Mathematics
Grade 11
15 min
Continuously compounded interest: word problems
Continuously compounded interest: word problems
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Introduction & Learning Objectives
Learning Objectives
Identify the principal, rate, and time from a word problem.
Apply the continuous compounding formula, A = Pe^(rt), to calculate the future value of an investment.
Use natural logarithms to solve for the time (t) required for an investment to reach a specific value.
Use natural logarithms to solve for the interest rate (r) needed to achieve a financial goal.
Calculate the initial principal (P) required to achieve a desired future value.
Interpret the results of calculations in the context of the original word problem.
Ever wonder how a small investment could grow into a fortune without you ever touching it? 📈 Let's explore the magic of interest that never sleeps!
This tutorial will teach you how to solve real-world financial problems using the formu...
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Key Concepts & Vocabulary
TermDefinitionExample
Principal (P)The initial amount of money invested or borrowed.If you deposit $2,000 into a new savings account, the principal is $2,000.
Interest Rate (r)The percentage of the principal that is earned or paid as interest over a period, expressed as a decimal for calculations.An annual interest rate of 6% must be written as r = 0.06 in the formula.
Time (t)The duration for which the money is invested or borrowed, almost always expressed in years.An investment held for 5 years and 6 months would have t = 5.5.
Future Value (A)The total amount of money in an account after a certain period, which includes the principal plus all the accumulated interest.If your $2,000 principal grows to $2,500 after some time, the future value is $2,500.
Continuous CompoundingThe mathemati...
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Core Formulas
Continuous Compounding Formula
A = P * e^(r*t)
Use this primary formula to find the future value (A) of an investment when you know the initial principal (P), the annual interest rate (r), and the time in years (t).
Solving for Time (t)
t = (ln(A/P)) / r
Derived from the main formula using natural logarithms. Use this to find how long it will take for an initial principal (P) to grow into a future value (A) at a given rate (r).
Solving for Rate (r)
r = (ln(A/P)) / t
Also derived from the main formula. Use this to find the annual interest rate required for an investment (P) to reach a specific value (A) in a set amount of time (t).
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Challenging
Account A starts with $5,000 and earns 7% compounded continuously. Account B starts with $8,000 and earns 5% compounded continuously. After how many years will the two accounts have the same balance?
A.18.5 years
B.23.5 years
C.25.0 years
D.They will never have the same balance.
Challenging
An investment of $10,000 is placed in an account where the interest is compounded continuously. The rate is 4% for the first 7 years, and then it increases to 6% for the next 5 years. What is the total interest earned over the entire 12-year period?
A.$7,860.43
B.$17,860.43
C.$7,551.23
D.$6,988.12
Challenging
A financial advisor uses the 'Rule of 72' to estimate that money will double in 9 years at an 8% interest rate (72/8 = 9). What is the more precise doubling time if the 8% interest is compounded continuously, and what is the approximate percentage error of the 'Rule of 72' estimate?
A.8.66 years, with an error of about 3.9%
B.9.00 years, with an error of 0%
C.8.33 years, with an error of about 7.5%
D.8.66 years, with an error of about 0.34%
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