Math 7
Mathematics homework help
Introduction
Course Objectives
This lesson will address the following course outcomes:
· 23. Determine the exponential function for a situation when given an initial value and either the growth/decay rate or a second function value. Interpret the initial value and growth rate of an exponential function. Include compound interest as one application.
Specific Objectives
Students will understand that
· compounding is repeated multiplication by a compounding factor.
· compounding is best expressed in terms of exponential growth, using exponential notation.
· exponential growth models the compounding of interest on an initial investment.
Students will be able to
· calculate the earnings on a principal investment with annual compound interest.
· write a formula for annual compound interest.
· compare and contrast linear and exponential models.
Certificates of Deposit
A Certificate of Deposit (CD) is a type of investment used by many people because it is very safe and predictable. CDs can be purchased through banks and other financial institutions. The U.S. Security and Exchange Commission gives the following information about CDs.
The ABCs of CDs
A CD is a special type of deposit account with a bank or thrift institution that typically offers a higher rate of interest than a regular savings account. Unlike other investments, CDs feature federal deposit insurance up to $250,000.
When you purchase a CD, you invest a fixed sum of money for fixed period of time—six months, one year, five years, or more—and, in exchange, the issuing bank pays you interest, typically at regular intervals. When you cash in or redeem your CD, you receive the money you originally invested, plus any accrued interest. If you redeem your CD before it matures, you may have to pay an “early withdrawal” penalty or forfeit a portion of the interest you earned …
At one time, most CDs paid a fixed interest rate until they reached maturity. But, like many other products in today’s markets, CDs have become more complicated. Investors may now choose among variable-rate CDs, long-term CDs, and CDs with other special features.
In this course, you will only be looking at situations in which the CD has a fixed interest rate. The period of time that you agree to leave your money in the CD is called the term. Terms can range from 3 months to multiple years. Some CDs require a minimum deposit, but others do not. The deposit, or amount you put into the CD, is called the principal.
CDs pay compound interest. Recall in previous assignments you have worked with situations that use simple interest. In simple interest, the total interest is calculated from the original principal. In compound interest, the interest is added to the principal after a set period of time. Then interest for the next period of time is calculated based on the new, higher balance in the account.
Here is an example:
You invest $100 at 5% interest compounded annually or each year. At the end of the year, you earn 5% of $100 or $5. This is added into the account so now the balance is $105. In the second year, you earn 5% of $105 or $5.25.
Following are some common compounding periods:
· Quarterly (4 times a year)
· Monthly (12 times a year)
· Daily (365 times a year)
There is also a type of compounding called continuous, but that will not be discussed in this course.
Five-year CD
Problem Situation 1: The Five-year CD
Suppose you invest $1,000 principal into a certificate of deposit (CD) with a five-year term that pays a 2% annual percentage rate (APR) interest. The compounding period is one year.
To calculate how much you have after a year:
Interest = Principal * interest rate. I=PrI=Pr
End amount = Principal + Interest. A=P+IA=P+I
Putting these together: A=P+PrA=P+Pr
#1 Points possible: 5. Total attempts: 5
How much money will you have in your account after the first year? Round to the nearest cent if necessary.
$
#2 Points possible: 12. Total attempts: 5
How much will you have at the end of the five-year term? Calculate the balance after each year (you already found the balance after 1 year above) Round to the nearest cent if necessary.
Year | Account Balance | |
2 years | $ | |
3 years | $ | |
4 years | $ | |
5 years | $ |
For the first year, you found the balance using the calculation 1000+1000(0.02)1000+1000(0.02)
Notice that 1000+1000(0.02)=1000⋅1+1000(0.02)1000+1000(0.02)=1000⋅1+1000(0.02) and we can factor out the Greatest Common Factor of 1000, so
1000⋅1+1000(0.02)=1000(1+0.02)=1000(1.02)1000⋅1+1000(0.02)=1000(1+0.02)=1000(1.02)
In year 2, you calculated: 1020+1020(0.02)1020+1020(0.02) . We can factor out the GCF of 1020, so
1020+1020(0.02)=1020⋅1+1020(0.02)=1020(1+0.02)=1020(1.02)1020+1020(0.02)=1020⋅1+1020(0.02)=1020(1+0.02)=1020(1.02)
Now, remember where that 1020 came from: 1000+1000(0.02)=1000(1.02)=10201000+1000(0.02)=1000(1.02)=1020 .
If we take the expression for the balance after 2years, 1020(1.02)1020(1.02) , and replace the 1020 with 1000(1.02)1000(1.02) then we find another way to calculate the balance after 2 years: 1020+1020(0.02)=1020(1.02)=1000(1.02)(1.02)1020+1020(0.02)=1020(1.02)=1000(1.02)(1.02)
#3 Points possible: 5. Total attempts: 5
The expression 1000(1.02)(1.02)1000(1.02)(1.02) could be written more compactly by using exponents. Rewrite the expression using exponents.
#4 Points possible: 5. Total attempts: 5
The amount in the account after 3 years could be calculated as 1040.40+1040(0.02)1040.40+1040(0.02) or 1040.40(1.02)1040.40(1.02) .
Or, we could use the same approach shown above and replace the 1040.40 with the expression for the value after 2 years you found in the previous problems. Make that replacement, then try to write it more compactly using exponents.
Hint: If you get stuck, we’ll offer some hints
#5 Points possible: 5. Total attempts: 5
Look at the simplified expressions you have found for the amount after two and three years. Do you notice a pattern in how those expressions look?
Use that pattern to develop a formula for the account balance after t years.
A =
Hint: If you get stuck, we’ll offer some hints
#6 Points possible: 20. Total attempts: 5
Using the formula you developed in the previous problem, fill in the table below for the account balance after the given number of years, if the principal = $1,000 and the interest rate APR = 2%. Round to the nearest cent if needed.
Year | Account Balance | |
10 years | $ | |
20 years | $ | |
40 years | $ | |
70 years | $ | |
100 years | $ |
#7 Points possible: 5. Total attempts: 5
Plot your results from the previous question. You won’t be able to plot the exact values, so you will need to estimate where the values will lie.
Clear All Draw:
#8 Points possible: 5. Total attempts: 5
Is your formula you developed earlier for the account balance linear? If not, what family does it belong to?
· Linear
· Exponential
· Quadratic
#9 Points possible: 5. Total attempts: 5
In an earlier problem, you came up with a formula that gave the account balance after t years. Look carefully at that formula, and consider where the numerical values in the formula came from.
Write a general formula that could be used to find the accrued amount(A) for a CD with annual compounding. Let P = the principal, r = the APR as a decimal, and t = number of years.
A =
Value of a CD
Problem Situation 2: The Value of a CD
In the last problem, the CD was compounded annually. In this problem, we will extend that for compounding periods of various durations.
Suppose you invest $1,000 principal in a two-year CD, advertised with an annual percentage rate (APR) of 2.4%, where compounding occurs monthly.
#10 Points possible: 8. Total attempts: 5
If the APR is 2.4% per year, how much is the interest per month (Remember Periodic Rate from 1.9)?
%
Now write this as a decimal.
#11 Points possible: 24. Total attempts: 5
Using your answer from the previous problem, and the concepts developed earlier in the lesson, complete the following table. Round values to the nearest cent if needed.
Period | Account Balance |
1 month | $ |
2 months | $ |
6 months | $ |
12 months | $ |
24 months | $ |
3 years | $ |
Hint: If you get stuck, we’ll provide some hints
In Question 10, you figured out the periodic rate (interest per month) by dividing the APR by 12, the number of times the interest is being compounded in one month. In Question 11, to find the account balance after 24 months, you used 24 months in the exponent.
In the next problem, we want to develop a general formula to calculate the value of any CD.
Let
P = the principal, r = the annual interest rate (APR) as a decimal, n = the number of compounding periods in a year (1 for annual, 12 for monthly, 52 for weekly, etc.) t = number of years.
So in the previous two questions, n = 12, since we were compounding monthly (12 times per year). Think about how you answered the last question: how did you find the interest rate per period? To find the value after 3 years, how did you figure out the number of months?
#12 Points possible: 5. Total attempts: 5
Write a general formula that can be used to calculate the value of any CD, using the variables defined above (P, t, r, and n).
A =
Hint: If you get stuck, we’ll provide some hints
HW 4.1
#1 Points possible: 5. Total attempts: 5
Which of the following was one of the main mathematical ideas of the lesson?
· Compound interest is an important concept in savings and investment.
· Exponential equations can be used to model percentage growth.
· Compound interest can be modeled with linear equations.
· Annual compounding is done once a year.
#2 Points possible: 1. Total attempts: 5
Explain what the following statements mean. Give examples to support your explanation.
· Multiplication is repeated addition.
· A power is repeated multiplication.
#3 Points possible: 1. Total attempts: 5
How is a linear model different from an exponential model?
#4 Points possible: 24. Total attempts: 5
In Lesson 3.7, you explored the Simple Interest formula, A=P+PrtA=P+Prt , which is used when interest does not compound. This is fairly rare, but is how Treasury Notes and some other specialized investments work.
In this problem we will compare Simple Interest to Compound Interest, which you learned in this Lesson. Recall from the lesson that when interest is compounded annually the compound interest formula is A=P(1+r)tA=P(1+r)t.
In both formulas, A is the Amount (total principal plus interest) you end up with, P is the Principal (starting amount), r is the annual interest rate (quoted as a percent, but used as a decimal), and t is the time in years.
a. Complete the table below to compare the amounts you would have if you invested $1000 at 5% with simple interest compared to $1000 at 5% compounded annually. Round to the nearest cent.
Simple Interest | Compound Interest | |
1 year | $ | $ |
2 years | $ | $ |
3 years | $ | $ |
4 years | $ | $ |
5 years | $ | $ |
b.
c. In the lesson you noticed that compound interest is an exponential model. What kind of model is simple interest?
d. In the long run, which method of receiving interest will result in more money being in the account, assuming the rate of interest is the same?
#5 Points possible: 12. Total attempts: 5
Which is the better investment for 10 years: investing $500 at 10% APR or $1,000 at 2% APR? Both investments have annual compounding.
After 10 years, $500 at 10% APR would grow to $
After 10 years, $1,000 at 2% APR would grow to $
The better investment is:
CDs are a very safe investment because they are usually insured by the U.S. government. (There are some CDs that are not insured so it is important to always check!) Because they are so safe, CDs earn low rates of interest. The amount earned on an investment is often called the return. In general, investments with higher risk also have the potential for higher rates of return. For example, if you invest in a stock, which is like buying a piece of a company, you might earn far more than you would with a CD. However, you also run the risk of losing all of your money.
Mutual funds are another type of investment used by many people. The U.S. Security and Exchange Commission defines a mutual fund as follows:1
A mutual fund is a company that brings together money from many people and invests it in stocks, bonds, or other assets. The combined holdings of stocks, bonds, or other assets the fund owns are known as its portfolio. Each investor in the fund owns shares, which represent a part of these holdings.
Mutual funds are an attractive investment to many people for several reasons.
· You can invest small amounts of money at a time.
· The fund is managed by a company that does all the work of researching and choosing specific investments.
· In general, there is less risk than owning a single stock because the money is spread across many investments.
· In general, there are higher rates of returns than those available in insured investments like CDs and savings accounts.
However, it is important to understand that mutual funds do still have risks. It is possible to lose some or all of your investment. Also, funds charge fees to pay for the management. Sometimes these fees can be very high.
Selecting investments is an important decision and should be researched carefully. Through this course, you will be introduced to a few basics concepts related to investment, but you should not make any decisions based only on the information presented here.
One of the most important concepts in investing is to take advantage of the power of compounding over time. The following example will help you explore this idea.
#6 Points possible: 16. Total attempts: 5
Lorenzo and Michael each decide to invest in a mutual fund to save for retirement. They each choose a mutual fund that has had an average annual return of 5.6% over the last decade. There is no guarantee that the mutual fund will continue to earn this same rate, but it can be used as an estimate of future returns. Use the information given below to estimate how much each man will have when he retires. Round to the nearest dollar.
· Lorenzo invests $2,000 when he is 25 years old.
· Michael invests $5,000 when he is 45 years old.
· Both men plan to retire at age 65.
Hint 1: The amount earned is how much they have beyond the initial investment. Hint 2: You can treat this problem as if the account compounds annually.
Amount at retirement | Amount earned | |
Lorenzo | $ | $ |
Michael | $ | $ |
#7 Points possible: 5. Total attempts: 5
In this problem you will compare the effects of different compounding periods on the interest an investment earns. Complete the table below using the values indicated. Show the formula you used, with the correct values, in the second column. In the third column, give the result, rounded to the nearest cent.
· Principal: $1,000
· APR: 4.5%
· Time: 10 years
Compounding Period | Equation Used for Calculation (with values inserted) | Amount Accrued After 10 Years |
Annual | A = | $ |
Quarterly | A = | $ |
Monthly | A = | $ |
Daily | A = | $ |