div id="linear-programming" class="section level2 unnumbered hasAnchor">
<h2>2.3 Linear Programming<a href="preliminaries.html#linear-programming" class="anchor-section" aria-label="Anchor link to header"></a></h2>
</div>
</div>
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