What is an Exponential Decay Model?

S3-SA1-0609

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

An Exponential Decay Model describes a process where a quantity decreases by a fixed percentage over equal time intervals. It means the amount reduces quickly at first, and then the rate of reduction slows down over time, but it never quite reaches zero. Think of it as losing a part of what's left, not a fixed amount.

Simple Example

Quick Example

Imagine you have 1000 rupees in a special account, and every year, the bank deducts 10% of the money remaining in your account as a 'maintenance fee'. The first year, 10% of 1000 (which is 100 rupees) is deducted. The next year, 10% of the remaining 900 rupees (which is 90 rupees) is deducted. The amount deducted keeps getting smaller, even though the percentage is the same.

Worked Example

Step-by-Step

Let's say a new phone costs 20,000 rupees and its value decreases by 20% every year.

Step 1: Initial Value = 20,000 rupees.

---Step 2: Calculate value after Year 1.
Decrease = 20% of 20,000 = (20/100) * 20,000 = 4,000 rupees.
Value after Year 1 = 20,000 - 4,000 = 16,000 rupees.

---Step 3: Calculate value after Year 2.
Decrease = 20% of 16,000 = (20/100) * 16,000 = 3,200 rupees.
Value after Year 2 = 16,000 - 3,200 = 12,800 rupees.

---Step 4: Calculate value after Year 3.
Decrease = 20% of 12,800 = (20/100) * 12,800 = 2,560 rupees.
Value after Year 3 = 12,800 - 2,560 = 10,240 rupees.

Answer: After 3 years, the phone's value will be 10,240 rupees.

Why It Matters

Understanding exponential decay helps you predict how things change over time in many fields. Data Scientists use it to model how information spreads or fades, and Engineers use it to understand how materials weaken. It's also vital in Finance for calculating depreciation, helping people like Chartered Accountants make smart decisions.

Common Mistakes

MISTAKE: Subtracting the same fixed amount each time. | CORRECTION: Remember, in exponential decay, you subtract a fixed *percentage* of the *current* amount, not the original amount.

MISTAKE: Confusing decay with growth. | CORRECTION: Decay means the quantity is getting smaller, while growth means it's getting larger. The core idea (fixed percentage change) is similar, but the direction is opposite.

MISTAKE: Calculating the percentage decrease incorrectly (e.g., just multiplying by the percentage). | CORRECTION: First calculate the amount of decrease (percentage * current value), then subtract this amount from the current value to find the new value.

Practice Questions

Try It Yourself

QUESTION: A car bought for 5,00,000 rupees depreciates (loses value) by 10% each year. What will its value be after 1 year? | ANSWER: 4,50,000 rupees

QUESTION: A radioactive substance has 100 grams. It decays by 50% every hour. How much will be left after 2 hours? | ANSWER: 25 grams

QUESTION: You start with 8000 bacteria in a lab. If they die off at a rate of 25% every 30 minutes, how many bacteria will be left after 1 hour? | ANSWER: 4500 bacteria

MCQ

Quick Quiz

Which of these scenarios best describes exponential decay?

A fixed amount of money is added to your account every month.

A plant grows taller by 2 cm every week.

The number of unread messages in your inbox reduces by 15% each day.

You eat 5 samosas every evening until they are all gone.

The Correct Answer Is:

Option C shows a percentage reduction of the remaining amount, which is the definition of exponential decay. Options A and B are fixed additions, and Option D is a fixed subtraction until zero.

Real World Connection

In the Real World

Exponential decay is used in many real-world situations. For example, when you charge your phone, the battery percentage drops faster when it's full and then slows down as it gets lower – this is often modeled using decay. Also, scientists at ISRO use it to understand the decay of radioactive elements in space or how signals from satellites weaken over distance.

Key Vocabulary

Key Terms

DECAY: To decrease or get smaller over time | PERCENTAGE: A fraction out of 100 | RATE: How fast something changes | DEPRECIATION: The reduction in value of an asset over time | EXPONENTIAL: Something that grows or decays at an increasingly rapid rate (or decreasingly rapid rate for decay)

What's Next

What to Learn Next

Great job understanding exponential decay! Next, you should explore 'What is an Exponential Growth Model?'. It's the opposite of decay, where quantities increase by a fixed percentage, and it will help you see how these two important ideas are related.