Practice Problems for Exponential Growth, Decay, and Compound Interest

🚀 Exponential Growth Practice Problems

Exponential growth occurs when a quantity increases at a rate proportional to its current value. The formula for exponential growth is:

$A = P(1 + r)^t$

Where:

• $A$ = the future value of the investment/loan, including interest
• $P$ = the principal investment amount (the initial deposit or loan amount)
• $r$ = the annual interest rate (as a decimal)
• $t$ = the number of years the money is invested or borrowed for

Problem 1

A population of bacteria doubles every hour. If the initial population is 100, what will the population be after 4 hours?

# Given:
initial_population = 100
doubling_time = 1  # hour
time = 4  # hours

# Formula:
# population = initial_population * (2**(time / doubling_time))

population = initial_population * (2**(time / doubling_time))
print(population)
# Output: 1600

📉 Exponential Decay Practice Problems

Exponential decay occurs when a quantity decreases at a rate proportional to its current value. The formula for exponential decay is:

$A = P(1 - r)^t$

Where:

• $A$ = the future value of the investment/loan, including interest
• $P$ = the principal investment amount (the initial deposit or loan amount)
• $r$ = the annual decay rate (as a decimal)
• $t$ = the number of years the money is invested or borrowed for

Problem 2

The half-life of a radioactive substance is 20 years. If you start with 500 grams, how much will remain after 60 years?

# Given:
initial_amount = 500  # grams
half_life = 20  # years
time = 60  # years

# Formula:
# remaining_amount = initial_amount * (0.5**(time / half_life))

remaining_amount = initial_amount * (0.5**(time / half_life))
print(remaining_amount)
# Output: 62.5

💰 Compound Interest Practice Problems

Compound interest is the interest calculated on the principal and the accumulated interest. The formula for compound interest is:

$A = P(1 + \frac{r}{n})^{nt}$

Where:

• $A$ = the future value of the investment/loan, including interest
• $P$ = the principal investment amount (the initial deposit or loan amount)
• $r$ = the annual interest rate (as a decimal)
• $n$ = the number of times that interest is compounded per year
• $t$ = the number of years the money is invested or borrowed for

Problem 3

If you invest $1000 at an annual interest rate of 5% compounded quarterly, what will be the amount after 10 years?

# Given:
principal = 1000  # dollars
annual_interest_rate = 0.05
compounded_per_year = 4  # quarterly
time = 10  # years

# Formula:
# amount = principal * (1 + (annual_interest_rate / compounded_per_year))**(compounded_per_year * time)

amount = principal * (1 + (annual_interest_rate / compounded_per_year))**(compounded_per_year * time)
print(amount)
# Output: 1643.619459792396

Problem 4

What principal amount should be invested to obtain $5000 after 5 years, if the interest rate is 6% compounded monthly?

# Given:
future_value = 5000  # dollars
annual_interest_rate = 0.06
compounded_per_year = 12  # monthly
time = 5  # years

# Formula:
# principal = future_value / (1 + (annual_interest_rate / compounded_per_year))**(compounded_per_year * time)

principal = future_value / (1 + (annual_interest_rate / compounded_per_year))**(compounded_per_year * time)
print(principal)
# Output: 3704.0907471222536