Korey is planning to open a comic book store. In his first year of operation, Korey expects to average $1,000 of profit each month. He then expects profits to increase by 6% each year for the next 4 years. How much does Korey expect to make in profits in his fifth year of operation?
[tex]$A=P(1+r)^t$[/tex]
a. $15,149.72
b. $16,058.71
c. $31,120,46
d. $32,987.69
Answer (2)
Korey expects to make approximately $15,149.72 in profits in his fifth year of operation. This amount is determined by calculating the profit after applying a 6% increase each year for four years, starting from an initial annual profit of $12,000. Therefore, the correct choice is (a) $15,149.72.
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Calculate the profit in the first year: 1 , 000/ m o n t h × 12 m o n t h s = $12 , 000 .
Apply the compound interest formula: A = P ( 1 + r ) t , where P = 12000 , r = 0.06 , and t = 4 .
Calculate the profit in the fifth year: A = 12000 ( 1.06 ) 4 = 15149.72352 .
The expected profit in the fifth year is approximately $15 , 149.72 .
Explanation
Calculate the first year's profit Korey expects to average $1,000 of profit each month in his first year. This means his total profit for the first year is $1,000/month * 12 months = $12,000.
Recognize the problem type He expects his profits to increase by 6% each year for the next 4 years. We want to find his profit in the fifth year. This is a compound interest problem, where the initial profit is $12,000, the interest rate is 6%, and the number of years is 4.
State the formula The formula for compound interest is A = P ( 1 + r ) t , where:
A is the amount of profit after t years
P is the initial profit
r is the interest rate (as a decimal)
t is the number of years
Plug in the values In this case, P = 12000 , r = 0.06 , and t = 4 . Plugging these values into the formula, we get: A = 12000 ( 1 + 0.06 ) 4 A = 12000 ( 1.06 ) 4
Calculate the final profit Calculating this value: A = 12000 ∗ ( 1.06 ) 4 A = 12000 ∗ 1.26247696 $A = 15149.72352
State the final answer Therefore, Korey expects to make approximately $15,149.72 in profits in his fifth year of operation.
Examples
Understanding compound growth is useful in many real-life situations. For example, when you invest money in a savings account or the stock market, the returns often grow at a certain percentage each year. Knowing how to calculate compound growth helps you estimate how much your investments will be worth in the future. This is also applicable when calculating population growth or even the spread of information through social networks, where the rate of spread can be modeled as a percentage increase over time.
