B. Exponential Functions · Financial Mathematics: Compound Interest and Annuities

Simple Interest and Compound Interest: Tables and Graphs

B3.1About 70 minutes

Learning goals

I can calculate simple interest with I = Prt and the amount with A = P + I.
I can calculate compound interest year by year as repeated simple interest.
I can build a table of values and a graph that compare simple and compound interest over time.
I can explain, using first differences and ratios, why simple interest grows linearly and compound interest grows exponentially.

Words to know

principal (P)
The money you invest or borrow at the start. Also called the present value (PV).

interest
Money paid for the use of money. Savers earn it; borrowers pay it.

interest rate (r)
The percent of the principal charged or paid per year, written as a decimal in formulas (5% = 0.05).

simple interest
Interest calculated only on the original principal: I = Prt.

compound interest
Interest calculated on the principal plus all interest already earned. Interest earns interest.

amount (A)
Principal plus interest at the end. Also called the future value (FV).

first differences
The changes between consecutive values in a table. Constant first differences mean a linear pattern.

Two cousins, one gift

Amira and Jonah each get $1 000 from their grandparents for their sixteenth birthdays. Both find an offer that pays 5% per year for 25 years.

Amira's offer pays simple interest: $50 every year, always calculated on the original $1 000.
Jonah's offer pays compound interest: each year's 5% is calculated on everything in the account, including the interest from earlier years.

After one year they both have $1 050. After 25 years, Amira has $2 250 and Jonah has $3 386.35. Same gift, same rate, same time, and a difference of more than $1 100. In this lesson you will see exactly where that difference comes from, using tables and graphs.

Remember from Grades 8 and 9

In Grade 8 you calculated simple interest with I = Prt, compound interest compounded once a year, and used the Rule of 72. In Grade 9 you used A = P(1 + i)ⁿ with different compounding periods (semi-annual, quarterly, monthly) and compared loans. In this Grade 11 unit you will go further: you will see compound interest as an exponential function, work backwards to find how much to invest now (present value), solve for the rate and the time, and study annuities, the regular deposits and payments behind savings plans and loans.

Simple interest

Simple interest is paid only on the principal, the money you started with.

I = Prt

I = interest in dollars
P = principal in dollars
r = annual interest rate as a decimal
t = time in years

The amount at the end is A = P + I = P + Prt.

For Amira: I = 1000 × 0.05 × t = 50t, so A = 1000 + 50t. Every year adds the same $50. That is a linear relation: its graph is a straight line with slope 50 (dollars per year) and vertical intercept 1000.

Compound interest as repeated simple interest

With compound interest, you do a simple interest calculation for one period at a time, and the interest is added to the balance before the next period starts. (In this lesson the interest is compounded annually, so one period is one year.)

Year 1: interest = 1000 × 0.05 = 50.00, balance = 1050.00
Year 2: interest = 1050 × 0.05 = 52.50, balance = 1102.50
Year 3: interest = 1102.50 × 0.05 = 55.13 (rounded), balance = 1157.63

Notice that adding 5% is the same as multiplying by 1.05. So each year's balance is the previous balance × 1.05:

1000 → 1000(1.05) → 1000(1.05)² → 1000(1.05)³ → …

After t years the value is 1000(1.05)ᵗ. You will meet the general formula in the next lesson.

Worked example

Example 1: Building a comparison table

Make a table showing the value of $1 000 at 5% per year with simple interest and with interest compounded annually, for years 0 to 5. Find the first differences of each.

Simple: A = 1000 + 50t. Values: 1000, 1050, 1100, 1150, 1200, 1250.
Compound: multiply by 1.05 each year. Values: 1000.00, 1050.00, 1102.50, 1157.63, 1215.51, 1276.28.
First differences, simple: 50, 50, 50, 50, 50. They are constant, so the relation is linear.
First differences, compound: 50.00, 52.50, 55.13, 57.88, 60.77. They grow, because each year's interest is 5% of a bigger balance.
Ratios for compound: 1050 ÷ 1000 = 1.05, 1102.50 ÷ 1050 = 1.05, and so on. A constant ratio is the sign of an exponential relation.

Simple interest: constant first differences of $50 (linear). Compound interest: first differences that grow, and a constant ratio of 1.05 (exponential).

Year	Simple interest value	Compound interest value	Difference (compound − simple)
0	$1 000.00	$1 000.00	$0.00
1	$1 050.00	$1 050.00	$0.00
2	$1 100.00	$1 102.50	$2.50
5	$1 250.00	$1 276.28	$26.28
10	$1 500.00	$1 628.89	$128.89
15	$1 750.00	$2 078.93	$328.93
20	$2 000.00	$2 653.30	$653.30
25	$2 250.00	$3 386.35	$1 136.35

$1 000 at 5% per year. The gap between the two options is tiny at first and then grows faster and faster.

Simple interest gives a straight line (it rises $50 every year). Compound interest gives a curve that bends upward (it rises a little more every year). They start together and separate over time.

Reading the graphs

Both graphs start at the principal, $1 000, when t = 0.
For the first year or two the two graphs are almost on top of each other. Over a short time, simple and compound interest are nearly the same.
Over a long time compound interest wins by a lot, because the interest itself keeps earning interest.

The bar graph below shows the interest earned in each single year. For simple interest, every bar is $50. For compound interest, each bar is 5% taller than the one before it.

Each year, simple interest pays the same $50. Compound interest pays 5% more than the year before: $50 in year 1, $52.50 in year 2, and $77.57 in year 10.

Worked example

Example 2: Simple and compound interest on a loan

Noor borrows $800 from a relative at 12% per year and repays everything after 3 years. How much more would she repay if the interest were compounded annually instead of simple?

Simple: I = Prt = 800 × 0.12 × 3 = 288. A = 800 + 288 = $1 088.00.
Compound, year by year: 800 × 1.12 = 896.00; 896 × 1.12 = 1003.52; 1003.52 × 1.12 = 1123.94 (rounded).
Or in one step: 800(1.12)³ = 800 × 1.404928 ≈ 1123.94.
Difference: 1123.94 − 1088.00 = 35.94.

With annual compounding Noor repays $1 123.94, which is $35.94 more than the $1 088.00 she would repay with simple interest. Compound interest works against a borrower.

Worked example

Example 3: Using a table to answer a question

For Amira and Jonah's $1 000 at 5%, after how many full years is Jonah's compound interest account worth at least $500 more than Amira's simple interest account?

From the table, after 15 years the difference is $328.93 and after 20 years it is $653.30, so the answer is between 15 and 20.
Year 17: 1000(1.05)¹⁷ ≈ 2292.02 and 1000 + 50(17) = 1850. Difference ≈ $442.02. Not yet.
Year 18: 1000(1.05)¹⁸ ≈ 2406.62 and 1000 + 50(18) = 1900. Difference ≈ $506.62. Yes.

After 18 years the compound interest account is at least $500 ahead (by about $506.62).

Just for fun

You may have seen a quote online saying Albert Einstein called compound interest "the eighth wonder of the world". Researchers who track down quotations have never found evidence that he said it. Compound interest really is impressive, but its fan club may have invented its most famous member.

Hands-on activity

Bean bank: watch the gap grow

You need: about 300 dried beans, buttons or pieces of dry cereal, two bowls, and paper. (Keep small objects away from young children.)

Use a very high rate, 50% per round, so the pattern is easy to see.

Put 16 beans in each bowl. Bowl A earns simple interest. Bowl B earns compound interest.
Each round, add 50% of the original 16 to bowl A (8 beans). Add 50% of what is in bowl B right now to bowl B.
Do 4 rounds. Record both bowls in a table after each round.
Find the first differences and the ratios for each bowl.

What to look for: Bowl A: 16, 24, 32, 40, 48 (always +8, linear). Bowl B: 16, 24, 36, 54, 81 (always ×1.5, exponential). After round 4, bowl B has 33 more beans than bowl A.

Extension: Build the same table for $1 000 at 5% for 25 years in a spreadsheet (such as Google Sheets or Excel). Put the year in column A, =1000+50A2 in column B, and =10001.05^A2 in column C. Fill down, then insert a line chart of columns B and C.

Your turn

Find the simple interest and the amount for $600 at 4% per year for 5 years.

Find the amount if the $600 in the last question earns 4% compounded annually for 5 years. How much more is this than with simple interest?

Use repeated simple interest to find the value of $2 000 at 3% compounded annually after 1, 2 and 3 years.

An account is worth $500, $530, $560, $590, $620 at the end of years 0 to 4. Is this simple or compound interest? What is the annual rate?

An account is worth $400.00, $432.00, $466.56, $503.88 at the end of years 0 to 3. Simple or compound? What is the rate?

On a graph of value against time, which option gives a straight line and which gives a curve? Explain why in one sentence each.

More practice: warm-up

Write 3.5% as a decimal.

Find the simple interest on $1 200 at 5% for 2 years.

What number do you multiply by to add 7% to a balance?

$300 earns 10% compounded annually. Find its value after 2 years.

True or false: after exactly one year with interest paid once a year, simple and compound interest give the same amount.

More practice: on your own

$5 000 is invested for 4 years at 6%. Find the amount with simple interest and with annual compounding. What is the difference?

Make a table for $250 at 20% per year for years 0 to 3, simple and compound. Show the first differences.

A simple interest investment grows from $800 to $944 in 3 years. Find the annual rate.

Write an equation for the value V after t years of $1 500 at 4%: (a) simple interest, (b) compounded annually. Which is linear?

Using the table in this lesson, how much of Jonah's $3 386.35 after 25 years is "interest on interest"? Explain.

Challenge

Option A: $2 000 at 6% simple interest. Option B: $2 000 at 5% compounded annually. Which is worth more after 10 years? After 20 years?

Explain why the first differences of a compound interest table are themselves in a geometric pattern (each one is a constant multiple of the last).

A simple interest loan and a compound interest loan (compounded annually) both charge 10% on $1 000. After how many full years does the compound loan cost at least $200 more than the simple loan?

Check your understanding

Quiz

Which formula gives simple interest?

$1 000 earns 5% compounded annually. What is the interest earned in the second year?

A table of account values has constant first differences of $40. What kind of interest is it?

Values of $200.00, $210.00, $220.50, $231.53 show:

The graph of value against time for compound interest is:

$400 is borrowed at 10% for 2 years. How much more is owed with annual compounding than with simple interest?

For parents and teachers

Big idea (B3.1): Students compare simple and compound interest on the same principal at the same rate, using tables of values and graphs. Simple interest grows by the same dollar amount each year (linear); compound interest grows by the same percent each year (exponential).

How to help: Ask your teen to explain the bean activity or the table to you. A good explanation mentions that compound interest adds interest to the balance, so next year's interest is bigger. If you have a savings account or GIC statement, look at it together and ask whether the interest is simple or compound and how often it is added.

Common difficulties:

Forgetting to change the percent to a decimal (5% → 0.05).
Adding 5% each year to the original principal when the question says compound (that is simple interest).
Rounding the balance too early. Keep all the digits in the calculator until the final answer.
Thinking compound interest is "a lot more" right away. Over 1 or 2 years the difference is small; it becomes large over long periods.

The rates in this lesson are examples for practice, not current offers.