Digital SAT Math Prep: Exponential Word Problems & Models

July 25, 2026

SAT Saturday Week 13: Demystifying Exponential Models

Welcome back to SAT Saturday!

A pair of acorns - to represent the SAT/PSAT tests. — Image by svklimkin from Pixabay

Now that we have locked down linear relationships, it is time to scale up. Today, we are stepping into the arena of Advanced Math, which accounts for roughly 35% of your total SAT Math score.

While linear models represent quantities that change by a constant amount over time, exponential models represent quantities that change by a constant percentage over time. This includes population growth, compound interest, and radioactive decay.

The College Board tests exponential equations in two distinct ways: checking if you understand what the percentage multiplier actually represents, and throwing tricky time-intervals at you to see if your exponent rules hold up.

Let’s dive in, check our algebraic foundations, and master the curves of exponential modeling!

This Week’s Advanced Math Challenges

Question 25: Exponential Growth Rates & Percentages

A wildlife biologist monitors the population of a specific bird species in a state park. The model below estimates the population, P, of the bird species t years after the initial observation.

Which of the following statements is the best interpretation of the number 1.045 in this context?

A. The bird population is estimated to increase by 4.5% each year.
B. The bird population is estimated to increase by 1.045% each year.
C. The initial population of the bird species was 1,045.
D. The bird population is estimated to increase by 45 birds each year.

Question 26: Exponential Decay & Time-Interval Adjustments

A technician places a sample of a certain radioactive isotope into a secure chamber. The mass of the sample, in grams, decreases exponentially over time. The sample loses half of its mass every 6 hours. If the initial mass of the sample is 320 grams, which equation models the mass, , in grams, remaining after hours?

Week 12 Answer Key & Explanations

Let’s check your work from last week’s linear algebra challenges!

Question 23 Analysis (Interpreting the Parts)

The Problem: A local solar energy company installs rooftop solar panels. The total cost, y, in dollars, for a customer to have solar panels installed and maintained for x months is modeled by the equation:

y = 125x + 3800

What is the best interpretation of 3,800 in this context?

A) The monthly maintenance fee charged by the company.
B) The total number of months the solar panels are under warranty.
C) The initial cost to purchase and install the solar panels.
D) The expected dollar amount saved on electricity each month.

🧭 The SAT Concept: Slope-Intercept Structure

The Digital SAT regularly tests whether you understand the structural components of the slope-intercept formula: y = mx + b

The Slope (m = 125): This represents the variable rate of change. Because it is multiplied by (months), it represents the recurring monthly maintenance fee of $125.
The y-intercept (b = 3,800): This represents the constant or starting value when x = 0. Since zero months have passed at this starting point, the $3,800 must represent the upfront, one-time initial cost of purchasing and installing the solar panels.
Correct Answer: C. The initial cost to purchase and install the solar panels.

Question 24 Analysis (Creating Linear Models)

The Problem: An online bookstore charges a flat shipping fee per order plus a fixed price for each paperback novel purchased. An order of 4 paperback novels costs a total of $31.00. An order of 7 paperback novels costs a total of $49.00. Which equation represents the total cost, C, in dollars, for an order of n paperback novels?

📐 The Mathematical Proof

Step 1 (Find the Slope/Rate per Book): Think of the given data as coordinate pairs , where is the number of books and is the total cost. This gives us two points: (4, 31) and (7, 49).

Now, calculate the slope (the rate per book):

use slope formula to determine the slope of this equation is 6

Since each paperback book costs $6, our slope must be 6. This instantly eliminates options B, C, and D!

Step 2 (Find the y-intercept/Flat Fee): Use our slope in the linear equation template with one of our coordinates, like (4, 31):

C = 6n + b

31 = 6(4) + b

31 = 24 + b

b = 7

The flat shipping fee is $7, yielding our final linear equation: C = 6n + 7

Correct Answer: A.

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