Solve for the unknown number of years in each of the following: (Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 8,732.16)

Present Value	Years	Interest Rate	Future Value
900	
12%	1,755
2,591	
10%	4,350
34,105	
15%	393,620
33,800	
22%	217,868

To solve for the unknown number of years in each case, we will use the formula for compound interest:

\[
FV = PV \times (1 + r)^n
\]
Where:
- \(FV\) is the future value,
- \(PV\) is the present value,
- \(r\) is the interest rate per period (in decimal form),
- \(n\) is the number of periods (years in this case).

We need to solve for \(n\), the number of years. Rearranging the formula to solve for \(n\):

\[
n = \frac{\log(FV / PV)}{\log(1 + r)}
\]

Let’s apply this to each set of values provided.

### 1. For \( PV = 900 \), \( FV = 1,755 \), and \( r = 12\% = 0.12 \):
\[
n = \frac{\log(1,755 / 900)}{\log(1 + 0.12)}
\]

### 2. For \( PV = 2,591 \), \( FV = 4,350 \), and \( r = 10\% = 0.10 \):
\[
n = \frac{\log(4,350 / 2,591)}{\log(1 + 0.10)}
\]

### 3. For \( PV = 34,105 \), \( FV = 393,620 \), and \( r = 15\% = 0.15 \):
\[
n = \frac{\log(393,620 / 34,105)}{\log(1 + 0.15)}
\]

### 4. For \( PV = 33,800 \), \( FV = 217,868 \), and \( r = 22\% = 0.22 \):
\[
n = \frac{\log(217,868 / 33,800)}{\log(1 + 0.22)}
\]

Let me now calculate these values.The number of years for each case is:

1. For \( PV = 900 \), \( FV = 1,755 \), and \( r = 12\% \), the time is **5.89 years**.
2. For \( PV = 2,591 \), \( FV = 4,350 \), and \( r = 10\% \), the time is **5.44 years**.
3. For \( PV = 34,105 \), \( FV = 393,620 \), and \( r = 15\% \), the time is **17.50 years**.
4. For \( PV = 33,800 \), \( FV = 217,868 \), and \( r = 22\% \), the time is **9.37 years**.

Would you like further details on any of these calculations?

Here are 5 related questions you can explore:
1. How does changing the interest rate affect the number of years needed?
2. What happens if the present value is increased while keeping other factors constant?
3. Can the formula for compound interest be applied to non-annual periods?
4. What is the difference between compound and simple interest in time calculation?
5. How does inflation impact the future value in real-world scenarios?

**Tip:** Remember, the higher the interest rate, the fewer years it takes to reach a given future value.

Solve for the unknown number of years in each of the following: (Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 8,732.16) Present Value, Years, Interest Rate, Future Value: 900, 12%, 1,755; 2,591, 10%, 4,350; 34,105, 15%, 393,620; 33,800, 22%, 217,868.

Learn how to solve for the unknown number of years using the compound interest formula with step-by-step examples. This problem covers key mathematical concepts such as logarithms and exponential growth, suitable for students in Grades 10-12. Examples include solving for time using interest rates of 12%, 10%, 15%, and 22%.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation