Compound Interest Formula on the Digital SAT

Quick Answer: The compound interest formula calculates the total amount of money after a certain time, factoring in interest earned on both the principal and previously accumulated interest. Always remember to convert your percentage rate to a decimal before plugging it into Desmos or solving algebraically.

graph TD
    A[Read Compound Interest Question] --> B{Compounding Frequency?}
    B -->|Annually| C[Use A = P\(1 + r\)^t]
    B -->|n times per year| D[Use A = P\(1 + r/n\)^\(nt\)]
    B -->|Continuously| E[Use A = Pe^\(rt\)]

What Is Compound Interest Formula?

Compound interest is a specific application of exponential growth. Unlike simple interest, which only pays out based on the original amount invested (the principal), compound interest pays you interest on your principal and on the interest you've already earned. On the College Board Digital SAT, these questions fall under the Advanced Math domain because they test your ability to model real-world scenarios using non-linear functions.

The standard formula you need to know is $A = P(1 + \frac{r}{n})^{nt}$. In this equation, $A$ represents the final amount, $P$ is the initial principal balance, $r$ is the interest rate expressed as a decimal, $n$ is the number of times interest is compounded per year, and $t$ is the number of years.

While linear equations grow at a constant rate, compound interest grows exponentially. This means the math behaves very differently from what you might see when working with the /sat/math/quadratic-formula or analyzing parabolas through /sat/math/factoring-quadratics. For complex calculations, especially when solving for $t$, leveraging the built-in Desmos Calculator on the testing interface is highly recommended.

Step-by-Step Method

1. Step 1 — Identify the given values from the word problem: Principal ($P$), rate ($r$), compounding frequency ($n$), and time ($t$).
2. Step 2 — Convert the percentage interest rate $r$ into a decimal by dividing by $100$ (e.g., $5\%$ becomes $0.05$).
3. Step 3 — Determine $n$ based on the text. "Annually" means $n=1$, "semi-annually" means $n=2$, "quarterly" means $n=4$, and "monthly" means $n=12$.
4. Step 4 — Plug the values into the formula $A = P(1 + \frac{r}{n})^{nt}$.
5. Step 5 — Use your calculator to solve for the missing variable, making sure to follow the order of operations by calculating the exponent first.

Desmos Shortcut

If the SAT asks you to find how long it takes for an account to reach a certain amount (solving for $t$), doing the algebra requires logarithms, which can be tedious and prone to errors. Instead, use Desmos.

Type the compound interest formula into Desmos as your first equation, using $x$ for time: y = 2000(1 + 0.05/4)^(4x). Then, type the target amount as your second equation: y = 3000. Click the point where the curve and the horizontal line intersect. The x-coordinate of that intersection is your answer!

Worked Example

Question: Sarah invests \2,000$in a savings account that earns$5%$annual interest compounded quarterly. How much money will be in the account after$3$ years? (Round to the nearest cent)

A) \2,300.00$B)$$2,315.25$C)$$2,321.51$D)$$2,330.00$

Solution:

First, identify your variables from the problem: $P = 2000$ $r = 0.05$ (converted $5\%$ to a decimal) $n = 4$ (compounded quarterly means $4$ times a year) $t = 3$

Now, plug these into the compound interest formula: $A = P(1 + \frac{r}{n})^{nt}$

$A = 2000(1 + \frac{0.05}{4})^{(4 \times 3)}$

Simplify the terms inside the parentheses and the exponent: $A = 2000(1 + 0.0125)^{12}$

$A = 2000(1.0125)^{12}$

Calculate the final amount: $A \approx 2000(1.16075)$

$A \approx 2321.51$

The correct answer is C.

Common Traps

1. Forgetting to convert percentages to decimals — Based on Lumist student data, 25% of errors on compound interest questions happen because students plug the whole percentage number directly into the formula (using $5$ instead of $0.05$). This results in astronomically incorrect answers.

2. Confusing growth and decay factors — Our data shows that 60% of students initially confuse the growth factor $(1+r)$ with the decay factor $(1-r)$. While compound interest is almost always growth, the SAT occasionally tests depreciation (like a car losing value), which requires you to subtract the rate instead.

3. Ignoring the compounding frequency — Just like missing the sign when putting an equation into /sat/math/vertex-form-quadratic, forgetting to adjust $n$ is a critical error. If the problem says "compounded monthly," you must divide your rate by $12$ and multiply your time by $12$.

FAQ

What is the formula for compound interest on the SAT?

The standard formula is $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the annual interest rate as a decimal, $n$ is the number of times interest is compounded per year, and $t$ is the time in years.

Do I need to memorize the compound interest formula for the Digital SAT?

Yes, you should memorize it. While the SAT provides a reference sheet with basic geometry formulas, the compound interest formula is not included.

How do I handle continuous compounding if it shows up?

If a question specifically states that interest is compounded continuously, use the formula $A = Pe^{rt}$, where $e$ is Euler's number. However, the standard compounding formula is much more common.

How many Compound Interest Formula questions are on the SAT?

Advanced Math makes up approximately 35% of SAT Math. On Lumist.ai, we have 22 practice questions specifically on this topic to help you prepare.