If there is one prayer that you should pray/sing every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)
- Samuel Dominic Chukwuemeka
It is the most powerful prayer.
A pure heart, a clean mind, and a clear conscience is necessary for it.
For in GOD we live, and move, and have our being.
- Acts 17:28
The Joy of a Teacher is the Success of his Students.
- Samuel Dominic Chukwuemeka
I greet you this day,
First: read the notes.
Second: view the videos.
Third: solve the questions/solved examples.
Fourth: check your solutions with my thoroughly-explained solutions.
Fifth: check your answers with my specially developed calculators and/or the Texas Instruments (TI) calculators.
I wrote the codes for the calculators using JavaScript, a client-side scripting language. Please use the
latest Internet browsers. The calculators should work.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may
contact me. However, if you are my student; please do not contact me here. Contact me via the school's system.
Thank you.
Samuel Dominic Chukwuemeka (SamDom For Peace) B.Eng.,A.A.T, M.Ed., M.S
Students will:
(1.) Discuss the topic of simple interest.
(2.) Solve applied problems involving simple interest.
(3.) Discuss the topic of compound interest.
(4.) Solve applied problems involving compound interest.
(5.) Discuss the topic of continuous compound interest.
(6.) Solve applied problems involving continuous compound interest.
(7.) Discuss the topic of annual percentage yield.
(8.) Solve applied problems involving annual percentage yield.
(9.) Discuss several investments for retirement.
(10.) Discuss the topic of ordinary annuity.
(11.) Solve applied problems involving ordinary annuity.
(12.) Discuss the topic of annuity due.
(13.) Solve applied problems involving annuity due.
(14.) Discuss the topic of amortization.
(15.) Solve applied problems involving amortization.
(16.) Discuss the topic of sinking fund.
(17.) Solve applied problems involving sinking fund.
(18.) Discuss the rule of 78.
(19.) Solve applied problems involving the rule of 78.
(20.) Discuss the buying of a car in the United States.
(21.) Discuss the buying of a home in the United States.
(22.) Solve finance problems using technology including my specially developed calculators and Texas Instruments (TI) calculators.
(1.) Use of prior knowledge
(2.) Critical Thinking
(3.) Interdisciplinary connections/applications
(4.) Technology
(5.) Active participation through direct questioning
(6.) Research
(1.) Monthly interest payment = Monthly interest rate * Average balance
(2.) Net monthly cash flow = Monthly income − Monthly expenses
$ (1.)\:\: SI = Prt \\[3ex] (2.)\:\: SI = A - P \\[3ex] (3.)\:\: P = \dfrac{SI}{rt} \\[5ex] (4.)\:\: t = \dfrac{SI}{Pr} \\[5ex] (5.)\:\: r = \dfrac{SI}{Pt} \\[5ex] (6.)\:\: A = P + SI \\[3ex] (7.)\:\: A = P(1 + rt) \\[3ex] (8.)\:\: P = \dfrac{A}{1 + rt} \\[5ex] (9.)\:\: t = \dfrac{A - P}{Pr} \\[5ex] (10.)\:\: r = \dfrac{A - P}{Pt} \\[5ex] (11.)\:\: SI = \dfrac{Art}{1 + rt} $
$
(1.)\:\: A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[7ex]
(2.)\:\: P = \dfrac{A}{\left(1 + \dfrac{r}{m}\right)^{mt}} \\[10ex]
(3.)\:\: r = m\left[\left(\dfrac{A}{P}\right)^{\dfrac{1}{mt}} - 1\right] \\[10ex]
(4.)\:\: r = m\left(10^{\dfrac{\log\left(\dfrac{A}{P}\right)}{mt}} - 1\right) \\[10ex]
(5.)\:\: t = \dfrac{\log\left(\dfrac{A}{P}\right)}{m\log\left(1 + \dfrac{r}{m}\right)} \\[7ex]
(6.)\:\: A = P + CI \\[3ex]
(7.)\:\: CI = A - P \\[3ex]
(8.)\:\: A = P(1 + i)^n \\[4ex]
(9.)\:\: P = \dfrac{A}{(1 + i)^n} \\[7ex]
(10.)\:\: i = \dfrac{r}{m} \\[5ex]
(11.)\:\: n = mt \\[3ex]
(12.)\;\; Total\;\;Return = \dfrac{A - P}{P} * 100\% \\[7ex]
(13.)\;\; Annual\;\;Return = \left(\dfrac{A}{P}\right)^{\dfrac{1}{t}} * - 1 \\[7ex]
$
Future Value (Amount) of Cash Flows (Principal) for Several Years
$
(13.)\:\:At\:\:the\:\:end\:\:of\:\:each\:\:year:\:\: FV = PV\left(1 +
\dfrac{r}{m}\right)^{m(last\:\:year - that\:\:year)} \\[7ex]
(14.)\:\: Total\:FV = \Sigma FV
$
Values of $m$
If Compounded: | m = |
---|---|
Annually |
$1$ ($1$ time per year) Also means every twelve months |
Semiannually |
2 (2 times per year) Also means every six months |
Quarterly |
4 (4 times per year) Also means every three months |
Monthly |
12 (12 times per year) Also means every month |
Weekly | 52 (52 times per year) |
Daily (Ordinary/Banker's Rule) | 360 (360 times per year) |
Daily (Exact) | 365 (365 times per year) |
$ (1.)\:\: A = Pe^{rt} \\[4ex] (2.)\:\: P = \dfrac{A}{e^{rt}} \\[7ex] (3.)\:\: t = \dfrac{\ln \left(\dfrac{A}{P}\right)}{r} \\[7ex] (4.)\:\: r = \dfrac{\ln \left(\dfrac{A}{P}\right)}{t} \\[7ex] (5.)\;\; Total\;\;Return = \dfrac{A - P}{P} * 100\% \\[7ex] (6.)\;\; Annual\;\;Return = \left(\dfrac{A}{P}\right)^{\dfrac{1}{t}} * - 1 $
$ (1.)\:\: APY = \left(1 + \dfrac{r}{m}\right)^m - 1 \\[7ex] (2.)\:\: r = m\left[(APY + 1)^{\dfrac{1}{m}} - 1\right] \\[7ex] (3.)\:\: r = m\left(\sqrt[m]{APY + 1} - 1\right) $
$ (1.)\:\: APY = e^r - 1 \\[4ex] (2.)\:\: r = \ln(APY + 1) $
(1.) Market Capitalization (Market Cap) = Total Number of Outstanding Shares * Current Share Price
$ (1.)\:\: FV = m * PMT * \left[\dfrac{\left(1 + \dfrac{r}{m}\right)^{mt} - 1}{r}\right] \\[10ex] (2.)\;\; FV = PMT * \dfrac{\left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right]}{\dfrac{r}{m}} \\[10ex] (3.)\:\: t = \dfrac{\log\left[\dfrac{r * FV}{m * PMT} + 1\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (4.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (5.)\:\: CI = FV - Total\:\:PMTs \\[5ex] (6.)\:\: FV = PMT * \left[\dfrac{(1 + i)^n - 1}{i}\right] \\[7ex] (7.)\:\: n = \dfrac{\log\left[\dfrac{i * FV}{PMT} + 1\right]}{\log(1 + i)} \\[10ex] (8.)\:\: s_{n\i} = \dfrac{m}{r} * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right] \\[7ex] (9.)\:\: s_{n\i} = \dfrac{(1 + i)^n - 1}{i} \\[5ex] (10.)\:\: FV = PMT * s_{n\i} \\[3ex] (11.)\:\: i = \dfrac{r}{m} \\[5ex] (12.)\:\: n = mt \\[3ex] (13.)\:\: Annual\:\:Fuel\:\:Expense = \dfrac{Annual\:\:Miles\:\:Driven}{Miles\:\:per\:\:Gallon} * Price\:\:per\:\:Gallon $
$ (1.)\:\: PMT = \dfrac{r * FV}{m * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right]} \\[10ex] (2.)\:\: t = \dfrac{\log\left[\dfrac{r * FV}{m * PMT} + 1\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (3.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (4.)\:\: CI = FV - Total\:\:PMTs \\[3ex] (5.)\:\: PMT = \dfrac{i * FV}{(1 + i)^n - 1} \\[7ex] (6.)\:\: n = \dfrac{\log\left[\dfrac{i * FV + PMT}{PMT}\right]}{\log(1 + i)} \\[10ex] (7.)\:\: s_{n\i} = \dfrac{m}{r} * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right] \\[7ex] (8.)\:\: s_{n\i} = \dfrac{(1 + i)^n - 1}{i} \\[5ex] (9.)\:\: i = \dfrac{r}{m} \\[5ex] (10.)\:\: n = mt $
$ (1.)\:\: PV = m * PMT * \left[\dfrac{1 - \left(1 + \dfrac{r}{m}\right)^{-mt}}{r}\right] \\[10ex] (2.)\:\: t = -\dfrac{\log\left[1 - \dfrac{r * PV}{m * PMT}\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (3.)\:\: PV = PMT * \left[\dfrac{1 - (1 + i)^{-n}}{i}\right] \\[7ex] (4.)\:\: n = \dfrac{\log \left[\dfrac{PMT}{PMT - i * PV}\right]}{\log(1 + i)} \\[10ex] (5.)\:\: a_{n\i} = \dfrac{m}{r} * \left[1 - \left(1 + \dfrac{r}{m}\right)^{-mt}\right] \\[7ex] (6.)\:\: a_{n\i} = \dfrac{1 - (1 + i)^{-n}}{i} \\[5ex] (7.)\:\: PV = PMT * a_{n\i} \\[3ex] (8.)\:\: i = \dfrac{r}{m} \\[5ex] (9.)\:\: n = mt \\[3ex] (10.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (11.)\:\: CI = Total\:\:PMTs - PV $
$ (1.)\:\: PMT = \dfrac{PV}{m} * \left[\dfrac{r}{1 - \left(1 + \dfrac{r}{m}\right)^{-mt}}\right] \\[10ex] (2.)\:\: t = -\dfrac{\log\left[1 - \dfrac{r * PV}{m * PMT}\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (3.)\:\: PMT = \dfrac{i * PV}{1 - (1 + i)^{-n}} \\[7ex] (4.)\:\: n = \dfrac{\log \left[\dfrac{PMT}{PMT - i * PV}\right]}{\log(1 + i)} \\[10ex] (5.)\:\: i = \dfrac{r}{m} \\[5ex] (6.)\:\: n = mt \\[3ex] (7.)\:\: Payoff = PMT * n * \left[\dfrac{1 - \left(1 + \dfrac{r}{n}\right)^{-k}}{r}\right] \\[10ex] (8.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (9.)\:\: CI = Total\:\:PMTs - PV \\[3ex] (10.)\:\: CI = PMT * m * t - PV \\[3ex] (11.)\:\: Number\:\:of\:\:payments = m * t \\[3ex] (12.)\:\: Down\:\:Payment = Given\:\:Rate * Purchase\:\:Price \\[3ex] (13.)\:\: Amount\:\:of\:\:Mortgage = Purchase\:\:Price - Down\:\:Payment \\[3ex] (14.)\:\: Payment\:\:for\:\:x\:\:points\:\:at\:closing = x\:\:as\:\:\% * Amount\:\:of\:\:Mortgage $
$ (1.)\:\: FV = m * PMT * \left[\dfrac{\left(1 + \dfrac{r}{m}\right)^{mt} - 1}{r}\right] * \left(1 + \dfrac{r}{m}\right) \\[10ex] (2.)\:\: PMT = \dfrac{r * FV}{(m + r) * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right]} \\[10ex] (3.)\:\: t = \dfrac{\log\left[\dfrac{r * FV}{PMT(m + r)} + 1\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (4.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (5.)\:\: CI = FV - Total\:\:PMTs \\[3ex] (6.)\:\: FV = PMT * \left[\dfrac{(1 + i)^n - 1}{i}\right] * (1 + i) \\[7ex] (7.)\:\: PMT = \dfrac{i * FV}{(1 + i)\left[(1 + i)^n - 1\right]} \\[7ex] (8.)\:\: n = \dfrac{\log\left[\dfrac{i * FV}{PMT(1 + i)} + 1\right]}{\log(1 + i)} \\[10ex] (9.)\:\: i = \dfrac{r}{m} \\[5ex] (10.)\:\: n = mt \\[3ex] (11.)\:\: CFV = P\left(1 + \dfrac{r}{m}\right)^{mt} + m * PMT * \left[\dfrac{\left(1 + \dfrac{r}{m}\right)^{mt} - 1}{r}\right] * \left(1 + \dfrac{r}{m}\right) \\[10ex] (12.)\:\: t = \dfrac{\log\left[\dfrac{rCFV + PMT(m + r)}{rP + PMT(m + r)}\right]}{m\log\left(1 + \dfrac{r}{m}\right)} $
$ \underline{Monthly} \\[3ex] (1.)\:\: UI = \dfrac{TI * k * (k + 1)}{n(n + 1)} \\[5ex] (2.)\:\: TP = n * PMT \\[3ex] (3.)\:\: TI = TP - LA \\[3ex] (4.)\:\: RF = \dfrac{UI}{TI} \\[5ex] (5.)\:\: RF = \dfrac{sum\:\:of\:\:digits\:\:for\:\:up\:\:to\:\:k}{sum\:\:of\:\:digits\:\:for\:\:up\:\:to\:\:n} \\[5ex] (6.)\:\: LAR = LA * RF \\[3ex] (7.)\:\: UI = TI * RF $
$ (1.)\:\: CP = \dfrac{CR * FV}{m} \\[7ex] (2.)\:\: YTM = \left(\dfrac{FV}{BP}\right)^{\dfrac{1}{t}} - 1 \\[7ex] (3.)\:\: BP = \dfrac{FV}{(YTM + 1)^t} \\[7ex] (4.)\:\: FV = BP * (YTM + 1)^t \\[5ex] (5.)\:\: t = \dfrac{\log\left(\dfrac{FV}{BP}\right)}{\log(YTM + 1)} $
$ (1.)\:\: CP = \dfrac{CR * FV}{m} \\[5ex] (2.)\:\: BP = \dfrac{FV * CR}{YTM} * \left[1 - \dfrac{1}{\left(1 + \dfrac{YTM}{m}\right)^{mt}}\right] + \dfrac{FV}{\left(1 + \dfrac{YTM}{m}\right)^{mt}} \\[10ex] (3.)\:\: YTM \approx \dfrac{m * t * CP + FV - BP}{t(FV + BP)} \\[7ex] (4.)\:\: Annualized\:\:YTM \approx \dfrac{2(m * t * CP + FV - BP)}{t(FV + BP)} \\[7ex] (5.)\:\: YTM \approx \dfrac{t * CR * FV + FV - BP}{t(FV + BP)} \\[7ex] (6.)\:\: Annualized\:\:YTM \approx \dfrac{2(t * CR * FV + FV - BP)}{t(FV + BP)} \\[7ex] $
Discuss savings account.
Discuss the importance and benefits of credit unions over banks.
Use these "personal experience" examples and others as time demands:
(1.) Alabama Teachers Credit Union
(2.) Wells Fargo
Simple Interest is a topic discussed in Financial Mathematics, Finite Mathematics, Business Mathematics,
Financial Management, and other finance/business classes.
Say you:
(1.) Wanted to save some money OR set aside some money for a rainy day
(2.) Opened a savings account with a financial institution such as a credit union or a bank
(3.) Deposited some money into that savings account
That initial sum of money you deposited is known as the Principal or Principal sum of money
or Investment
The financial institution uses your money for financial transactions including lending loans to
individuals and businesses among others.
That money you deposited (Principal) is supposed to give you an extra money over a certain period of
time.
That extra money you get over a period of time due to the initial money you deposited is known as
Simple Interest or Dividend or Yield or Return
This is because the individuals or businesses that borrowed the loans from the financial institution (which used your money)
pay back those loans with some extra money.
So, the financial institution gives you a portion of that interest.
Teacher: Why would the financial institution charge the extra money (interest) on the loans they give to
an individual or business?
Student: Because they traded with your money.
Teacher: Partly, yes. But not fully.
Student: Why not fully?
What are the other reasons?
Teacher: Assume someone, say your friend asks you to lend him some money.
Assume you wanted to use that money to purchase land.
But you realize that your friend's need for your money is probably a higher priority
You decide to lend him the money if he promises to pay back within a month
This is because you really wanted to purchase that land because it was listed for a reasonable price you could afford
Your friend did not pay back the money in a month.
Did not pay back in two, three, four, ..., ten months.
He pays you back the money in a year
You found out that the land you wanted to purchase has appreciated in value, and hence was listed at a higher price
Would you be happy if your friend pays you the exact amount he borrowed from you?
Student: Not really
Teacher: Why?
Student: Because I can no longer purchase the land for the amount I loaned him
Teacher: Makes sense.
But, what if you could still purchase the land for that amount or for a lesser amount?
Student: Well, as a Christian; I probably would not be angry
Teacher: Why?
Student: Because of Psalm 15:5...lending money without interest
Teacher: I understand.
But, do you understand the concept of charging interest on money borrowed, or giving interest on money deposited?
Student: Yes... because of inflation
Because what you would have done with the money has probably appreciated in value
As someone taught by Mr. C (this is my advice, you are encouraged to think for yourselves and decide for
yourselves); you should:
Look for credit unions with free savings accounts: no conditions except the one-time $\$25.00$
membership fee (which is refunded to you upon closing the account)
Look for credit union(s) with okay interest rates on savings accounts.
Small interest is better than no interest.
Again, take note: no conditions except the membership fee.
Look for credit union(s) with free checking accounts. Absolutely no conditions except the membership
fee.
Look for credit union(s) with okay interest rates on checking accounts.
This may have some conditions, but it should be fair conditions that you can meet without worries.
Regardless of any conditions for receiving an interest on the checking account, there should definitely
be no penalty whatsoever.
Depending on the instruction time, give some examples of these credit unions
Some students may ask the advantage of using banks over credit unions.
Be prepared to answer those questions.
Say someone deposited:
an initial sum of money, $P$ in dollars, naira, yen, etc.
into a financial institution that gives an interest rate, $r$ in percent
then, after some time, $t$ in years
the person will earn a simple interest, $SI$ in dollars, naira, yen, etc.
where the $Simple\:\:Interest = Principal * rate * time$
The total amount the person receives will now be the sum of the principal and the simple interest
earned.
$Amount = Principal + Simple\:\:Interest$
OR
Say someone borrowed:
an initial sum of money, $P$ in dollars, naira, yen, etc.
from a financial institution that charges an interest rate, $r$ in percent
then, after some time, $t$ in years
the person will owe a simple interest, $SI$ in dollars, naira, yen, etc.
where the $Simple\:\:Interest = Principal * rate * time$
The total amount the person will pay back will now be the sum of the principal and the simple
interest earned.
$Amount = Principal + Simple\:\:Interest$
The Principal is the initial sum of money deposited in a financial institution for the
purpose of earning an interest. OR
the initial sum of money borrowed from a financial institution for the purpose of paying an interest.
It is also known as Investment
The Simple Interest is the sum of money earned when a principal is deposited, or charged when a principal is borrowed; at
a certain rate over a period of time.
Please write all the formulas that you see under Simple Interest and let us
solve some examples
To explain the concept of Compound Interest, let us analyze the two cases:
Case 1 and Case 2 and the two tables: Table 1 and Table 2
Case 1:
Say you deposited an initial sum of money, $1000.00 in a financial institution that has an interest
rate of 10%
How much would you have after 3 years?
$Year$ | $P(\$)$ | $r(\%)$ | $t(years)$ | $SI(\$)$ | $A(\$)$ |
---|---|---|---|---|---|
$1$ | $1000$ | $0.1$ | $1$ | $100$ | $1100$ |
2 | $1000$ | $0.1$ | $1$ | $100$ | $1200$ |
$3$ | $1000$ | $0.1$ | $1$ | $100$ | $1300$ |
Interest after $3$ years = $100 + 100 + 100 = \$300$
Amount after $3$ years = $\$1300$
Teacher: Is this familiar to you - what we just did?
Student: Why would you do it by using a Table?
It is much easier to use the Simple Interest Formula!
Teacher: That is correct. However, observe the second table: Table 2
I want to teach you something 😊
Case 2:
Say you deposited an initial sum of money, $1000.00 in a financial institution that has an interest
rate of 10% compounded once (one time) per year.
How much would you have after 3 years?
$Year$ | $P(\$)$ | $r(\%)$ | $t(years)$ | $CI(\$)$ | $A(\$)$ |
---|---|---|---|---|---|
$1$ | $1000$ | $0.1$ | $1$ | $100$ | $1100$ |
2 | $1100$ | $0.1$ | $1$ | $110$ | $1210$ |
$3$ | $1210$ | $0.1$ | $1$ | $121$ | $1331$ |
Interest after $3$ years = $100 + 110 + 121 = \$331$
Amount after $3$ years = $\$1331$
Teacher: What did you notice?
Student: The interest is $\$331$
It is $\$31$ greater than the interest in Table 1
Teacher: Correct!
What else?
Student: The amount in Table 2 is also greater than the amount in Table 1 by $\$31$
Teacher: That is right.
Is that all?
This leads to ...
Compound Interest
As noticed from Table 2:
The amount at the end of Year 1 is the principal at the beginning of Year 2
The amount at the end of Year 2 is the principal at the beginning of Year 3
This implies that the interest is compounded.
Compound Interest is the interest compounded on a principal sum of money over a period of time.
It is the interest on the principal plus interest.
In the example we just did, (Table 2), the interest was compounded every year.
In other words, the interest was compounded annually or yearly
In other words, the interest was compounded once every year
If the interest is compounded annually, it is compounded once (1) time a year (every twelve months)
If the interest is compounded semiannually, it is compounded twice (2) times a year (every six months)
If the interest is compounded quarterly, it is compounded four (4) times a year (every three months)
If the interest is compounded monthly, it is compounded twelve (12) times a year (every month)
If the interest is compounded weekly, it is compounded fifty two (52) times a year
If the interest is compounded ordinary daily, it is compounded three hundred and sixty (360) times a year
If the interest is compounded exact daily, it is compounded three hundred and sixty five (365) times a year
Student: What if we want to compound it 1000 times a year, what is called?
Teacher: We can. But, these are the standard compounding periods.
We can compound it as many times as you want per year.
Student: Which interest is greater - compounding the same amount annually versus semiannually?
Teacher: You are asking great questions.
What do you think?
Student: I think compounding it semiannually will give more interest than compounding it annually
Teacher: You are correct.
Student: So, I guess some people or most people would want it to be compounded a million times a
year.
Is there any limit on how many times you can compound in a year?
Teacher: Great question!
We shall answer that question when we do Continuous Compound Interest
As you see,
There is a difference between "daily - ordinary (ordinary days)" and "daily - exact (exact days)" in a
year.
Daily - Ordinary is 360 days in a year.
It is also known as Banker's Rule.
It is taken as a year consisting of 12 months in which each month consists of 30 days.
Exact - Ordinary is 365 days in a year.
Teacher: Rather than using Table 2, is there any way we could have calculated the compound
interest and the compound amount?
What if we wanted to compound annually for a period of 30 years?
Student: That would be a "yuge" table
Teacher: LOL...
So, yes, we can do it another way.
Student: Is is a simpler way?
Teacher: Of course. Algebraically: By Formula
Let us derive that formula. Let us do some Algebra.
Let us use that Table and use variables with it.
Year | $P(\$)$ | $r(\%)$ | $t(years)$ | $CI(\$)$ | $A(\$)$ |
---|---|---|---|---|---|
$1$ | $P$ | $r$ | $t$ | $Prt$ |
$P + Prt$ $P(1 + rt)$ |
2 | $P(1 + rt)$ | $r$ | $t$ | $Prt(1 + rt)$ |
$P(1 + rt) + Prt(1 + rt)$ $P(1 + rt)[1 + rt]$ $P(1 + rt)^2$ |
$3$ | $P(1 + rt)^2$ | $r$ | $t$ | $Prt(1 + rt)^2$ |
$P(1 + rt) + Prt(1 + rt)$ $P(1 + rt)^2[1 + rt]$ $P(1 + rt)^3$ |
Teacher: For Year 4, what will the amount be?
Student: $A = P(1 + rt)^4$
Teacher: For 5 years (Year 5), what will it be?
Student: $A = P(1 + rt)^5$
Teacher: And for $t$ years;
Student: $A = P(1 + rt)^t$
Teacher: That is correct.
Keep in mind that this is a case where the interest is compounded annually: one time per year
Assume the interest is compounded semiannually (two times per year)
What factors in that table will be affected?
Student: The rate will be affected
Teacher: How will it be affected?
Student: For example: a rate of 4% per year implies a rate of 2% every six months
Teacher: Say: the variable: m represents the number of compounding periods per year
Student: The rate will be: $\dfrac{r}{m}$
Teacher: That is correct.
What other factor is affected?
Student: The time
Time of 1 for one year implies time of 2 every six months
Teacher: In terms of the number of compounding periods of year, what will be the formula for the time?
Student: The time will be: $mt$
Teacher: So, what do you think the formula will be considering the number of compounding periods per
year?
Student: $A = P\left(1 + \dfrac{r}{m}\right)^{mt}$
Teacher: Correct.
This is known as the Compound Amount Formula
The Compound Amount Formula is:
$A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex]$
Say someone:
deposited an initial sum of money, $P$
in a financial institution that gives an interest rate of $r\%$,
compounded $m$ times per year;
then, after $t$ years,
the person will earn an amount, $A$
OR
Say someone:
borrowed an initial sum of money, $P$
from a financial institution that charges an interest rate of $r\%$,
compounded $m$ times per year;
then, after $t$ years,
the person will pay back an amount, $A$
Student: Is there any case when the simple interest will be equal to the compound interest?
Teacher: Yes...
What do you suggest?
Student: If you have the same principal, for example $\$$1000
the same interst rate, for example 10%
the time must be 1 year
then, the simple interest will be the same as the compound interest compounded annually
I've done the calculation with some numbers.
But, I appreciate if you can do it without numbers.
I'm trying to be very comfortablw with Algebra
Teacher: Sure... let's do it.
When Simple Interest Equals Compound Interest
The principal must be the same
The interest rate must be the same
The time must be one year.
The interest must be compounded annually (for the case of the compound interest)
$
\underline{Simple Interest} \\[3ex]
SI = Prt \\[3ex]
t = 1 \\[3ex]
SI = P * r * 1 \\[3ex]
SI = Pr \\[5ex]
\underline{Compound\;\;Interest} \\[3ex]
compounded\;\;annually \implies m = 1 \\[3ex]
t = 1 \\[3ex]
A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex]
A = P\left(1 + \dfrac{r}{1}\right)^{1 * 1} \\[5ex]
A = P(1 + r)^1 \\[3ex]
A = P(1 + r) \\[3ex]
CI = A - P \\[3ex]
CI = P(1 + r) - P \\[3ex]
CI = P + Pr - P \\[3ex]
CI = Pr \\[3ex]
$
Please write all the formulas that you see under Compound Interest and let us
solve some examples
The total return on an investment is the relative change in the value of the investment.
It shows the percentage change as the value of the investment changes.
Assume an investment grows an initial principal, P to an amount A.
The total return is the percentage change in the investment value.
The annual return on an investment is the annual percentage yield (APY) that would give the same overall growth over t years.
Teacher: Now, we can get to your question.
Student: Yes...
I would like to know if there is a limit to how much money one can earn.
I mean...who would want his/her money to be compounded annually when he/she makes more money
when the same amount of money is compounded semiannually?
Say you deposit $\$1$ in a financial institution that has an annual interest rate of $100\%$ for $1$
year.
Let us calculate the Amount for several options of times: annually, semiannually, quarterly,
monthly, weekly, ordinary-daily, exact-daily, and several times per year.
We want to see how much you would get for each option of time for that same principal of
$1 for that same time of 1 year.
Recall the Compound Amount Formula:
$
A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex]
P = \$1 \\[3ex]
r = 100\% = \dfrac{100}{100} = 1 \\[5ex]
t = 1\:\: year \\[3ex]
$
Let us calculate and observe. 😊
Compounded: | m | A |
---|---|---|
Annually | 1 | $$ A = 1 * \left(1 + \dfrac{1}{1}\right)^{1 * 1} \\[5ex] A = 1(1 + 1)^1 \\[3ex] A = 1(2)^1 \\[3ex] A = 1(2) \\[3ex] A = \$2.00 $$ |
Semiannually | 2 | $$ A = 1 * \left(1 + \dfrac{1}{2}\right)^{2 * 1} \\[5ex] A = 1(1 + 0.5)^2 \\[3ex] A = 1(1.5)^2 \\[3ex] A = 1(2.25) \\[3ex] A = \$2.25 $$ |
Quarterly | 4 | $$ A = 1 * \left(1 + \dfrac{1}{4}\right)^{4 * 1} \\[5ex] A = 1(1 + 0.25)^4 \\[3ex] A = 1(1.25)^4 \\[3ex] A = 1(2.44140625) \\[3ex] A = \$2.44 $$ |
Monthly | 12 | $$ A = 1 * \left(1 + \dfrac{1}{12}\right)^{12 * 1} \\[5ex] A = 1(1 + 0.0833333333)^{12} \\[3ex] A = 1(1.0833333333)^{12} \\[3ex] A = 1(2.61303529) \\[3ex] A = \$2.61 $$ |
Weekly | 52 | $$ A = 1 * \left(1 + \dfrac{1}{52}\right)^{52 * 1} \\[5ex] A = 1(1 + 0.0192307692)^{52} \\[3ex] A = 1(1.019230769)^{52} \\[3ex] A = 1(2.692596954) \\[3ex] A = \$2.69 $$ |
Daily - Ordinary | 360 | $$ A = 1 * \left(1 + \dfrac{1}{360}\right)^{360 * 1} \\[5ex] A = 1(1 + 0.0027777778)^{360} \\[3ex] A = 1(1.002777778)^{360} \\[3ex] A = 1(2.714516025) \\[3ex] A = \$2.71 $$ |
Daily - Exact | 365 | $$ A = 1 * \left(1 + \dfrac{1}{365}\right)^{365 * 1} \\[5ex] A = 1(1 + 0.002739726)^{365} \\[3ex] A = 1(1.002739726)^{365} \\[3ex] A = 1(2.714567482) \\[3ex] A = \$2.71 $$ |
500 times per year | 500 | $$ A = 1 * \left(1 + \dfrac{1}{500}\right)^{500 * 1} \\[5ex] A = 1(1 + 0.002)^{500} \\[3ex] A = 1(1.002)^{500} \\[3ex] A = 1(2.715568521) \\[3ex] A = \$2.72 $$ |
1,000 times per year | 1,000 | $$ A = 1 * \left(1 + \dfrac{1}{1000}\right)^{1000 * 1} \\[5ex] A = 1(1 + 0.001)^{1000} \\[3ex] A = 1(1.001)^{1000} \\[3ex] A = 1(2.716923932) \\[3ex] A = \$2.72 $$ |
10,000 times per year | 10,000 | $$ A = 1 * \left(1 + \dfrac{1}{10000}\right)^{10000 * 1} \\[5ex] A = 1(1 + 0.0001)^{10000} \\[3ex] A = 1(1.0001)^{10000} \\[3ex] A = 1(2.718145927) \\[3ex] A = \$2.72 $$ |
100,000 times per year | 100,000 | $$ A = 1 * \left(1 + \dfrac{1}{100000}\right)^{100000 * 1} \\[5ex] A = 1(1 + 0.00001)^{100000} \\[3ex] A = 1(1.00001)^{100000} \\[3ex] A = 1(2.718268237) \\[3ex] A = \$2.72 $$ |
1,000,000 times per year | 1,000,000 | $$ A = 1 * \left(1 + \dfrac{1}{1000000}\right)^{1000000 * 1} \\[5ex] A = 1(1 + 0.000001)^{1000000} \\[3ex] A = 1(1.000001)^{1000000} \\[3ex] A = 1(2.718280469) \\[3ex] A = \$2.72 $$ |
10,000,000 times per year | 10,000,000 | $$ A = 1 * \left(1 + \dfrac{1}{10000000}\right)^{10000000 * 1} \\[5ex] A = 1(1 + 0.0000001)^{10000000} \\[3ex] A = 1(1.0000001)^{10000000} \\[3ex] A = 1(2.718281693) \\[3ex] A = \$2.72 $$ |
Teacher: Did you notice anything?
Student: THANK GOD for cents (2 decimal places)
Teacher: Exactly!
Imagine if money was not rounded to 2 decimal places
Student: It would continue...and...continue...and...continue...
Teacher: That is right.
What did you notice?
Student: The final amount is $\$2.72$
Would it continue that way even if you compound it a trillion times per year?
Teacher: Yes, it would still give an amount of $\$2.72$
So, no matter how many times you compound $\$1$ in $1$ year,
Student: The amount at the end of $1$ year would be $\$2.72$
Teacher: and the compound interest would be ...
Student: $2.72 - 1.00 = \$1.72$
Teacher: Correct!
Did you notice any constant in the last four that we did: from $m = 10,000$ up to to $m =
10,000,000$?
Student: Yes, the first four digits is: 2.718
Observation:
The amount from when we compounded $\$1$ (a dollar), $10,000$ (ten thousand) times up to when we
compounded it $10,000,000$ (ten million) times in $1$ year, the first four digits were constant.
The amount was approximately $2.718$
This constant is known as the Euler number, by Leonhard Euler (a Swiss mathematician) OR
Napier's constant, John Napier (a Scottish/English mathematician).
Teacher: To be honest, any one of you could have discovered this constant.
You have learned what they discovered.
Now, think and work towards discovering your own.
Let me show you this constant in your calculator.
The constant is denoted by $e$
$e \approx 2.718$
The logarithm to base, $e$ is known as Natural Logarithm or Napierian Logarithm
The logarithm to base $e$ of $x$ is written as: $\log_e{x}$ or $\ln{x}$
Write down all the formulas that you see under Continuous Compound Interest and let us
solve some examples
The total return on an investment is the relative change in the value of the investment.
It shows the percentage change as the value of the investment changes.
Assume an investment grows an initial princial, P to an amount A.
The total return is the percentage change in the investment value.
The annual return on an investment is the annual percentage yield (APY) that would give the same
overall growth over Y years.
Teacher: Everyone would want a compounded interest rate. Right?
Student: To be honest, I would want a continuous compounded interest rate.
Teacher: That is right. It is still compounded though 😊
So, how do we solve this problem?
Would there be any need for simple interest rate?
Student: I guess not.
Teacher: So, how about finding a simple interest rate that would give the same compound
amount in one year as if we used the compound interest compounded annually?
Student: May you please elaborate?
Teacher: Let us review this example:
Say Mr. C deposits $\$1000$ in a financial institution that gives an interest rate of $7\%$.
How much will be in his account after $1$ year?
Student:
$
P = \$1000 \\[3ex]
r = 7\% = 0.07 \\[3ex]
t = 1\:year \\[3ex]
SI = 1000(0.07)(1) = \$70 \\[3ex]
A = 1000 + 70 = \$1070 \\[3ex]
Mr.\:C\:\:will\:\:have\:\:\$1070\:\:after\:\:1\:year \\[3ex]
$
Teacher: That is correct!
Say Mr. C deposits $\$1000$ in a financial institution that gives an interest rate of $7\%$.
How much will be in his account after $1$ year if the interest rate is compounded annually?
Student:
$
P = \$1000 \\[3ex]
r = 7\% = 0.07 \\[3ex]
t = 1\:year \\[3ex]
m = 1...compounded\:\:annually \\[3ex]
A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex]
mt = 1(1) = 1 \\[3ex]
\dfrac{r}{1} = \dfrac{0.07}{1} = 0.07 \\[5ex]
A = 1000(1 + 0.07)^1 \\[5ex]
A = 1000(1.07)^1 \\[3ex]
A = 1000(1.07) \\[3ex]
A = \$1070 \\[3ex]
Mr.\:C\:\:will\:\:have\:\:\$1070\:\:after\:\:1\:year \\[3ex]
It\:\:is\:\:the\:\:same\:\:amount\:\:after\:\:1\:\:year \\[3ex]
$
Teacher: That is correct!
Teacher: So, the question is this.
Can we find a simple interest rate that will give $\$1070$ in one year?
In other words, can we find a simple interest rate that will give that compound amount of
$\$1070$ in one year?
Student: Well, we did already.
That simple interest rate of $7\%$ gave the same amount of $\$1070$ after one year.
Teacher: That is correct... in this case.
The simple interest rate that will give the same compound amount after one year is known as the
Annual Percentage Yield for Compound Interest
Annual Percentage Yield is also known as APY
Student: Do we have the Annual Percentage Yield for Continuous Compound Interest as well?
Teacher: Yes, we do!
How would you define it?
Student: I guess it will be the simple interest rate that will give the same continuous
compound amount after one year.
Teacher: Correct!
Let us find out.
Say Mr. C deposits $\$1000$ in a financial institution that gives an interest rate of $7\%$.
How much will be in his account after $1$ year if the interest rate is continously compounded?
Student:
$
P = \$1000 \\[3ex]
r = 7\% = 0.07 \\[3ex]
t = 1\:year \\[3ex]
A = Pe^{rt} \\[3ex]
rt = 0.07(1) = 0.07 \\[3ex]
A = 1000 * e^{0.07} \\[5ex]
A = 1000(1.07250818) \\[3ex]
A = 1072.50818 \\[3ex]
A \approx \$1072.51 \\[3ex]
Mr.\:C\:\:will\:\:have\:\:\$1072.51\:\:after\:\:1\:year \\[3ex]
$
Teacher: That is correct!
The simple interest rate of $7\%$ will not work this time.
This means that it is not the APY for the Continuous Compound Interest
Student: How do we find that APY?
Just to be sure, how do we find the simple interest rate that will give $\$1072.51$ after one year?
Teacher: Try the simple interest rate of $7.2508\%$
Say Mr. C deposits $\$1000$ in a financial institution that gives an interest rate of $7.2508\%$.
How much will be in his account after $1$ year?
Student:
$
P = \$1000 \\[3ex]
r = 7.2508\% = 0.072508 \\[3ex]
t = 1\:year \\[3ex]
SI = 1000(0.072508)(1) = \$72.08 \\[3ex]
A = 1000 + 72.08 = \$1072.08 \\[3ex]
Mr.\:C\:\:will\:\:have\:\:\$1072.08\:\:after\:\:1\:year \\[3ex]
$
It is almost the same...just a few cents difference
It seems you gave me a rounded value, Sir.
But, you asked not not to round.
Teacher: You are right. It is a rounded value.
You are right. I asked you not to round, if I did not ask you to round. 😊
Here, try this value...the simple interest rate of $7.25081813\%$
Student:
$
P = \$1000 \\[3ex]
r = 7.25081813\% = 0.0725081813 \\[3ex]
t = 1\:year \\[3ex]
SI = 1000(0.0725081813)(1) = \$72.5081813 \\[3ex]
A = 1000 + 72.5081813 = \$1072.50818 \\[3ex]
Mr.\:C\:\:will\:\:have\:\:\$1072.51\:\:after\:\:1\:year \\[3ex]
$
How did you get that rate? This is interesting!
Teacher: Yes it is! Welcome to APY!!!
The Annual Percentage Yield is also known as APY or Effective Interest Rate or
Effective Rate or True Interest Rate
The Annual Percentage Yield for Compound Interest is defined as the simple interest rate
that will yield the same amount as the compound amount after one year.
APR and APY are the same with annual compounding.
However, with more than one compounding per year, the APY is always greater than the APR. It reflects the cumulative effects of several compoundings during the year.
Thus, the more compoundings during the year, the greater the APY.
It does not depend on the starting principal. It gives the percentage increase in the balance.
Let us derive the formula.
This means that:
Amount at Simple Interest after one year is equal to Amount at Compound Interest after one year.
$
\underline{Simple\:\:Interest} \\[3ex]
A = P(1 + rt) \\[3ex]
After\:\:1\:\:year \rightarrow t = 1\:year \\[3ex]
A = P(1 + r * 1) \\[3ex]
A = P(1 + r) \\[3ex]
\underline{Compound\:\:Interest} \\[3ex]
A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex]
After\:\:1\:\:year \rightarrow t = 1\:year \\[3ex]
A = P\left(1 + \dfrac{r}{m}\right)^{m * 1} \\[5ex]
A = P\left(1 + \dfrac{r}{m}\right)^{m} \\[5ex]
Equate\:\:both\:\:amounts \\[3ex]
A = A \\[3ex]
P(1 + r) = P\left(1 + \dfrac{r}{m}\right)^{m} \\[5ex]
But\:\:the\:\:r\:\:in\:\:Simple\:\:Interest = APY \\[3ex]
P(1 + APY) = P\left(1 + \dfrac{r}{m}\right)^{m} \\[5ex]
Divide\:\:both\:\:sides\:\:by\:\:P \\[3ex]
1 + APY = \left(1 + \dfrac{r}{m}\right)^{m} \\[5ex]
Subtract\:\:1\:\:from\:\:both\:\:sides \\[3ex]
\therefore APY = \left(1 + \dfrac{r}{m}\right)^{m} - 1 \\[5ex]
This\:\:is\:\:the\:\:APY\:\:for\:\:Compound\:\:Interest
$
The Annual Percentage Yield for Continuous Compound Interest is defined as the simple interest
rate that will
yield the same amount as the continuous compound amount after one year.
Let us derive the formula.
This means that:
Amount at Simple Interest after one year is equal to Amount at Continuous Compound Interest after one
year.
$
\underline{Simple\:\:Interest} \\[3ex]
A = P(1 + rt) \\[3ex]
After\:\:1\:\:year \rightarrow t = 1\:year \\[3ex]
A = P(1 + r * 1) \\[3ex]
A = P(1 + r) \\[3ex]
\underline{Continuous Compound\:\:Interest} \\[3ex]
A = Pe^{rt} \\[3ex]
After\:\:1\:\:year \rightarrow t = 1\:year \\[3ex]
A = Pe^{r * 1} \\[3ex]
A = Pe^r \\[3ex]
Equate\:\:both\:\:amounts \\[3ex]
A = A \\[3ex]
P(1 + r) = Pe^r \\[3ex]
But\:\:the\:\:r\:\:in\:\:Simple\:\:Interest = APY \\[3ex]
P(1 + APY) = e^r \\[3ex]
Divide\:\:both\:\:sides\:\:by\:\:P \\[3ex]
1 + APY = e^r \\[3ex]
Subtract\:\:1\:\:from\:\:both\:\:sides \\[3ex]
\therefore APY = e^r - 1 \\[5ex]
This\:\:is\:\:the\:\:APY\:\:for\:\:Continuous\:\:Compound\:\:Interest
$
Write down all the formulas that you see under Annual Percentage Yield and let us solve some examples
Write down all the formulas that you see under Investment for Retirement and
let us
solve some examples: Part 1
and
solve some examples: Part 2
An annuity is defined as a sequence of equal periodic payments.
If the payments are made at the end of each period, it is known as an ordinary annuity
If the payments are made at the beginning of each period, it is known as an annuity due
An Ordinary Annuity is a sequence of equal periodic payments in which each payment is made at the end of each period.
The periods could be yearly, monthly (most common), biweekly (also common), and weekly among others.
The Future Value of an Ordinary Annuity is the sum of all the periodic payments and all the earned interests.
Applications of the Future Value of an Ordinary Annuity are seen in: college savings accounts, IRAs (Individual Retirement Accounts), 401(k), 403(b), and other investments that
require periodic payments.
For example: If someone is contributing a certain percentage of his salary or a certain amount of his salary every month to a 401(k) investment account, how much would the
person have when he retires at a certain age, or when he stops contributing after some time?
Write down all the formulas that you see under Future Value of Ordinary Annuity and Sinking Fund and let us solve some examples
An installment loan is a loan or money that is borrowed and agreed to be
paid back in installments.
Typically, equal periodic payments are made over the term of the loan (duration of the loan).
The periodic payments could be weekly payments, bi-weekly payments, monthly payments, or some other
periodic payments as the case may be.
Installment loans allow consumers to finance high price purchases of some things that they would
otherwise not be able to afford immediately. These include cars and homes among others.
Amortization is defined as the repayment of a loan by installments.
In the United States, most installment loans are based on monthly payments, and the interest on those
loans is compounded on a monthly basis.
Write down all the formulas that you see under Amortization and the Present Value of Ordinary Annuity and let us solve some examples
Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology.
Retrieved from https://www.samuelchukwuemeka.com
Bennett, J. O., & Briggs, W. L. (2019).
Using and Understanding Mathematics: A Quantitative Reasoning Approach. Pearson.
Blitzer, R. (2015). Thinking Mathematically (6th ed.).
Boston: Pearson
Cleaves, C. S., Hobbs, M. J., & Noble, J. J. (2014). Business Math (10th ed.).
Upper Saddle River, NJ: Prentice Hall.
Tan, S. (2015). Finite Mathematics for the Managerial, Life, and Social Sciences (11th ed.).
Boston: Cengage Learning.
Microsoft Office Clip Art. (n.d.). Retrieved from
https://support.office.com/en-us/article/add-clip-art-to-your-file-0a01ae25-973c-4c2c-8eaf-8c8e1f9ab530?legRedir=true&CorrelationId=9c8bb412-68e2-4db0-9704-c9af116e2521&ui=en-US&rs=en-US&ad=US
Authority (NZQA), (n.d.). Mathematics and Statistics subject resources. www.nzqa.govt.nz. Retrieved December 14,
2020, from https://www.nzqa.govt.nz/ncea/subjects/mathematics/levels/
CrackACT. (n.d.). Retrieved from http://www.crackact.com/act-downloads/
CMAT Question Papers CMAT Previous Year Question Bank - Careerindia. (n.d.). https://www.careerindia.com. Retrieved
May 30, 2020, from https://www.careerindia.com/entrance-exam/cmat-question-papers-e23.html
CSEC Math Tutor. (n.d). Retrieved from https://www.csecmathtutor.com/past-papers.html
TI Products | Calculators and Technology | Texas Instruments. (n.d.). Education.ti.com. Retrieved March 18, 2023, from https://education.ti.com/en/products
Desmos. (n.d.). Desmos Graphing Calculator. https://www.desmos.com/calculator
Durante, A. (2022, October 18). 2023 Tax Brackets. Tax Foundation. https://taxfoundation.org/2023-tax-brackets/
DLAP Website. (n.d.). Curriculum.gov.mt.
https://curriculum.gov.mt/en/Examination-Papers/Pages/list_secondary_papers.aspx
Free Jamb Past Questions And Answer For All Subject 2020. (2020, January 31). Vastlearners.
https://www.vastlearners.com/free-jamb-past-questions/
Geogebra. (2019). Graphing Calculator - GeoGebra. Geogebra.org. https://www.geogebra.org/graphing?lang=en
GCSE Exam Past Papers: Revision World. Retrieved April 6, 2020, from
https://revisionworld.com/gcse-revision/gcse-exam-past-papers
HSC exam papers | NSW Education Standards. (2019). Nsw.edu.au.
https://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/resources/hsc-exam-papers
JAMB Past Questions, WAEC, NECO, Post UTME Past Questions. (n.d.). Nigerian Scholars. Retrieved February 12, 2022,
from https://nigerianscholars.com/past-questions/
KCSE Past Papers by Subject with Answers-Marking Schemes. (n.d.). ATIKA SCHOOL.
Retrieved June 16, 2022, from https://www.atikaschool.org/kcsepastpapersbysubject
Myschool e-Learning Centre - It's Time to Study! - Myschool. (n.d.). https://myschool.ng/classroom
Netrimedia. (2022, May 2). ICSE 10th Board Exam Previous Papers- Last 10 Years. Education Observer.
https://www.educationobserver.com/icse-class10-previous-papers/
NSC Examinations. (n.d.). www.education.gov.za.
https://www.education.gov.za/Curriculum/NationalSeniorCertificate(NSC)Examinations.aspx
School Curriculum and Standards Authority (SCSA): K-12. Past ATAR Course Examinations. Retrieved December 10, 2021,
from https://senior-secondary.scsa.wa.edu.au/further-resources/past-atar-course-exams
West African Examinations Council (WAEC). Retrieved May 30, 2020, from
https://waeconline.org.ng/e-learning/Mathematics/mathsmain.html
Papua New Guinea: Department of Education. (n.d.). www.education.gov.pg. Retrieved November 24, 2020, from
http://www.education.gov.pg/TISER/exams.html
51 Real SAT PDFs and List of 89 Real ACTs (Free) : McElroy Tutoring. (n.d.).
Mcelroytutoring.com. Retrieved December 12, 2022,
from https://mcelroytutoring.com/lower.php?url=44-official-sat-pdfs-and-82-official-act-pdf-practice-tests-free