Problem 38
Question
Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to solve these compound interest problems. Round to the nearest tenth. How long does it take for a \(\$ 1500\) investment to earn \(\$ 200\) interest if it is invested at \(10 \%\) interest compounded semiannually?
Step-by-Step Solution
Verified Answer
It takes approximately 1.25 years to earn $200 interest.
1Step 1: Understand the Problem
We need to determine the time it takes for an investment to earn a specified amount of interest. We are given these values: Principal amount \(P = \\(1500\), interest earned \(I = \\)200\), and interest rate \(r = 10\%\) compounded semiannually, i.e., twice a year. We need to find \(t\) in the equation \(A = P\left(1+\frac{r}{n}\right)^{nt}\).
2Step 2: Calculate the Total Amount
The interest earned is \(\\(200\), so the total amount \(A = P + I = \\)1500 + \\(200 = \\)1700\). This will be substituted into the formula to solve for \(t\).
3Step 3: Set Up the Compound Interest Formula
The given formula is \(A = P\left(1+\frac{r}{n}\right)^{nt}\). Substitute \(A = 1700\), \(P = 1500\), \(r = 0.10\), and \(n = 2\) (since the interest is compounded semiannually). This step looks as follows:\[1700 = 1500\left(1 + \frac{0.10}{2}\right)^{2t}\]
4Step 4: Simplify the Expression
Calculate \(1 + \frac{0.10}{2} = 1.05\). Substituting this into the equation gives:\[1700 = 1500 \times 1.05^{2t}\]
5Step 5: Isolate the Exponential Term
Divide both sides by 1500 to isolate the exponential term:\[\frac{1700}{1500} = 1.05^{2t}\]Simplify the left side to get:\[1.1333 \approx 1.05^{2t}\]
6Step 6: Solve for Time \(t\) Using Logarithms
Take the logarithm of both sides to solve for \(t\):\[\log(1.1333) = 2t \cdot \log(1.05)\]Solve for \(t\):\[t = \frac{\log(1.1333)}{2 \cdot \log(1.05)}\]
7Step 7: Calculate the Value of \(t\)
Using a calculator, compute the logarithms and divide:\[t \approx \frac{0.0531}{2 \times 0.0212} \approx \frac{0.0531}{0.0424} \approx 1.25\]Therefore, it takes approximately 1.25 years to earn \$200 in interest.
Key Concepts
Interest RateExponential GrowthLogarithms
Interest Rate
An interest rate is the percentage at which interest is paid by borrowers for the use of money they borrow from a lender. In the context of compound interest, it plays a crucial role in determining how much an investment will grow over time.
When we talk about a 10% interest rate, it means that for every \(100, \)10 will be added to your initial amount every year. However, in our problem, the interest is compounded semiannually. This means the interest will be calculated twice a year at half the annual rate, i.e., 5% each time.
When we talk about a 10% interest rate, it means that for every \(100, \)10 will be added to your initial amount every year. However, in our problem, the interest is compounded semiannually. This means the interest will be calculated twice a year at half the annual rate, i.e., 5% each time.
- The principal is the original sum of money invested or loaned.
- The rate (\(r=0.10\) for 10%) indicates the percentage of the principal that is paid each year.
- Compounding frequency (\(n=2\)) refers to how often the applied interest is calculated.
Exponential Growth
Exponential growth is an essential concept when discussing compound interest. Unlike simple interest, where the interest is calculated solely on the principal original sum, compound interest grows exponentially. This means that each period, interest is earned not only on the initial principal but also on the accumulated interest from previous periods.
The formula for compound interest:\[A = P\left(1+\frac{r}{n}\right)^{nt}\]shows how your investment grows over time:- \(A\) denotes the total amount after time \(t\)- \(P\) is the principal amount- \(r\) is the annual interest rate (in decimal form)- \(n\) is the number of times the interest is compounded per year- \(t\) represents the time the money is invested for
The term \(\left(1+\frac{r}{n}\right)^{nt}\) illustrates the nature of exponential growth. With each compounding period, the growth rate of the interest becomes more significant, impacting your returns markedly over time. The difference might be subtle in the short term, but over years, this exponential nature makes a noticeable difference.
The formula for compound interest:\[A = P\left(1+\frac{r}{n}\right)^{nt}\]shows how your investment grows over time:- \(A\) denotes the total amount after time \(t\)- \(P\) is the principal amount- \(r\) is the annual interest rate (in decimal form)- \(n\) is the number of times the interest is compounded per year- \(t\) represents the time the money is invested for
The term \(\left(1+\frac{r}{n}\right)^{nt}\) illustrates the nature of exponential growth. With each compounding period, the growth rate of the interest becomes more significant, impacting your returns markedly over time. The difference might be subtle in the short term, but over years, this exponential nature makes a noticeable difference.
Logarithms
Logarithms are invaluable for solving equations where the unknown variable is in the exponent. In our step-by-step solution, we used logarithms to isolate and solve for the time variable \(t\) in the compound interest equation.
When dealing with compound interest problems, you often encounter equations where the total amount and base are known, but you need to find the time it takes (in this case, for the investment to grow by a certain amount). Here’s where logarithms come in handy:\[\log(A) = nt \cdot \log\left(1+\frac{r}{n}\right) \]Breaking this down, we simply transfer from exponential terms to a multiplication by using the properties of logarithms:
When dealing with compound interest problems, you often encounter equations where the total amount and base are known, but you need to find the time it takes (in this case, for the investment to grow by a certain amount). Here’s where logarithms come in handy:\[\log(A) = nt \cdot \log\left(1+\frac{r}{n}\right) \]Breaking this down, we simply transfer from exponential terms to a multiplication by using the properties of logarithms:
- \(\log(1.1333) = 2t \cdot \log(1.05)\) isolates \(t\) after simplification
- We divide to solve for \(t\): \(t = \frac{\log(1.1333)}{2 \cdot \log(1.05)}\)
Other exercises in this chapter
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