The value of a savings account, in dollars, $V(r)$, at the end of 2 years is represented by the interest, expressed as a decimal. What is the value of $V(r)$ for $r=0.03$?
$501.06
$530.45
$500.45
$509.00
Answer (2)
Assume an initial deposit of P and consider simple and compound interest formulas.
Test the answer choices with both simple and compound interest.
With compound interest, V ( 0.03 ) = P ( 1 + 0.03 ) 2 = P ( 1.0609 ) .
If V ( 0.03 ) = 530.45 , then P = 530.45/1.0609 = 500 , and V ( 0.03 ) = 500 ( 1.03 ) 2 = 530.45 . The final answer is 530.45 .
Explanation
Understanding the Problem We are given a problem where the value of a savings account, V ( r ) , at the end of 2 years depends on the interest rate r , expressed as a decimal. We need to find the value of V ( r ) when r = 0.03 . The possible answers are $501.06, $530.45, $500.45, 509.00. S in ce t h e p ro b l e m d oes n o tp ro v i d e an e x pl i c i t f or m u l a f or V(r) , w e m u s t d e d u ce i t f ro m t h e an s w erc h o i ces . L e t ′ s a ss u m e t h e ini t ia l d e p os i t i s P . A f t er 2 ye a rs w i t hanin t eres t r a t e r , t h e v a l u eo f t h e a cco u n t i s V(r)$. We can consider two common types of interest: simple interest and compound interest.
Possible Interest Types Simple Interest: V ( r ) = P ( 1 + 2 r ) .
Compound Interest: V ( r ) = P ( 1 + r ) 2 . We can test each of the answer choices to see if we can determine P and whether the interest is simple or compound.
Checking Answer Choices If we assume simple interest, V ( 0.03 ) = P ( 1 + 2 ( 0.03 )) = P ( 1.06 ) . If V ( 0.03 ) = 530.45 , then P = 530.45/1.06 = 500.4245... . This is close to 500. If we assume compound interest, V ( 0.03 ) = P ( 1 + 0.03 ) 2 = P ( 1.03 ) 2 = P ( 1.0609 ) . If V ( 0.03 ) = 530.45 , then P = 530.45/1.0609 = 500 .
Final Calculation If P = 500 and compound interest is used, then V ( 0.03 ) = 500 ( 1.03 ) 2 = 500 ( 1.0609 ) = 530.45 . Therefore, the value of the savings account V ( 0.03 ) is $530.45.
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing how your savings will grow over time with compound interest helps you estimate your future wealth. Similarly, when taking out a loan, understanding compound interest allows you to calculate the total cost of borrowing, including the principal and accumulated interest. This knowledge empowers you to make sound choices about investments, loans, and other financial products, ensuring you can achieve your long-term financial goals.
The value of the savings account V ( 0.03 ) , at an interest rate of 0.03 after 2 years, is found to be $530.45. This is confirmed by using the compound interest formula. Therefore, the correct answer is $530.45.
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