Problem 4
Question
In the formula \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\) for compound interest the letters \(P, r, n,\) and \(t\) stand for _____, _____, _____, and _____, respectively, and \(A(t)\) stands for _____. So if \(\$ 100\) is invested at an interest rate of 6\(\%\) compounded quarterly, then the amount after 2 years is _____.
Step-by-Step Solution
Verified Answer
The amount is approximately \$112.67.
1Step 1: Identify Variables in the Formula
The formula for compound interest is given by \(A(t)=P\left(1+\frac{r}{n}\right)^{nt}\). Here, \(P\) represents the principal amount (the initial investment), \(r\) represents the annual interest rate expressed as a decimal, \(n\) represents the number of times compound interest is applied per year, and \(t\) represents the number of years. \(A(t)\) represents the amount of money accumulated after \(t\) years, including interest.
2Step 2: Assign Given Values to the Variables
From the exercise, we are given:- \(P = \$100\) (the principal amount)- \(r = 6\% = 0.06\) (the annual interest rate as a decimal)- \(n = 4\) (quarterly compounding means 4 times a year)- \(t = 2\) (the number of years the money is invested)We need to use these values in the compound interest formula.
3Step 3: Calculate the Compound Interest
Substitute the known values into the formula \(A(t)=P\left(1+\frac{r}{n}\right)^{nt}\):\[A(2) = 100\left(1+\frac{0.06}{4}\right)^{4 \times 2}\] First, calculate \(\frac{0.06}{4} = 0.015\). So the formula becomes\[A(2) = 100\left(1 + 0.015\right)^{8}\].Next, calculate \(1 + 0.015 = 1.015\).Now compute \(1.015^8\):\[1.015^8 \approx 1.12668\].Finally, calculate \(100 \times 1.12668 = 112.668\).
4Step 4: Round to the Nearest Cent
Since money is typically rounded to the nearest cent, round \(112.668\) to \$112.67.
Key Concepts
Interest RatePrincipal AmountAnnual CompoundingInterest Calculation
Interest Rate
When you hear the term "interest rate," it means the percentage at which your investment grows over time. It's expressed as a percentage like 6%.
Mathematically, in compound interest calculations, the rate appears as a decimal. So for 6%, you write it as 0.06. This rate indicates how much interest will be added to the principal over a given period. The interest rate affects how fast your money grows.
Here’s a fun fact: Higher interest rates mean your money grows faster. But remember, when borrowing money, high rates mean higher costs for you! Look closely at rates before investing or borrowing money.
Mathematically, in compound interest calculations, the rate appears as a decimal. So for 6%, you write it as 0.06. This rate indicates how much interest will be added to the principal over a given period. The interest rate affects how fast your money grows.
Here’s a fun fact: Higher interest rates mean your money grows faster. But remember, when borrowing money, high rates mean higher costs for you! Look closely at rates before investing or borrowing money.
Principal Amount
The principal amount is the original sum of money placed in an investment or deposited to earn interest.
In the formula for compound interest, it is represented by the letter \(P\).
This is your starting point of growth: the amount from which interest is calculated. Without a principal, there would be nothing to earn interest on!
If you invest more as a principal, then you have the potential to earn more interest in the future, thanks to how compound interest works.
The principal is crucial; it’s your seed that will grow larger with added interest over time.
In the formula for compound interest, it is represented by the letter \(P\).
This is your starting point of growth: the amount from which interest is calculated. Without a principal, there would be nothing to earn interest on!
If you invest more as a principal, then you have the potential to earn more interest in the future, thanks to how compound interest works.
The principal is crucial; it’s your seed that will grow larger with added interest over time.
Annual Compounding
Annual compounding refers to how often interest is added to your investment's balance within a year.
With annual compounding, the \(n\) in the compound interest formula represents how many times interest compounds in a year.
If interest is added more frequently, like quarterly, bi-annually, or monthly, the total amount can grow faster as interest compounds on previously earned interest.
In our exercise, with quarterly compounding, the interest compounds four times a year. This increases the total amount because each time the interest is added, it becomes part of the principal for the next calculation.
With annual compounding, the \(n\) in the compound interest formula represents how many times interest compounds in a year.
If interest is added more frequently, like quarterly, bi-annually, or monthly, the total amount can grow faster as interest compounds on previously earned interest.
In our exercise, with quarterly compounding, the interest compounds four times a year. This increases the total amount because each time the interest is added, it becomes part of the principal for the next calculation.
Interest Calculation
Interest calculation can seem tricky at first, but it's really about using all the pieces of the formula together.
Using the compound interest formula \(A(t)=P\left(1+\frac{r}{n}\right)^{nt}\):
To find this, plug in your known values and solve. As you do this, notice how each step builds upon the last: the rate’s influence on growth, how often it compounds, and how they all affect the final amount.
It’s just puzzle pieces fitting together to show you how your investment grows!
Using the compound interest formula \(A(t)=P\left(1+\frac{r}{n}\right)^{nt}\):
- \(P\) is the principal, or initial amount of money.
- \(r\) is the interest rate as a decimal.
- \(n\) is the number of times interest is compounded annually.
- \(t\) is the total time in years.
To find this, plug in your known values and solve. As you do this, notice how each step builds upon the last: the rate’s influence on growth, how often it compounds, and how they all affect the final amount.
It’s just puzzle pieces fitting together to show you how your investment grows!
Other exercises in this chapter
Problem 4
Find the solution of the exponential equation, rounded to four decimal places. \(10^{-x}=4\)
View solution Problem 4
(a) We can expand \(\log \left(\frac{x^{2} y}{z}\right)\) to get _____. (b) We can combine \(2 \log x+\log y-\log z\) to get _____.
View solution Problem 5
Find the solution of the exponential equation, rounded to four decimal places. \(e^{-2 x}=7\)
View solution Problem 5
Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$ f(x)=4^{x} ; \quad f(0.5), f(\sqrt{2}), f(-\pi), f\l
View solution