To find the simple interest and the future value for a principal amount of [tex]$20,000$[/tex], an annual interest rate of 14%, and a time period of 3 months, you can follow these steps:
### Step 1: Convert the annual interest rate to a decimal
The annual interest rate provided is 14%. To convert this to a decimal, divide 14 by 100:
[tex]\[ \text{rate} = \frac{14}{100} = 0.14 \][/tex]
### Step 2: Convert the time period from months to years
Since interest rates are usually annual, we need to convert the given time period from months to years. There are 12 months in a year, so:
[tex]\[ \text{time in years} = \frac{3 \text{ months}}{12 \text{ months/year}} = 0.25 \text{ years} \][/tex]
### Step 3: Calculate the simple interest
Using the simple interest formula [tex]\( I = P \times r \times t \)[/tex], where [tex]\( P \)[/tex] is the principal amount, [tex]\( r \)[/tex] is the rate, and [tex]\( t \)[/tex] is the time in years:
[tex]\[ I = 20,000 \times 0.14 \times 0.25 \][/tex]
### Step 4: Perform the multiplication to find the simple interest
Multiply the principal amount by the rate and then by the time:
[tex]\[ I = 20,000 \times 0.14 \times 0.25 \][/tex]
[tex]\[ I = 20,000 \times 0.035 \][/tex]
[tex]\[ I = 700.0000000000001 \][/tex]
So, the simple interest is:
[tex]\[ I = \$ 700.00 \][/tex]
### Step 5: Calculate the future value
The future value is the sum of the principal amount and the simple interest. Add the calculated simple interest to the principal amount:
[tex]\[ \text{Future value} = \text{Principal} + \text{Simple interest} \][/tex]
[tex]\[ \text{Future value} = 20,000 + 700.0000000000001 \][/tex]
[tex]\[ \text{Future value} = 20,700.00 \][/tex]
Thus, the future value is:
[tex]\[ \text{Future value} = \$ 20,700.00 \][/tex]
### Summary
- The simple interest is [tex]\(\$ 700.00\)[/tex].
- The future value is [tex]\(\$ 20,700.00\)[/tex].
