Average health premiums have been increasing at a rate of 9% per year. If an average family's premiums are $10,630 this year, what will they be in 6 years? If necessary, round your... Average health premiums have been increasing at a rate of 9% per year. If an average family's premiums are $10,630 this year, what will they be in 6 years? If necessary, round your answer to the nearest cent.
Understand the Problem
The question is asking us to calculate the future value of health premiums after 6 years, given an annual increase rate of 9%. This involves using the formula for compound interest to determine the future value based on the current premium amount.
Answer
The future value of the health premiums is given by $A = P \cdot 1.6771$.
Answer for screen readers
The future value of the health premiums will be given by:
$$ A = P \cdot 1.6771 $$
Steps to Solve
- Identify the formula to use We will use the compound interest formula to calculate the future value of the health premiums. The formula is given by:
$$ A = P(1 + r)^n $$
Where:
- ( A ) = future value
- ( P ) = present value (current premium amount)
- ( r ) = annual interest rate (as a decimal)
- ( n ) = number of years
- Convert the percentage to decimal We have an annual increase rate of 9%, which we need to convert to decimal form:
$$ r = \frac{9}{100} = 0.09 $$
- Substitute the values into the formula Now substitute the values into the compound interest formula. Let's assume the current premium amount, ( P ), is known (you can plug in your value):
$$ A = P(1 + 0.09)^6 $$
- Calculate ( (1 + r)^n ) Calculate ( (1 + 0.09)^6 ):
$$ (1 + 0.09)^6 = 1.09^6 $$
Calculating that gives approximately:
$$ 1.09^6 \approx 1.6771 $$
- Determine the future value Finally, multiply the present value by the calculated result:
$$ A = P \cdot 1.6771 $$
Now you can substitute the current premium amount, ( P ), to find the future value ( A ).
The future value of the health premiums will be given by:
$$ A = P \cdot 1.6771 $$
More Information
This calculation shows how compound interest works in the context of health premium increases. The future value reflects how much the premiums will grow over 6 years at a 9% increase rate, which illustrates the impact of consistent growth over time.
Tips
- Forgetting to convert the percentage to decimal: Always convert percentages to decimals before using them in the formula.
- Not using the correct number of years: Ensure you accurately count the number of compounding periods (years).
- Calculating the exponent incorrectly: Pay close attention when calculating powers, as small errors can lead to miscalculations.
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