Imagine putting $1,000 into a savings account and watching it grow over time.
Sounds pretty good, right?
But here’s something interesting: How quickly your money grows depends on how the interest is calculated.
That’s where simple interest and compound interest come into play.
Both concepts can appear on the SAT, often in word problems involving money, percentages, and exponential growth.
The good news? Once you understand the difference between them, the formulas become much easier to use.
Simple interest earns money on your original investment. Compound interest earns money on your original investment AND the interest you’ve already earned.
There’s also one very important distinction we’ll explore: the difference between calculating the interest earned and calculating the total amount of money.
First, What Is Interest?
Interest is money that is earned or charged over time, usually based on a percentage.
For example, suppose you deposit $1,000 into a savings account that earns 5% interest per year.
Since 5% of $1,000 is $50, you would earn $50 in interest during the first year.
But there’s a difference between the interest you earned and the total amount in your account.
Interest earned: $50
Total amount: $1,050
This distinction is extremely important on the SAT because a problem might ask for either one!
Simple Interest: The Easy One
Simple interest is calculated using only the original amount of money, called the principal.
That means the amount of interest earned each year stays the same.
The Simple Interest Formula
I = Prt
Here’s what each letter means:
- I = Interest earned
- P = Principal (the original amount)
- r = Annual interest rate, written as a decimal
- t = Time in years
This formula calculates only the interest earned, not the total amount in the account.
For example, an interest rate of 5% becomes:
r = 0.05
Remember, whenever you’re using these formulas, you must convert percentages into decimals.
The Total Amount Formula for Simple Interest
But what if the SAT asks how much money you have altogether, including your original investment?
In that case, we use:
A = P + I
Since I = Prt, we can substitute:
A = P + Prt
Factor out P:
A = P(1 + rt)
Here, A represents the total amount of money.
Important SAT Tip: The formula I = Prt calculates the interest earned. The formula A = P(1 + rt) calculates the total amount, including the original principal.
Both formulas describe simple interest, but they answer different questions!
SAT Problem Type #1: Calculating Simple Interest
Suppose you deposit $2,000 into an account earning 4% simple interest annually.
Part A: How much interest will you earn after 3 years?
Step 1: Identify the information.
P = 2000
r = 0.04
t = 3
Step 2: Use the simple interest formula.
I = Prt
I = 2000 × 0.04 × 3
Step 3: Calculate.
I = 80 × 3
I = $240
You earned $240 in interest.
But that’s not the total amount in your account!
Part B: How much money will you have altogether?
Now we want the total amount, including the original $2,000.
We can simply add the principal and interest:
A = P + I
A = 2000 + 240
A = $2,240
Or we could use the total amount formula directly:
A = P(1 + rt)
A = 2000(1 + 0.04 × 3)
A = 2000(1.12)
A = $2,240
Both approaches give us the same answer.
The important thing is knowing which quantity the SAT is asking for.
Why Is It Called Simple Interest?
Because the interest stays the same every year.
In our example:
4% of $2,000 is $80.
So the account earns:
- Year 1: $80
- Year 2: $80
- Year 3: $80
Total interest:
80 + 80 + 80 = 240
Final amount:
2000 + 240 = 2240
Notice that the interest never increases.
SAT Tip: Simple interest produces linear growth because the same amount of money is added during each equal time period.
Compound Interest: Interest That Earns More Interest
Now things get a little more interesting.
With compound interest, the interest you earn gets added to your account balance.
Then, during the next period, you earn interest on that larger balance.
In other words, your interest starts earning interest!
The Compound Interest Total Amount Formula
A = P(1 + r/n)⁽ⁿᵗ⁾
Here’s what each letter means:
- A = Total amount, including principal and interest
- P = Principal
- r = Annual interest rate as a decimal
- n = Number of times interest is compounded per year
- t = Time in years
That extra variable, n, is important.
It tells us how often the interest is added to the account.
Notice the exponent, nt. This means n multiplied by t.
What If We Only Want the Compound Interest Earned?
Here’s another important distinction.
The compound interest formula above calculates the total amount, not just the interest earned.
If we want only the interest earned, we must subtract the original principal.
I = A − P
Or, combining everything into one formula:
I = P(1 + r/n)⁽ⁿᵗ⁾ − P
So remember:
- A = P(1 + r/n)⁽ⁿᵗ⁾ gives the total amount.
- I = A − P gives the compound interest earned.
SAT Tip: Don’t confuse the total balance with the interest earned. This distinction matters for both simple and compound interest problems.
What Does “Compounded” Mean?
Compound interest can be calculated at different intervals.
Here are some common examples:
| Compounding frequency | Value of n |
|---|---|
| Annually (once per year) | 1 |
| Semiannually (twice per year) | 2 |
| Quarterly (four times per year) | 4 |
| Monthly (12 times per year) | 12 |
| Daily (typically 365 times per year) | 365 |
The more frequently interest compounds, the more opportunities the balance has to grow.
SAT Problem Type #2: Calculating Compound Interest
Suppose you invest $1,000 at an annual interest rate of 6%, compounded annually.
Part A: How much money will you have after 3 years?
Step 1: Identify the information.
P = 1000
r = 0.06
n = 1
t = 3
Step 2: Substitute into the total amount formula.
A = P(1 + r/n)⁽ⁿᵗ⁾
A = 1000(1 + 0.06/1)³
Step 3: Simplify.
A = 1000(1.06)³
A = 1000(1.191016)
A ≈ $1,191.02
After 3 years, your account will contain approximately $1,191.02.
Part B: How much compound interest did you actually earn?
Remember, the $1,191.02 includes your original $1,000.
To find the interest earned:
I = A − P
I = 1191.02 − 1000
I ≈ $191.02
So the results are:
- Total amount: $1,191.02
- Interest earned: $191.02
That’s a distinction the SAT could easily test!
Why Does Compound Interest Grow Faster?
Let’s compare simple interest and compound interest using the same starting amount.
Suppose you invest $1,000 at 10% annual interest for 3 years.
| Year | Simple Interest Balance | Compound Interest Balance |
|---|---|---|
| Start | $1,000 | $1,000 |
| Year 1 | $1,100 | $1,100 |
| Year 2 | $1,200 | $1,210 |
| Year 3 | $1,300 | $1,331 |
With simple interest, you earn $100 each year.
With compound interest, you earn $100 in the first year, $110 in the second year, and $121 in the third year.
After three years:
Simple Interest
Total amount: $1,300
Interest earned: $300
Compound Interest
Total amount: $1,331
Interest earned: $331
The difference in interest earned is $31.
That might not seem like much, but over longer periods, compound interest can create much larger differences.
Think of simple interest as climbing a staircase with equal-sized steps. Compound interest is more like climbing a staircase where the steps keep getting taller.
SAT Problem Type #3: Compounding More Than Once Per Year
Here’s where SAT questions can become a little trickier.
Suppose you invest $5,000 at 8% annual interest, compounded quarterly, for 2 years.
Part A: What is the total amount after 2 years?
Step 1: Identify the values.
P = 5000
r = 0.08
n = 4
t = 2
Remember, quarterly means four times per year.
Step 2: Substitute.
A = 5000(1 + 0.08/4)⁽⁴ × ²⁾
Step 3: Simplify the fraction and exponent.
A = 5000(1 + 0.02)⁸
A = 5000(1.02)⁸
Step 4: Calculate.
A ≈ 5000(1.171659)
A ≈ $5,858.30
The investment grows to approximately $5,858.30.
Part B: How much interest was earned?
Now subtract the original principal:
I = A − P
I = 5858.30 − 5000
I ≈ $858.30
Notice that the exponent was 8, not 2.
That’s because interest compounded four times per year for two years, giving us eight compounding periods.
SAT Problem Type #4: Finding the Interest Rate
Sometimes the SAT gives you the starting amount and final amount but asks you to find the interest rate.
For example:
A savings account starts with $800 and grows to $920 after 3 years using simple interest.
What is the annual interest rate?
This time, let’s use the simple interest formula directly.
Step 1: Find the interest earned.
I = A − P
I = 920 − 800
I = 120
Step 2: Use the simple interest formula.
I = Prt
120 = 800 × r × 3
Step 3: Simplify.
120 = 2400r
Divide both sides by 2400:
r = 0.05
Convert to a percentage:
r = 5%
The annual interest rate is 5%.
Notice how useful it was to first calculate the interest earned and then use I = Prt.
This approach is often easier than working with the total amount formula.
SAT Problem Type #5: Recognizing Exponential Growth
Here’s another important SAT connection.
Compound interest is an example of exponential growth.
Suppose the value of an investment is modeled by:
A(t) = 2000(1.05)ᵗ
What does this equation tell us?
The number 2000 represents the starting amount.
The number 1.05 represents a growth factor of 5% per year.
Why?
Because:
1.05 = 1 + 0.05
That means the investment increases by 5% each year.
And the exponent t tells us how many years have passed.
Notice that A(t) represents the total value of the investment, not just the interest earned.
If you wanted to calculate only the interest earned after t years, you would subtract the original $2,000:
I(t) = 2000(1.05)ᵗ − 2000
SAT Tip: In exponential growth equations, a growth factor of 1.05 means a 5% increase, not a 105% increase.
This idea appears in many SAT problems, even when the question isn’t specifically about money.
Common SAT Mistakes to Avoid
Here are some of the biggest mistakes students make with interest problems.
Mistake #1: Confusing Interest Earned With the Total Amount
This is one of the most important things to watch for.
For simple interest:
I = Prt calculates the interest earned.
A = P(1 + rt) calculates the total amount.
For compound interest:
A = P(1 + r/n)⁽ⁿᵗ⁾ calculates the total amount.
I = A − P calculates the interest earned.
Always read the question carefully to determine what you’re being asked to find.
Mistake #2: Forgetting to Convert Percentages
If the interest rate is 7%, don’t substitute 7 into the formula.
Use:
r = 0.07
Mistake #3: Forgetting the Compounding Frequency
If interest compounds monthly:
n = 12
If it compounds quarterly:
n = 4
And remember that n appears in two places in the compound interest formula.
Mistake #4: Forgetting That Time Is Measured in Years
If a problem says 6 months, use:
t = 0.5
If it says 18 months, use:
t = 1.5
Mistake #5: Confusing Linear and Exponential Growth
Simple interest is linear because it adds the same amount each year.
Compound interest is exponential because the balance is multiplied by the same growth factor each compounding period.
A Quick Interest Formula Cheat Sheet
Here are the four formulas worth knowing.
1. Simple Interest Earned
I = Prt
Use this when you want to calculate only the interest earned using simple interest.
2. Total Amount With Simple Interest
A = P(1 + rt)
Use this when you want the original principal plus the simple interest earned.
3. Total Amount With Compound Interest
A = P(1 + r/n)⁽ⁿᵗ⁾
Use this when you want the total balance after compound interest has been added.
4. Compound Interest Earned
I = A − P
Or, written as one formula:
I = P(1 + r/n)⁽ⁿᵗ⁾ − P
Use this when you want only the compound interest earned, without including the original principal.
The easiest way to remember the difference: I means interest only. A means the amount altogether.
Final Thoughts
Simple and compound interest problems may look intimidating at first, especially when percentages, exponents, and fractions all appear in the same equation.
But the underlying ideas are straightforward.
Simple interest means you earn the same amount of interest during each equal time period.
Compound interest means your interest earns additional interest, allowing your balance to grow faster.
And perhaps most importantly, always distinguish between the interest earned and the total amount.
When you encounter one of these problems on the SAT, start by identifying what the question is asking you to calculate.
Is it asking for the interest alone? Or does it want the total balance?
Once you know that, choosing the correct formula becomes much easier.
And remember, these formulas aren’t just useful for the SAT. Understanding how interest works can help you make smarter decisions about savings, investments, and borrowing money throughout your life.
Learn the formulas, understand the difference, and you’ll be ready when interest rates show up on test day.