To save for a car when he turns 18, Pascale deposited $500 each year into a savings account with a 7.5% interest rate compounded annually.
Year | Beginning Balance | Interest Earned | Ending Balance
---|---|---|---
1 | $500.00 | $37.50 | $537.50
2 | $1,037.50 | $77.81 | $1,115.31
3 | $1,615.31 | $121.15 | $1,736.46
4 | $2,236.46 | $167.73 | $2,404.19
5 | | |
Using the formula [tex]$A=P(1+r)^t$[/tex], what is the value of the account at the end of the fifth year?
A. $3,071.92
B. $3,122.00
C. $3,851.77
D. $4,140.65
Answer (2)
Calculate the beginning balance for year 5: 2404.19 + 500 = 2904.19 .
Calculate the interest earned in year 5: 2904.19 × 0.075 = 217.81 .
Calculate the ending balance for year 5: 2904.19 + 217.81 = 3122.00 .
The value of the account at the end of the fifth year is $3 , 122.00 .
Explanation
Understanding the Problem We are asked to find the value of the account at the end of the fifth year, given that Pascale deposits $500 each year into a savings account with a 7.5% interest rate compounded annually. We are given a table showing the beginning balance, interest earned, and ending balance for the first four years. We need to complete the table for the fifth year.
Calculating Beginning Balance for Year 5 To find the beginning balance for year 5, we add the deposit of $500 to the ending balance of year 4. From the table, the ending balance of year 4 is 2404.19. S o , t h e b e g innin g ba l an ce f orye a r 5 i s : 2404.19 + 500 = 2904.19 $
Calculating Interest Earned in Year 5 Next, we calculate the interest earned in year 5 by multiplying the beginning balance of year 5 by the interest rate (7.5% or 0.075): 2904.19 × 0.075 = 217.81425 Rounding to two decimal places, the interest earned is $217.81.
Calculating Ending Balance for Year 5 Finally, we calculate the ending balance for year 5 by adding the beginning balance of year 5 and the interest earned in year 5: 2904.19 + 217.81 = 3122.00
Final Answer Therefore, the value of the account at the end of the fifth year is $3122.00.
Examples
Saving money with compound interest is like planting a tree. Each year, your tree grows, and the next year, it grows even more because it's growing off a bigger base. Similarly, with compound interest, you earn interest not only on your initial deposit but also on the interest you've already earned. This problem demonstrates how consistent savings combined with compound interest can help you reach your financial goals, like saving for a car or a down payment on a house. Understanding this principle encourages you to start saving early and consistently to maximize the benefits of compound interest over time.
The value of Pascale's savings account at the end of the fifth year is $3,122.00. This was calculated by determining the beginning balance, calculating the interest earned, and then finding the ending balance. The correct option is B. $3,122.00.
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