Mathematics
Grade 12
15 min
Compound interest word problems
Compound interest word problems
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1
Introduction & Learning Objectives
Learning Objectives
Model real-world financial scenarios using compound and continuous interest formulas.
Solve for any variable (A, P, r, t) in compound interest problems, including using logarithms to solve for time.
Differentiate between problems requiring discrete compounding versus continuous compounding.
Analyze and compare different investment scenarios to determine the better financial option.
Interpret the results of compound interest calculations in the context of a word problem.
Calculate the effective annual rate (EAR) to compare different compounding frequencies.
Ever wondered how a small $1,000 investment could grow into a million dollars? 💰 It's not magic; it's the mathematical power of an exponential function called compound interest.
This tutoria...
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Key Concepts & Vocabulary
TermDefinitionExample
Principal (P)The initial amount of money invested or borrowed.If you deposit $5,000 into a new savings account, the principal (P) is $5,000.
Annual Interest Rate (r)The percentage of the principal earned or paid per year. For calculations, this percentage must be converted to a decimal.An interest rate of 4.5% per year means r = 0.045.
Compounding Frequency (n)The number of times that interest is calculated and added to the principal within one year.If interest is compounded quarterly, it is calculated 4 times per year, so n = 4. If compounded monthly, n = 12.
Time (t)The total duration of the investment or loan, always expressed in years.An investment lasting for 30 months must be expressed as t = 2.5 years.
Future Value (A)The total amount of money in the account a...
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Core Formulas
Compound Interest Formula (Discrete Compounding)
A = P(1 + r/n)^{nt}
Use this formula when interest is compounded a specific number of times per year (e.g., annually, semi-annually, quarterly, monthly, daily).
Continuous Compounding Formula
A = Pe^{rt}
Use this formula only when the problem explicitly states that interest is compounded 'continuously'. Here, 'e' is Euler's number, an irrational constant approximately equal to 2.71828.
Solving for Time using Logarithms
t = \frac{\ln(A/P)}{n \cdot \ln(1 + r/n)}
This is a rearrangement of the discrete compound interest formula, used when you need to find the time (t) it takes for an investment to reach a certain future value. A similar process is used for the continuous formula: t = ln(A/P)/r.
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Challenging
An investment of $1,000 is made in an account with an interest rate of 5% compounded continuously. What is the instantaneous rate of change of the investment's value at the 10-year mark?
A.$50.00 per year
B.$164.87 per year
C.$82.44 per year
D.$100.00 per year
Challenging
For any investment compounded continuously at a rate 'r', what is the exact ratio of the time it takes to triple (t_triple) to the time it takes to double (t_double)?
A.1.5
B.ln(3/2)
C.ln(3) / ln(2)
D.e^3 / e^2
Challenging
The formula for continuous compounding, A = Pe^(rt), can be derived from the discrete compounding formula, A = P(1 + r/n)^(nt), by considering the behavior as the compounding frequency increases. Which limit expression correctly represents this derivation?
A.lim (t→∞) P(1 + r/n)^(nt)
B.lim (r→∞) P(1 + r/n)^(nt)
C.lim (P→∞) P(1 + r/n)^(nt)
D.lim (n→∞) P(1 + r/n)^(nt)
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