Answer :
Sure, let's solve this question step-by-step.
1. Understand the Problem:
- You start with a principal amount of [tex]$100,000. - The interest rate is 8% per year. - We want to find out how long it will take for this amount to double. 2. Identify the Formula: - For simple interest, the formula to find the amount after a certain time is: \[ A = P(1 + rt) \] where: - \(A\) is the final amount, - \(P\) is the principal amount (initial investment), - \(r\) is the interest rate (expressed as a decimal), - \(t\) is the time in years. 3. Rearrange the Formula to Solve for Time (t): - We know that we want to double our money, so \(A = 2P\). - Substitute \(A\) into the formula: \[ 2P = P(1 + rt) \] - Simplify the equation by dividing both sides by \(P\): \[ 2 = 1 + rt \] - Rearrange to solve for \(t\): \[ t = \frac{2 - 1}{r} \] \[ t = \frac{1}{r} \] 4. Substitute the Interest Rate: - The interest rate is 8%, which is 0.08 as a decimal. - Substitute \(r = 0.08\) into the equation: \[ t = \frac{1}{0.08} \] 5. Calculate \(t\): - Perform the division: \[ t = \frac{1}{0.08} = 12.5 \text{ years} \] 6. Final Answer: - It will take 12.5 years for $[/tex]100,000 to double at a simple interest rate of 8%.
So, the detailed solution shows that it will take 12.5 years for your principal of \$100,000 to double at an 8% simple interest rate.
1. Understand the Problem:
- You start with a principal amount of [tex]$100,000. - The interest rate is 8% per year. - We want to find out how long it will take for this amount to double. 2. Identify the Formula: - For simple interest, the formula to find the amount after a certain time is: \[ A = P(1 + rt) \] where: - \(A\) is the final amount, - \(P\) is the principal amount (initial investment), - \(r\) is the interest rate (expressed as a decimal), - \(t\) is the time in years. 3. Rearrange the Formula to Solve for Time (t): - We know that we want to double our money, so \(A = 2P\). - Substitute \(A\) into the formula: \[ 2P = P(1 + rt) \] - Simplify the equation by dividing both sides by \(P\): \[ 2 = 1 + rt \] - Rearrange to solve for \(t\): \[ t = \frac{2 - 1}{r} \] \[ t = \frac{1}{r} \] 4. Substitute the Interest Rate: - The interest rate is 8%, which is 0.08 as a decimal. - Substitute \(r = 0.08\) into the equation: \[ t = \frac{1}{0.08} \] 5. Calculate \(t\): - Perform the division: \[ t = \frac{1}{0.08} = 12.5 \text{ years} \] 6. Final Answer: - It will take 12.5 years for $[/tex]100,000 to double at a simple interest rate of 8%.
So, the detailed solution shows that it will take 12.5 years for your principal of \$100,000 to double at an 8% simple interest rate.
