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Math Lecture Notes
Interest

In-Class Activity link to here link to index

Six Years of Rent Increases

Consider an apartment whose monthly rent starts at $500, and every year it increases by 5%.

1. Complete the table below.

rent increase table

1. Year 2: Starting amount $525.00, Increase $26.25, Ending amount $551.25

Year 3: Starting amount $551.25, Increase $27.56, Ending amount $578.81

Year 4: Starting amount $578.81, Increase $28.94, Ending amount $607.75

Year 5: Starting amount $607.75, Increase $30.39, Ending amount $638.14

Year 6: Starting amount $638.14, Increase $31.91, Ending amount $670.05

2. What is the rent at the end of the sixth year?

2. The ending amount for the sixth year was $670.05.

3. Why does the amount of increase keep changing?

3. Five percent of a bigger number is a bigger dollar amount. (Just like 5% of a millionaire's income is a lot more than 5% of my income!)

Each year the starting number grows, so 5% times that number grows too.

4. What was the overall percent change?

4. The dollar amount of the overall change was $670.05 − $500 = $170.05.

Then we use the percent change formula which says percent change = change ÷ original, followed by RIP LOP to put the decimal format answer into percent format.

percent change = $170.05 ÷ $500 ≈ 0.34 = 34%.

Notice that if the increase had remained at $25 per year then the overall change would have been 6 years × 5% = 30%. The growth of the annual increases added up to an extra 4% overall change.


We can simplify our table using the one plus trick.

5. Complete the table again, this time using the one plus trick in each row.

rent increase table

That was a lot nicer!


Don't stop now! We can make the table even simpler!

The year two starting amount of $525 was calculated using $500 × 1.05. What happens if instead of the dollar amount we put that calculation in the place of the year two starting amount? And keep doing that for every year's starting amount?

6. Complete the table below, using the initial $500, incremental exponents, and the one plus trick.

rent increase table

That was the nicest of all!


Now we have a table that shows us a pattern.

7. What is the formula for the ending amount y if we know the year number n?

rent increase table

7. The formula is y = $500 × 1.05n

Facts link to here link to index

Simple Interest

Interest is a fee the lender charges the borrower.

For your savings account or a government bond, you are loaning money to the bank or government, and you collect the interest. For your debts, a bank or credit card is loaning money to you, and they collect the interest.

The initial amount of a loan is called the principal.

The fee is usually charged monthly annually. Each time the fee is charged is called a payout.

By tradition, the fee is always written as an annual interest rate that is multiplied by the size of the loan. This allows various loans to be more easily compared.

Annual Payout Example

8. A loan has a 3% annual interest rate. Payouts happen annually. If the principal is $100, what is the dollar amount of the first payout?

8. We ask "What is 3% of $100?" 0.03 × $100 = $3

But it does mean that for a loan with more than one payout per year, that annual interest rate must be divied up into equal size portions.

A loan with monthly payouts would have a monthly rate equal to one-twelfth of the listed annual interest rate. A loan with quarterly payouts would have a monthly rate equal to one-quarter of the listed annual interest rate.

Monthly Payout Example

9. A loan has a 3% annual interest rate. Payouts happen monthly. If the principal is $100, what is the dollar amount of the first payout?

9. First we divide the annual 3% rate into monthly portions. 0.03 ÷ 12 = 0.0025 per month

We ask "What is 0.0025 of $100?" 0.0025 × $100 = $0.25

Simple Interest Growth

Simple interest is when the fee remains the same with every payout. Its amount is fixed when the loan begins, and never changes.

The most common real-life examples of simple interest are payday loans, bank certificates of deposits, and bond investments.

The picture to the left shows loan with simple interest for which the borrower never makes any payments. The loan starts on the left side with only the principal (the blue bar). As time goes on we move to the right. Each year a fee is added to the loan (each payout is a new green bar). All the green bars have the same height. The amount of the loan grows at a steady pace.

If the first example above was simple interest then every year the loan amount would increase by a $3 payout.

If the second example above was simple interest then every month the loan amount would increase by a $0.25 payout.

Monthly Simple Interest Example

10. A loan has a 3% annual interest rate. Payouts happen monthly. The loan uses simple interest. If the principal is $100, what is the total amount of interest for the first two years?

10. First we divide the annual 3% rate into monthly portions. 0.03 ÷ 12 = 0.0025 per month

Then we find the payout amount. We ask "What is 0.0025 of $100?" 0.0025 × $100 = $0.25

Then we find the total number of payouts. Monthly for two years means 12 × 2 = 24 payouts

Then we copy that $0.25 payout amount for 24 payouts. $0.25 × 24 = $6


Wait a moment! We didn't need to divy up the year into months!

It was a waste of time to divide by 12 with the rate, and also multiply by 12 with how many payouts.

That example highlights a very convenient truth!

For simple interest we only need to find the total annual interest (principal × annual interest rate) and then multiply by the number of years.

Thinking in Years with a Simple Interest Example

11. A loan has a 3% annual interest rate. Payouts happen monthly. The loan uses simple interest. If the principal is $100, what is the total amount of interest for the first 18 months?

11. First we change the loan's length of time to years. 18 ÷ 12 = 1.5 years

Then we just multiply the principal, annual interest rate, and number of years.
Simple Interest = $100 principal × 0.03 annual rate × 1.5 years = $4.50

In other words, for simple interest we can be lazy and always pretend the payouts happen annually.

If the payouts are actually monthly then we could divide up the annual interest into its twelve equal parts, but we would end up smooshing them back together when we found the total value of all the payouts.

If the payouts are actually quarterly then we could divide up the annual interest into its four equal parts, but we would end up smooshing them back together when we found the total value of all the payouts.

So the product of the principal and annual interest rate equals the total annual interest—whether or not in real life it is one payout or several. Then multiplying by the number of years gives the total interest for the overall period.

We can write this as a formula. But remember that it is not the kind of formula that is discovered by looking at patterns. It is merely the fact of how simple interest is defined.

Simple Interest Formula

Simple Interest = Principal × Annual Interest Rate × Years

Unfortunately, this formula is traditionally written in a more confusing manner.

Traditional Simple Interest Formula

I = P × r × t


Pleae be careful with this version of the formula!

Remember that the interest it gives is simple interest.

Remember that the rate is an annual interest rate that needs RIP LOP.

Remember that the time is always measured in years and might need unit conversion.

Be alert! (Snakes don't eat lerts.) So far all the problems asked for the amount of interest as the answer. Other problems could ask for the new total.

New Total Examples

12. An loan has a 2% annual simple interest rate. The principal is $750. How much is it worth five years later?

12. Simple Interest = $750 principal × 0.02 annual rate × 5 years = $75

But wait! This problem asked not for the interest, but the new total. We have to add the principal back on. $750 principal + $75 interest = $825 total

13. An loan has a 5% annual simple interest rate. The principal is $400. How much is it worth 200 days later?

13. Simple Interest = $400 principal × 0.05 annual rate × (200 ÷ 365) years ≈ $10.96

But wait! This problem asked not for the interest, but the new total. We have to add the principal back on. $400 principal + $10.96 interest = $410.96 total

Appreciation and Depriciation

Similar to interest is a concept called appreciation. This is when an investment increases in value not because a fee is charged but just because the investment is worth more after time goes by. People often see appreciation in the value of their home, in the value of old furniture or jewelry, and in the value of the stocks in which they invest.

The opposite of appreciation is depriciation, when something's value goes down.

In a slightly confusing manner, people use the simple interest formula for both interest and appreciation/depriciation. Just pretend that the change from appreciation/depriciation is a fee even though it really is not.

Example 14

Scrooge McDuck

14. Scrooge McDuck puts one million dollars into an investment that appreciates 3% its first year. The second year the investment depreciates by 3%. Scrooge is sad, and thinks he is financially back where he started.

When his account statement arrives in the mail he is surprised to see that he has less money than he started with! What happened?

14. The first year his principal was $1,000,000 and his gain from appreciation was:
$1,000,000 principal × 0.03 rate × 1 time = $30,000.

The second year his principal was $1,030,000 and his loss from depreciation was:
$1,030,000 principal × 0.03 rate × 1 time = $30,900.

Overall, he lost $900.

Remember, that same 3% of a larger amount is itself larger!

videos

Chapter 6 Test, Problem 14: What is the simple interest on a principal of $120 at the annual interest rate of 7.1% for one year?

Chapter 6 Test, Problem 15: A city orchestra invests $5,200 at 6% annual simple interest. How much is in the account after half a year?

Formulas link to here link to index

Compound Interest

The English word "compound" means two different things. It can refer to an item made up of multiple and different parts (for example, a medicine that is a compound of ingredients). It can also refer to an item made greater by a certain type of thing (for example, strong winds compounding the difficulty of putting out a forest fire, or young siblings whose tendency to get into trouble is compounded when their cousins visit).

Compound Interest Growth

Compound interest uses the word "compound" in both ways. Lots of parts make the loan a lot bigger.

A loan that uses compound interest still has a principal amount, an annual interest rate, payouts, and an overall length of time.

However, the payouts no longer stay the same size. At the moment of each payout, the annual interest rate is applied to the current loan size then.

The picture to the left shows loan with simple interest for which the borrower never makes any payments. The loan again starts on the left side with only the principal (the blue bar). As time goes on we move to the right. The first year's fee is again a percentage of the principal (the first and lowest green bar). Now the second's year payout is based on both the principal (adding a second green bar) and the first year's interest (adding the first and lowest purple bar). In the third year there are four bars that contribute to the third payout. And so forth. The amount of the loan grows at an increasing pace, faster and faster.

For simple interest we could be lazy and always pretend the payouts happen annually. Perhaps in real life there were several that were evenly distributed throughout the year. The math allowed us to easily smoosh a year's worth together because they were all the same size.

For compound interest we cannot be lazy. Every payout is a different amount! We must consider each of them individually.

Here is a sample monthly compound interest spreadsheet. Why is the annual interest rate of 22% divided by twelve?

compound interest table

Remember that rounding in the middle of a math problem usually creates an incorrect answer. The table above uses one-twelfth of the annual interest rate as a decimal and does not round this number! The calculations would be wrong if it did. (Try it!)

That table should remind you of our in-class activity about six years of rent increases. You already invented the compound interest formula! (For the rent increase in-class activity, the Principal was $500 and the interest rate was 5%.)

The Compound Interest Formula

Final Amount = Principal × (1 + interest rate per payout)number of payouts

Notice that this formula gives us the final amount. If we were asked to only find the interest, we would need to subtract away the principal.

Compound Interest Example

15. A loan has a 3% annual interest rate. Payouts happen monthly. The loan uses compound interest. If the principal is $100, what is the total amount of interest for the first 18 months?

15. First we find the interest rate per payout. 0.03 ÷ 12 =0.0025 per payout

Then we use the compound interest formula:
Final Amount = Principal × (1 + interest rate per payout)number of payouts
     = $100 × 1.002518
     ≈ $104.60

Example 11 had the same annual interest rate, principal, and number of months to find $4.50 of simple interest. By compounding the interest we earned an extra $0.10!

Most real-life compound interest with bank accounts and credit cards happens monthly. But there are other options.

Payout Frequencies Example

16. Complete this chart to explore various frequences if payouts.

NameMeaningPayouts per Year
annuallyonce per year
semiannuallytwice per year
quarterlyonce per quarter
monthlyonce per month
semimonthlytwice per month
biweeklyevery other week
weeklyonce per week

16. annual = 1/year, semiannual = 2/year, quarterly = 4/year, monthly = 12/year, semimonthly = 24/year, biweekly = 26/year, weekly = 52/year.

In real life, most loans have compound interest with monthly payouts, but the formula does not apply because of deposits and withdrawls. In real life no one makes a single initial deposit/payment and then passively watches the interest grow!

Annual Effective Interest

A special case of the compound interest formula happens when the principal is $1, the time for each payout is less than a year, and the number of payouts exactly fits a single year. This situation would measure how much compounding happens each year.

In other words, there really is interest compounding during that year, but we are going to "summarize" the increase as if it were simple interest. Accountants call this the annual effective interest rate.

We can write the normal compound interest formula for this situation.

Final Amount = $1 × (1 + interest rate per payout)number of payouts

Then we make two changes. First, multiplying by $1 of principal does nothing, so we can leave that out. Second, we are really looking for the interest, not the final amount, so we subtract the $1 principal at the end.

The Annual Effective Interest Rate Formula

The number of payouts must fit a single year.

Annual Effective Interest Rate = (1 + interest rate per payout)number of payouts − 1

Annual Effective Interest Rate Example

17. A credit card has a 24% annual interest rate. Payouts happen monthly. The loan uses compound interest. What is the annual effective interest rate?

17. First we find the interest rate per payout. 0.24 ÷ 12 = 0.02 per payout

Then we use the annual effective interest rate formula:
AEIR = (1 + interest rate per payout)number of payouts − 1
     = 1.0212 − 1
     ≈ 0.268
     = 26.8%

So this credit card behaves like a simple interest rate of 26.8% regarding its effect on your wallet!

Homework link to here link to index

More Practice

31. Calculate the simple interest on $500 invested at 14% annually for three years and three months.

32. A business buys a copy machine for $300 by borrowing that $300 with a nine month loan of 15% simple interest. What is the total cost (the copy machine plus the loan's interest)?

33. Someone borrows $5,000 for 120 days at 9.25% annual simple interest. How much interest must she pay?

34. How much was borrowed at 17% annual simple interest for six months if the interest was $85?

35. Louie borrows $1,500 for three months. At the end of that time he pays back the $1,500 and also $30 interest. What was the simple interest rate? (Hint: you may rearrange the order of three numbers being multiplied without changing the result.)

36. Dale buys a painting for $1,000 and its value increases to $1,030. What is the percent appreciation? Chip buys a share of a mutual fund for $100 and its value increases to $130. What is the percent appreciation? Why did the two items have different percent appreciation if they both increased by $30?

31. Three years and three months is 3.25 years. Simple Interest = Principal × annual rate × time in years = P × r × t = $500 × 0.14 × 3.25 = $226.80

32. Nine months is 0.75 years. Simple Interest = P × r × t = $300 × 0.15 × 0.75 = $33.75. Add back the principal to get a total cost of $300 + $33.75 = 333.75.

We could have used the one plus trick.

Total = P × (1 + r) × t = $300 × 1.15 × 0.75 = $333.75.

33. Simple Interest = P × r × t = $5,000 × 0.0925 × (120 ÷ 365) = $152.05

34. Now we are solving for a missing P.

   I   = P × r × t

  $85  = P × 0.17 × 0.5

  $85  = P × 0.085

$1,000 = P

35. Now we are solving for a missing r. Three months is 0.25 years.

   I = P × r × t

 $30 = $1,500 × r × 0.25

 $30 = r × $1,500 × 0.25

 $30 = r × $375

0.08 = r

  8% = r

36. This is not a simple interest problem! There is no time measurement. We are only comparing two examples of percent change.

Dale's percent change = change ÷ original = $30 ÷ $1,000 = 0.03 = 3%

Chip's percent change = change ÷ original = $30 ÷ $100 = 0.3 = 30%

The same total change is ten times more significant when the original amount is only one-tenth as big.

payday advance loan neon sign

B8. Oregon Senate Bill 1105 limits payday loan interest rates to 36% or less. Before that bill was passed, payday loans in Oregon often had interest rates of 120%. Nationally, payday loans can have interest rates as high as 7,000%. The average payday loan is a two-week advance on $350. How much interest would be owed with an annual simple interest rate of 36%? 120%? 7,000%?

Einstein

37. If Ms. Caruso invests $2,000 today at 8% annual simple interest how much will she have after 30 years?

38. If Ms. Caruso invests $2,000 today at 8% annual interest, compounded annually, how much will she have after 30 years?

39. If Dr. Julius invests $2,000 today at 8% annual interest, compounded quarterly, how much will he have after 30 years?

40. If Dr. Julius invests $2,000 today at 8% annual interest, compounded monthly, how much will he have after 30 years?

41. Blue Bank offers five-year certificates of deposit that pay 6.5% compounded quarterly. Red Bank offers five-year certificates of deposit that pay 7% compounded semiannually. Which is a better deal? If Rosa Klebb has $30,000 to spend on a certificate of deposit, how much would each option be worth for her at the end of five years?

42. Mr. Largo invests $1,000 for seven years at 6% annual interest compounded monthly. Additionally, at the beginning of the fifth year she deposits an additional $2,000 into the account. What is the investment worth at the end of all seven years?

43. In seven years you want to have $20,000 to buy a vacation as a retirement present for yourself. What is the present value of this gift if the investment you make today will earn 11% annual interest compounded semiannually?

44. If you gave a 20-year-old sibling a $100 wedding present and they invested it at 11% annual compound interest, how much would it be worth when they are 60 years old?

37. Simple Interest = Principal × annual rate × time in years = P × r × t = $2,000 × 0.08 × 30 = $4,800. Then add back the prinicipal: $2,000 + $4,800 = $6,800

We could have used the one plus trick.

Total = P × (1 + r) × t = $2,000 × 1.08 × 30 = $6,800.

38. For compound interest, Final Amount = Principal × (1 + interest rate per payout)number of payouts = $2,000 × 1.0830 = $20,125.31

If Ms. Caruso switches from simple interest to compound interest it would make a huge difference!

39. Now the interest rate per payout is 0.08 ÷ 4 = 0.02, and the number of payout is 30 × 4 = 120. Final Amount = Principal × (1 + interest rate per payout)number of payouts = $2,000 × 1.02120 = $21,530.33

40. Now the interest rate per payout is 0.08 ÷ 12 = 0.006666... To avoid rounding in the middle we should keep that division happening in the formula. The number of payout is 30 × 12 = 360. Final Amount = Principal × (1 + interest rate per payout)number of payouts = $2,000 × (1 + 0.08 ÷ 12)360 = $21,871.46

More frequent compounding will earn more money for Dr. Julias, but not by a huge amount.

41. For Blue Bank we do Final Amount = Principal × (1 + interest rate per payout)number of payouts = $30,000 × 1.0162520 = $41,412.59. For Red Bank we do Final Amount = Principal × (1 + interest rate per payout)number of payouts = $30,000 × 1.03510 = $42,317.96. Red bank is a better deal. The income increase more from a higher percentage rate than from more frequent compounding.

The result of the previous problem is not always true. Over a long time span, more frequent compounding can be more significant than a half-percent rate increase. You could use a spreadsheet to check how long a time span is needed for Blue Bank to be a better deal.

42. At the end of four years we have Final Amount = Principal × (1 + interest rate per payout)number of payouts = $1,000 × 1.00548 = $1,270.49. Then it is the beginning of the fifth year, and the $2,000 deposin brings the current account balance up to $3,270.49. The account grows for years five, six, and seven (three more years). So the Very Final Amount = Principal × (1 + interest rate per payout)number of payouts = $3,270.49 × 1.00536 = $3,913.73.

43. Now we are solving for the principal.

Final Amount = Principal × (1 + interest rate per payout)number of payouts

     $20,000 = P × 1.117

     $20,000 = P × 2.0761

      $9,633 ≈ P

Notice how 1.117 ≈ 2. Any principal multiplied by 1.117 would double (with slight rounding). Since the stock market averages an 11% rate of increase, we get a rule of thumb that its value doubles every seven years.

44. Final Amount = Principal × (1 + interest rate per payout)number of payouts = $100 × 1.1140 = $6,500.09

45. If you invest $1 for a year, with compound interest of 6% happening monthly, what is that annual effective interest rate?

46. If you invest $1 for a year, with compound interest of 6% happening weekly, what is that annual effective interest rate?

45. Because of the monthly compounding we have an interest rate per payout of 0.06 ÷ 12 = 0.005. We have 12 payouts during the year. So the Annual Effective Interest Rate = (1 + interest rate per payout)number of payouts − 1 = 1.00512 − 1 = 0.06167 ≈ 6.17%.

The compounding raises the effective interest to slightly more than 6%. Not hugely more, but it would become significant over time!

46. Because of the weekly compounding we have an interest rate per payout of 0.06 ÷ 52 = 0.00115385. We have 52 payouts during the year. So the Annual Effective Interest Rate = (1 + interest rate per payout)number of payouts − 1 = 1.0011538552 − 1 = 0.06179 ≈ 6.18%.

More frequent compounding does raise the effective interest even more, but only barely. It would take a long time for that change to become significant. That is why banks and credit cards stick with monthly interest instead of competing with each other by also offering options with weekly compounding.

Random Homework Problems

Homework

Each time you load the page these problems change!

LCC Math 25 Packet Problems

Simple interest problems are in the packet on pages:
  • PF-45
  • PF-50
  • PF-52 to PF-56

Compound interest problems are in the packet on pages:
  • PF-46 to PF-49
  • PF-51
  • PF-57 to PF-59