Questions: If 26,000 is invested in an account for 20 years. Calculate the total interest earned at the end of 20 years if the interest is: (a) 8% simple interest: 41,600 (b) 8% compounded annually: 95,184.89 (c) 8% compounded quarterly: 100,761.42 (d) 8% compounded monthly: Round your answers to the nearest cent.

Solution Steps

To solve the problem, we need to calculate the total interest earned for different types of interest rates.

1. For simple interest, use the formula \( I = P \times r \times t \).
2. For compound interest, use the formula \( A = P \times (1 + \frac{r}{n})^{n \times t} \) and then subtract the principal \( P \) to get the interest.

Given:

• Principal \( P = 26000 \)
• Rate \( r = 0.08 \)
• Time \( t = 20 \) years

We will calculate: (a) Simple interest (b) Compound interest compounded annually (c) Compound interest compounded quarterly (d) Compound interest compounded monthly

The formula for simple interest is given by: \[ I = P \times r \times t \] where:

• \( P = 26000 \)
• \( r = 0.08 \)
• \( t = 20 \)

Substituting the values, we get: \[ I = 26000 \times 0.08 \times 20 = 41600 \]

The formula for compound interest is given by: \[ A = P \times \left(1 + \frac{r}{n}\right)^{n \times t} \] where:

• \( P = 26000 \)
• \( r = 0.08 \)
• \( t = 20 \)
• \( n = 1 \) (compounded annually)

Substituting the values, we get: \[ A = 26000 \times \left(1 + \frac{0.08}{1}\right)^{1 \times 20} = 121184.8857 \]

The compound interest is: \[ I = A - P = 121184.8857 - 26000 = 95184.8857 \]

Using the same formula for compound interest with:

• \( n = 4 \) (compounded quarterly)

Substituting the values, we get: \[ A = 26000 \times \left(1 + \frac{0.08}{4}\right)^{4 \times 20} = 126761.4181 \]

The compound interest is: \[ I = A - P = 126761.4181 - 26000 = 100761.4181 \]

Using the same formula for compound interest with:

• \( n = 12 \) (compounded monthly)

Substituting the values, we get: \[ A = 26000 \times \left(1 + \frac{0.08}{12}\right)^{12 \times 20} = 128096.8720 \]

The compound interest is: \[ I = A - P = 128096.8720 - 26000 = 102096.8720 \]

Final Answer