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Magazine Chapter 4: Problem 65
Use the model \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) The variable A represents the future value of P dollars invested at an interest rate \(r\) compounded \(n\) times per year for \(t\) years. If 4000 is put aside in a money market account with interest reinvested monthly at 2.2%, find the time required for the account to earn 1000. Round to the nearest month.
Short Answer
Expert verified
10 months.
Step by step solution
01
Identify Variables
Before substituting the values into the formula, identify each variable. Here, \( P = 4000 \), \( A = 4000 + 1000 = 5000 \), \( r = 0.022 \), and since the interest is compounded monthly, \( n = 12 \).
02
Substitute Known Values
Substitute the identified values into the compound interest formula: \[ 5000 = 4000 \left(1 + \frac{0.022}{12}\right)^{12t} \]
03
Simplify the Formula
Simplify inside the parentheses first: \[ 1 + \frac{0.022}{12} = 1 + 0.00183333333\approx 1.001833 \] So, the equation becomes: \[ 5000 = 4000 \left(1.001833\right)^{12t} \]
04
Isolate the Exponential Term
Divide both sides of the equation by 4000 to isolate the exponential term: \[ \frac{5000}{4000} = \left(1.001833\right)^{12t} \] This simplifies to: \[ 1.25 = \left(1.001833\right)^{12t} \]
05
Apply Natural Logarithm
Take the natural logarithm of both sides of the equation to get rid of the exponent: \[ \ln(1.25) = \ln\left(\left(1.001833\right)^{12t}\right) \] Using the power rule of logarithms, this simplifies to: \[ \ln(1.25) = 12t \cdot \ln(1.001833) \]
06
Solve for t
To find \( t \), divide both sides by \( 12 \cdot \ln(1.001833) \): \[ t = \frac{\ln(1.25)}{12 \cdot \ln(1.001833)} \] Using a calculator to find the values of the natural logarithms, \( \ln(1.25) \approx 0.223143 \) and \( \ln(1.001833) \approx 0.0018320 \): \[ t = \frac{0.223143}{12 \cdot 0.001832} \approx 10.16 \] Since the problem asks to round to the nearest month, \( t \)=10 months.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
compound interest
Compound interest refers to the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. Unlike simple interest, which is calculated only on the principal amount, compound interest allows your investment to grow faster over time because you earn interest on your interest.
The formula for compound interest is:
\( A = P \left(1 + \frac{r}{n}\right)^{nt} \)
Where:
The formula for compound interest is:
\( A = P \left(1 + \frac{r}{n}\right)^{nt} \)
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (in decimal).
- \(n\) is the number of times that interest is compounded per unit \(t\).
- \(t\) is the time the money is invested or borrowed for in years.
natural logarithm
The natural logarithm, often represented as \(\ln\), is a special type of logarithm with the base of the constant e (approximately 2.71828). It is widely used in mathematics, especially in areas involving growth processes and decay.
For example, when solving for \(t\) in compound interest problems, the natural logarithm helps to isolate the time variable. Taking the natural logarithm of both sides of an equation where a variable is exponentiated allows for the simplification of that variable.
Using the power rule of logarithms which states \(\ln(a^b) = b \cdot \ln(a)\), one can solve equations involving exponents. This is exactly what we did in the exercise to solve for \(t\) in the formula:
\[ \ln(1.25) = 12t \cdot \ln(1.001833) \] Understanding natural logarithms can make complex financial calculations much more manageable.
For example, when solving for \(t\) in compound interest problems, the natural logarithm helps to isolate the time variable. Taking the natural logarithm of both sides of an equation where a variable is exponentiated allows for the simplification of that variable.
Using the power rule of logarithms which states \(\ln(a^b) = b \cdot \ln(a)\), one can solve equations involving exponents. This is exactly what we did in the exercise to solve for \(t\) in the formula:
\[ \ln(1.25) = 12t \cdot \ln(1.001833) \] Understanding natural logarithms can make complex financial calculations much more manageable.
monthly compounding
Monthly compounding refers to the process where interest is calculated and added to the principal balance of an investment or loan every month. This is different from annual compounding where interest is only added once a year.
When interest is compounded monthly, the formula to compute the future value of an investment becomes:
\( A = P \left(1 + \frac{r}{12}\right)^{12t} \)
Here, the annual rate \(r\) is divided by 12 to get the monthly rate, and the exponent \(12t\) reflects the number of months.
Monthly compounding benefits investors because interest compounds more frequently, meaning more instances where interest is earned on the current principal and previous interest. This leads to a higher amount of money accumulated over time compared to less frequent compounding periods.
When interest is compounded monthly, the formula to compute the future value of an investment becomes:
\( A = P \left(1 + \frac{r}{12}\right)^{12t} \)
Here, the annual rate \(r\) is divided by 12 to get the monthly rate, and the exponent \(12t\) reflects the number of months.
Monthly compounding benefits investors because interest compounds more frequently, meaning more instances where interest is earned on the current principal and previous interest. This leads to a higher amount of money accumulated over time compared to less frequent compounding periods.
future value calculation
Future value calculation allows investors to predict how much their investments will grow over time with a given interest rate and compounding period. The general formula for the future value of an investment is:
\( A = P \left(1 + \frac{r}{n}\right)^{nt} \)
To find out how long it will take for an investment to grow to a specific future value, we need to rearrange the formula and solve for \(t\).
In our exercise, we started with a principal \(P = 4000\), aimed for a future value \(A = 5000\), had an annual interest rate \(r = 0.022\), and compounded monthly \(n = 12\).
By substituing these values and solving for \(t\), we found that it would take approximately 10 months to grow to the desired future value.
Calculating future value is crucial for setting financial goals and making informed decisions about investments and savings.
\( A = P \left(1 + \frac{r}{n}\right)^{nt} \)
To find out how long it will take for an investment to grow to a specific future value, we need to rearrange the formula and solve for \(t\).
In our exercise, we started with a principal \(P = 4000\), aimed for a future value \(A = 5000\), had an annual interest rate \(r = 0.022\), and compounded monthly \(n = 12\).
By substituing these values and solving for \(t\), we found that it would take approximately 10 months to grow to the desired future value.
Calculating future value is crucial for setting financial goals and making informed decisions about investments and savings.
