Section 1

Section 1 Introduction

How are your financial gardening skills? Are you interested in seeing your money grow? Don’t let your money sit neglected and withering on the vine. Invest! Banks, credit unions, governments, and businesses will all pay you when you deposit cash with them. They’ll pay you interest!

In the Section 1 Project: Capturing Your Interest!, you will explore various investments that earn interest. While working through the lessons, you will investigate how to make simple interest and compound interest work. You will discover how each type of interest is calculated and look at similarities and differences. Then, armed with this knowledge, you too can plant the seed for your financial money tree.

Lesson 1: Simple Interest

Focus

You may have heard the old saying “Money doesn’t grow on trees!” But your money can grow! When you put your hard-earned money into a savings account, buy a guaranteed investment certificate (GIC), or invest in regular-interest savings bonds, you earn interest. This is true since a bank, a business, or a government is effectively renting your money for their use.

The reverse is also true—a bank or business will charge you interest if they loan you money to make a purchase or pay a bill.

In this lesson you will examine one form of interest called simple interest. This is interest calculated on an amount of money that does not include interest earned.

Lesson Question

In this lesson you will investigate the following question:

• How can you solve financial problems involving simple interest?

Assessment

Your assessment for this lesson may include a combination of the following:

• course folder submissions from the Try This and Share sections of the lesson
  
  
• your contribution to the Mathematics 20-3 Glossary Terms and the Formula Sheet
  
  
• Lesson 1 Assignment (Save a copy of your lesson assignment to your course folder now.)
  
  
• the Project Connection

In this course you may come across Self-Check questions, Try This questions, and other activities that may or may not be assessed.

Remember that these questions and activities provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place those answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance.

Materials and Equipment

• calculator

Time

This lesson has been designed to take 150 minutes; however, it may take more or less time depending on how well you are able to understand the lesson concepts. It is important that you progress at your own pace based on your learning needs.

Launch

This section checks to see if you have the necessary background knowledge and skills required to successfully complete Lesson 1.

Complete the following Are You Ready? questions. If you have difficulty or any questions, visit Refresher for a review or contact your teacher.

An important skill in solving simple interest problems is working with percentage; specifically converting back and forth between percentage and decimals.

Are You Ready?

1. Convert each percentage to a decimal numeral.
   
   
   1. 50%
   2. 4.5%
   3. 0.4%
   Answer
   
   
2. Convert each decimal to a percentage.
   
   
   1. 0.25
   2. 3.2
   3. 0.001
   Answer
   
   
3. Calculate 16% of 72. Answer

If you answered the Are You Ready? questions without problems, move on to Discover.

If you found the Are You Ready? questions difficult, complete Refresher to review these topics.

Refresher

If you don’t know the answers for Are You Ready?, or require more information, study the following information about percentages.

This data clarifies why changing a percent to a decimal numeral means moving the decimal in the percent two places to the left.

The following clarifies why changing a decimal to a percent means moving the decimal point two places to the right.

2. Use proportional reasoning. Let the required percentage be n%.
   
   
   1. 
      
      So, 0.25 = 25%
      
      
   2. 
      
      So, 3.2 = 320%
      
      
   3. 
      
      So, 0.001=0.1%
      
      

Go back to Are You Ready? and try the questions again. Contact your teacher if you continue to have difficulty.

Discover

In this section you will derive the formula to find simple interest.

Try This 1

Use the Simple Interest Calculator to help investigate how simple interest is calculated. Click on the play button.

PART 1: Manipulating the Interest Rate (r)

1. Using the Simple Interest Calculator, what is the interest earned on $100 for one year at
   
   
   1. 4%
   2. 10%
      
      
2. Without using the calculator, can you predict how much simple interest you will earn when you invest $100 for one year at an interest rate of 20%? Check your answer using your calculator.
   
   
3. How do you think the Simple Interest Calculator calculated the values for questions 1 and 2?

PART 2: Manipulating the Time (t)

1. Using the Simple Interest Calculator, what is the interest earned on $100 at an interest rate of 5% for
   
   
   1. 1 year
   2. 2 years
      
      
2. Without using the calculator, can you predict how much simple interest you will earn when you invest $100 for 10 years at an interest rate of 5%? Check your answer using your calculator.
   
   
3. How do you think the Simple Interest Calculator calculated the values for questions 1 and 2?

PART 3: Putting It All Together

1. Based on your observations from Parts 1 and 2, suggest a formula for finding interest. Use the following symbols to build your formula.
   
   
   • I for interest
   • P for principal
   • r for the annual interest rate
   • t for the time in years

Save your answers to your course folder.

Share 1

As you are working on this Share section, refer to the Share Rubric. This information will help you understand what your teacher expects of you.

Share your responses to the Part 3 question from Try This 1 with a classmate or group. 

• How did your formula for finding interest compare with other students’ formulas? Why might there have been similarities and differences between the formulas?
  
  
• Use your formulas to find what the total interest on $6000 at 5% would be after ten years. Use the Simple Interest Calculator to check if your formulas are correct.

If required, save a copy of your discussion and calculation in your course folder.

Explore

In Discover you may have noticed that simple interest calculations depend on three factors:

• The first factor is the amount borrowed or invested. This amount is called the principal.

• A second factor determining interest is the interest rate. Banks advertise the interest rate as an annual rate—the rate charged or earned for one year.

• A third factor is the term or time, expressed in years, of the loan or investment.

In Discover you used these factors to help develop a formula for simple interest. The formula may have looked similar to the following one. Roll-over the formula to identify the variables.

I = Prt

Watch Marty’s Notebook to see how the simple interest formula can be used. Access this applet by clicking on the play button. Notice the extra step required when the time given is in months.

To find the total amount of money an investment is worth, the principal and all the interest made must be added together.

You may have noticed that this formula was used in the final step in the example you just watched. You will see this formula used again at the end of question 1 of Self-Check 1.

Self-Check 1

1. Jolene Danchuk has just invested $1000 in a 90-day GIC at her bank. The GIC earns simple interest at 3% per annum. How much will Jolene’s investment be worth after 90 days? Answer

Self-Check 2

1. Suu Kyi invested $4000 in a five-year regular interest bond. She will receive $200 in simple interest over the five-year term. What is the annual interest rate on the bond? Use the rearranged formula, . Answer
   
   
2.  
   1. Rearrange the formula for simple interest to find the term t if P, r, and I are given. Answer
      
      
   2. Charlie earned simple interest of $200 on a $2000 investment. The interest rate was 4% per annum. What was the term of the investment? Answer
      
      
3.  
   1. Rearrange the simple interest formula to find the principal, P. Answer
      
      
   2. Mabel borrowed money at 8% simple interest from the bank to purchase furniture for her new apartment. At the end of six months, she paid the loan off. She was charged $200 in interest. How much money did she borrow? Answer

Lesson 1 Summary

In this lesson you examined simple interest on loans and investments. To solve simple interest problems, you discovered that you had to know three of the four quantities: interest, annual interest rate, principal, and the term in years.

Interest on loans and investments has been part of human civilization for thousands of years. In ancient Babylon, regulations governed the interest lenders could legally charge. The same is true today in Canada. All charges and interest must be disclosed to the borrower, so that the borrower knows from the beginning how much interest the loan will cost. Remember to read the fine print before signing on the dotted line! And keep in mind that only legitimate borrowing is dealt with in this lesson.

Lesson 2: Compound Interest

Focus

Training for physical endurance in a race is a long-term process. Saving money through investing is also a long-term process. Most people do not become rich overnight. When training for a race, an athlete will focus on increasing strength and power. Similarly, using interest to increase your savings is using the power of interest.

In Lesson 1 you looked at the power of simple interest. In this lesson you will look at an even bigger powerhouse: compound interest.

Lesson Question

In this lesson you will investigate the following question:

• How are financial problems involving compound interest solved?

Assessment

Your assessment for this lesson may include a combination of the following:

• course folder submissions from the Try This and Share sections of the lesson
  
  
• your contribution to the Mathematics 20-3 Glossary Terms and the Formula Sheet
  
  
• Lesson 2 Assignment (Save a copy of your lesson assignment to your course folder now.)
  
  
• the Project Connection

Materials and Equipment

• calculator

Launch

This section checks to see if you have the necessary background knowledge and skills required to successfully complete Lesson 2.

Complete the following Are You Ready? questions. If you have difficulty or any questions, visit Refresher for a review or contact your teacher.

An important skill in solving compound interest problems is working with exponents.

Are You Ready?

1. What is the value of 43? Without using your calculator, explain how you found the answer. Answer

2. Use your calculator to evaluate the power (1.04)4. Answer
   
   
3. What keystrokes did you use in question 2 to find your answer? Answer
   
   

4. Evaluate.
   
   

   1. (1.03)3 Answer
   2. (1.025)10 Answer
   3. (1.07)4x3 Answer

If you answered the Are You Ready? questions without problems, move on to Discover.

If you found the Are You Ready? questions difficult, complete Refresher to review these topics.

Refresher

If you don’t know the answers for Are You Ready?, or require more information, click on the play button to review exponents on the Power page of the Mathematics Glossary.

Go back to Are You Ready? and try the questions again. Contact your teacher if you continue to have difficulty.

Discover

Try This 1

Jeremy and Sara both receive a graduation gift of $1000. Instead of spending this money, they decide to invest the cash. But both Jeremy and Sara are unsure about investing the money with simple or with compound interest.

Click on the play button in the Simple Interest Versus Compound Interest applet to explore the difference between simple interest and compound interest. Pay close attention to what is happening to the interest after each compounding period.

1. In Year 1, both investments earn interest. What is the difference between what happens with this interest between the simple interest investment and the compound interest investment?
   
   
2. After 15 yr, who earns more money?

Save your answers to your course folder.

Share 1

 With a partner or in a group, compare your answers to Try This 1. Discuss the following question:

• Why did one investment earn more than the other investment?

Explore

In Discover you may have noticed the following differences between simple interest and compound interest.

In Try This 1, both investments are calculated annually—for simple interest this is always the case, so time must be converted to years. However, compound interest may be paid out more than once a year, even though the interest is quoted as annual interest. These shorter periods of time where interest is paid out are called compounding periods.

Due to compounding periods, the compound interest formula is a bit more complicated than simple interest.

• A is the final amount of the investment.
• P is the principal.
• r is the annual interest rate.
• n is the number of compounding periods in a year.
• t is the term in years.
  

Example

Ravia lives in Fort Vermilion, Alberta. She has just invested $1000 in a five-year, compound interest bond. The annual interest rate is 6%, compounded annually. Use the compound interest formula to see how much her investment will be worth in five years.

Solution

Identify the variables from the question.	P = $5000 r = 6% = 0.06 t = 5 n = 1
Substitute the known variables into the compound interest formula to find the final amount of the investment (A).

Self-Check 2

1. Based on what you have investigated in this lesson, determine which investment will earn the most interest.
   
   
   1. $2500 compounded monthly at 3% per annum, or $2500 compounded yearly at 3% per annum Answer
      
      
   2. $1500 compounded semi-annually at 4% per annum, or $1500 compounded quarterly at 4% per annum Answer
      
      
2.  
   1. Do question 5 from “Build Your Skills” on page 273 of MathWorks 11. Answer
      
      
   2. What was the interest rate per compounding period? Answer
      
      
3. Do question 6 from “Build Your Skills” on page 273 of MathWorks 11. Answer
   
   
4. Do question 7 from “Build Your Skills” on page 273 of MathWorks 11. Answer

Lesson 2 Summary

In this lesson you examined compound interest on investments. To solve compound interest problems, you discovered that you had to know the annual interest rate, the number of compounding periods per year, the principal, and the investment term in years.

Compound interest is a much more powerful tool in making your money grow than simple interest because of the multiple compounding periods and reinvestment of the interest earned. Will you harness the power of compound interest?

Lesson 3: The Rule of 72

Focus

Albert Einstein is among the world’s greatest physicists and mathematicians. He’s the father of relativity and is popularly associated with the formula E = mc2. His formula links mass and energy and the harnessing of nuclear power.

In a related matter, Einstein claimed, “The most powerful force in the universe is compound interest.” In this lesson you will investigate the power of compound interest and discover and use a rule for estimating the length of time needed to double an investment.

Lesson Question

In this lesson you will investigate the following question:

• How can the Rule of 72 be used to solve compound interest problems?

Assessment

Your assessment for this lesson may include a combination of the following:

• course folder submissions from the Try This and Share sections of the lesson

• your contribution to the Mathematics 20-3 Glossary Terms and the Formula Sheet

• Lesson 3 Assignment (Save a copy of your lesson assignment to your course folder now.)

• the Project Connection

Materials and Equipment

• calculator

Launch

This section checks to see if you have the necessary background knowledge and skills required to successfully complete Lesson 3.

Complete the Are You Ready? questions. If you have difficulty or any questions, visit Refresher for a review or contact your teacher.

Are You Ready?

In this lesson you will apply your mental arithmetic skills. One skill in particular is identifying factors of composite numbers such as 72.

1. What is meant by the term factor? Answer
   
   
2. List all the factors of 72. Answer
   
   
3. What is a composite number? Answer

If you answered the Are You Ready? questions without problems, move on to Discover.

If you found the Are You Ready? questions difficult, complete Refresher to review these topics.

Refresher

If you don't know the answers for Are You Ready?, or require more information, review the Natural Numbers applet on factors. Click on the play button to get going.

Go back to Are You Ready? and try the questions again. Contact your teacher if you continue to have difficulty.

Discover

In this section you will discover how to quickly estimate the time required to double an investment at a given rate of compound interest.

When Oscar entered Grade 1, his grandmother invested $2000 towards his education. The money was invested at 8% compounded annually over 12 years. This year, Oscar will use this money when he attends the First Nations University of Canada at Regina.

The following spreadsheet tracks the growth of this investment over 12 years.

© Microsoft Corporation. All Rights Reserved. Used with permission from Microsoft Corporation.

Try This 1

1. At the end of which investment year does the original $2000 double in value?
   
   
2. Divide the number 72 by your answer to question 1. How does this answer compare to the interest rate of this investment?
   
   
3. The answer to question 2 illustrates the Rule of 72. Can you suggest what is meant by the Rule of 72?
   
   
4. About how many years would it take an investment to double in value at 6% compounded annually? Why?

Save your answers to your course folder.

Share 1

Share your responses to the questions in Try This 1 with a classmate or group.

• How did your interpretation of the Rule of 72 compare with the opinions of other students?
  
  
• Why might there have been similarities and differences between the interpretations?

If required, save a copy of your discussion in your course folder.

Explore

In Discover you investigated the Rule of 72. You discovered that by using the Rule of 72, and given an annual interest rate that is compounded annually, you can estimate the time required for a given investment to double in value.

approximate number of years for the investment to double =

Example

During the ski season, Rose works as a ski instructor at Mont Tremblant in Québec. She has just invested $2000 of her savings at 4% compounded annually. Use the Rule of 72 to estimate the number of years it will take for Rose’s investment to double.

Solution

The annual interest rate = 4%

Use the Rule of 72:

Divide 72 by 4.

Rose’s investment will double in about 18 years.

You have seen the rule of 72 explained using the following formula:

approximate number of years for the investment to double =

But in Try This 1 you divided 72 by 9, which is the number of years it took the investment to double—not the interest rate! This is because the formula can be rearranged to estimate the annual interest rate if you know the doubling time for an investment.

interest rate as a percent =

In the next example you will explore how the Rule of 72 can be used to estimate the annual interest rate from the doubling time for an investment.

Example

Ron, who is 15 years old, loves snowmobiling. He has his eye on a model that costs $6000, and Ron would like to own the snowmobile by the time he turns 25. If he has $3000 in savings today, at what rate of interest should Ron invest the $3000 so that he will have $6000 in ten years?

Solution:

doubling time = 10 years

Use the Rule of 72:

Ron should look for an investment that earns at least 7.2%.

Try This 2

• Search the Internet for a “Rule of 72 Calculator.” Find a calculator you like and use it to investigate the question in the next bullet.
  
  
• You have seen that the interest rate affects the doubling time of the investment, but does the amount of money you invest have any effect on the investment’s doubling time?

Save your responses to your course folder.

Share 2

Share your Rule of 72 Calculator link, found in Try This 2, with a partner or group. Discuss the following question:

• What effect does the amount of money you invest have on the doubling time of the investment?

If required, save a copy of the discussion to your course folder.

Caution: Remember that the Rule of 72 uses percentages instead of the decimal equivalent. If you are given a rate of 0.06, you will first rewrite the rate as 6%.

Self-Check 1

1. Complete the following table for compound interest investments using the Rule of 72. The first example, 4%, is completely done for your reference.
   
   

   Interest Rate	Doubling Time	Your Calculation
   4%	18 years	72 ÷ 4 = 18
   	24 years
   12%
   0.09
   0.02
   	8 years
   		72 ÷ 6 = 12

   
   Answer
   
   
2. Karl has invested $1000 in a compound-interest bond earning interest at the rate of 0.024 per annum. Karl applied the Rule of 72 as follows.
   
   72 ÷ 0.024 = 3000
   
   At that rate, Karl says it will take 3000 years to double his investment. Is Karl correct? Why or why not? Answer

Lesson 3 Summary

In Focus you read that Einstein claimed that compound interest was the greatest power in the universe. Perhaps Einstein’s famous formula E = mc2 was really E = mc72!

In this lesson you explored the Rule of 72. You now have a tool for relating compound interest rates and the corresponding times required to double investments. Use this information to help you make wise investment decisions.

Lesson 4: Comparing Simple and Compound Interest and Their Graphs

Focus

Have you leafed through the financial pages of a newspaper or checked out an online financial newspaper or site? There you will see a variety of graphs indicating trends in markets, interest rates, the dollar, inflation, and the performance of individual stocks.

You have likely heard the saying, “A picture is worth a thousand words!” Nowhere is this comment more accurate than when investors want to see, at a glance, how their money is doing. Are the graphs rising or falling, and how steeply?

In this lesson you will compare simple and compound interest and graphically track the performance of these types of investments.

Lesson Questions

In this lesson, you will investigate the following questions:

• What is the difference in interest earned by simple versus compound interest?

• How do the graphs of simple and compound interest compare?

Assessment

Your assessment for this lesson may include a combination of the following:

• course folder submissions from the Try This and Share sections of the lesson
  
  
• your contribution to the Mathematics 20-3 Glossary Terms and the Formula Sheet
  
  
• Lesson 4 Assignment (Save a copy of your lesson assignment to your course folder now.)
  
  
• the Project Connection

Materials and Equipment

• calculator
• quarter-inch graph paper
• straight edge

Launch

This section checks to see if you have the necessary background knowledge and skills required to successfully complete Lesson 4.

Complete the Are You Ready? questions. If you have difficulty or any questions, visit Refresher for a review or contact your teacher.

Are You Ready?

In this lesson you will apply your knowledge of simple interest and compound interest.

1. Annie Lutaaq has just sold one of her carvings. With part of this money, Annie has bought a $100 bond that pays her simple interest at 5% per annum.
   
   
   1. If the term of her investment is eight years, how much interest will she earn? Answer
      
      
   2. If the principal and interest are added together, what amount will Annie have at the end of eight years? Answer
      
      
2.  
   1. If Annie had invested her $100 in a compound interest bond, paying 5% compounded annually, how much would she have had at the end of eight years? Answer
      
      
   2. How much more interest would Annie have earned by investing her money in the compound interest bond instead of in the simple interest bond? Answer

If you answered the Are You Ready? questions without problems, move on to Discover.

If you found the Are You Ready? questions difficult, complete Refresher to review these topics.

Refresher

If you do not know the answers to the Are You Ready? questions or require more information, click on the play button to review comparing simple and compound interest. Choose the item called Simple and Compound Interest from the list.

Go back to Are You Ready? and try the questions again. Contact your teacher if you continue to have difficulty.

Discover

As you have already seen in Are You Ready?, compound interest can earn more interest than simple interest. In the following activity you will investigate the difference in interest earnings between simple and compound interest. You will visually represent this difference with a graph.

Try This 1

Compare what happens to $1000 over ten years if this money is placed in a simple interest investment, and what occurs if the cash is placed in a compound interest investment. You will use an interest rate of 20% for both investments—this rate is impossible to achieve but helps to emphasize the difference between compound and simple interest.

Fill in the remainder of this table by calculating the interest made and the total amount of money the investment is worth after each year. Remember that the interest will be added into the principal for compound interest but not for simple interest.

1. Based on the table, in the first year how much did each of the investments earn? Which one earned more?
   
   
2. By the end of ten years, did the simple interest investment or the compound interest investment earn more interest?
   
   
3. Using graph paper or a computer program, create a double-bar graph of the total amounts for the simple interest GIC and the compound interest GIC.
   
   
   
4. If you connected the tops of the bars for the simple interest investment, would a straight line or a curved line be best?
   
   
5. If you connected the tops of the bars for the compound interest investment, would a straight line or a curved line be best?
   
   
6. Use your graph to predict the accumulated total for the simple and compound interest in the eleventh year.

Save a copy of your responses to your course folder.

Share 1

Share your answers to Try This 1 with a partner or group. After you have shared your answers, discuss the following:

• How does the shape of the bar tops in a simple interest graph differ from a compound interest graph?
  
  
• Suggest reasons that could explain this difference in shape.

If required, save a record of your discussion in your course folder.

Lesson 4 Summary

Successful investing means being able to see the big picture. In this lesson the big picture was graphically comparing financial growth through simple and compound interest. You discovered that the bars in a simple interest graph lined up in a straight line, while those for compound interest swept upward to the right in a curve that grew steeper and steeper. This characteristic of compound interest is due to exponential growth.

Section 1 Project: Capturing Your Interest!

You have probably read that Canadians don’t save enough money for retirement. But saving for retirement isn’t the only reason to set some money aside. You may wish to buy a car or house in the future. Also, having a nest egg protects you in emergencies. Savings give you the freedom to make choices such as training for a new job, moving to a different city, travelling, or helping your family.

But saving is only one part of the equation. Once you have saved some money, how will you invest these funds? Banks and credit unions offer a variety of investment choices. In this project you will explore two of those choices—simple interest and compound interest.

For this project you will assume you have $10 000 to invest. What will you do?

Goal

Your goal for this project is to explore simple and compound-interest options for a $10 000 investment.

Time

The aim is to complete this project in less than hours. Each lesson in the section will direct you to complete at least one of the steps.

Steps

1. Contact a local bank or credit union (in person, online, or use the guided search information in bank links from the Module 1: Section 2 Project: Banking 101).
   
   
   • What is the best rate of interest you might receive on an investment of $10 000? These choices might include bonds, savings accounts, and Guaranteed Investment Certificates.
     
     
   • Report your findings by filling out the following information.
   
   Type of Investment:______________________
   
   Best Interest Rate:___________________________
   
   Interest Period: __________________________
   
   Term of Investment: _________________________
   
   
2. Assume you are able to invest at the best rate of simple interest for five years. Use the simple interest formula, I = Prt, and A = P + I to determine the sum (interest and principal) you would have at the end of five years. Show all your steps.
   
   
3. Assume you are able to invest at the best rate of compound interest for five years. Use the compound-interest formula, , to determine the sum (interest and principal) you would have at the end of five years. Show all your steps.
   
   
4. Apply the Rule of 72 to estimate how long it will take to double your money.
   
   
5. Based upon the rates you used, which investment option (simple or compound) is the better choice? Explain your answer.
   
   
6. Use the Simple Interest - Compound Interest Calculator to create graphs showing the growth of each investment (simple and compound) over the five-year term. The interest rates are set at 5% per annum. Change these amounts to the rates you used in your calculations. Attach a copy of each graph, and include them in your project summary.
   
   
   
   

How You Will Be Evaluated

Your teacher will use this rubric to evaluate your report.

Section 1 Summary

“Oh, they don’t have to worry; they have pots of money!” Is this a familiar phrase? Have you considered that perhaps they have overflowing pots of money due to wise investments?

During the Section 1 Project: Capturing Your Interest!, you examined how simple and compound interest offered by banks can make your own money grow differently. After exploring investment growth, you are now in a position to make informed decisions about your own savings.

In Lessons 1 and 2 you explored the formulas for simple and compound interest. You applied the Rule of 72 in Lesson 3 to find how long it would take to double your money. And in Lesson 4 you examined the growth of investments over time by analyzing graphs. All the skills you acquired are fundamental to your financial growth. It’s now your turn to watch your money tree grow bigger and stronger.