Which expression represents how much money you will have in one year if $6 \%$ is compounded quarterly on a $\$800$ for one year?
A. $(800 \times 0.06 \times 3)+800$
B. $800 \times 0.06 \times 1 / 4$
C. $(800 \times 0.06 \times 1 / 4)+800$
D. $800 \times 0.06 \times 3
Answer (2)
The problem involves calculating the future value of an investment with compound interest.
The compound interest formula is A = P ( 1 + n r ) n t , where P = 800 , r = 0.06 , n = 4 , and t = 1 .
Substituting the values, the correct expression is 800 ( 1 + 4 0.06 ) 4 = 800 ( 1.015 ) 4 .
None of the provided options match the correct compound interest calculation, indicating a possible error in the options.
Explanation
Understanding the Problem We are asked to find the expression that represents the amount of money you will have in one year if 6% is compounded quarterly on a $800 for one year. This is a compound interest problem.
Recalling the Compound Interest Formula The formula for compound interest is: A = P ( 1 + n r ) n t where:
A is the final amount
P is the principal amount (initial investment)
r is the annual interest rate (as a decimal)
n is the number of times that interest is compounded per year
t is the number of years
Identifying the Given Values In this problem:
P = 800
r = 0.06 (6% as a decimal)
n = 4 (compounded quarterly, which means 4 times per year)
t = 1 (for one year)
Applying the Formula Substituting these values into the formula, we get: A = 800 ( 1 + 4 0.06 ) 4 × 1 A = 800 ( 1 + 0.015 ) 4 A = 800 ( 1.015 ) 4
Analyzing the Options Now let's analyze the given options:
( 800 × 0.06 × 3 ) + 800 : This represents simple interest calculated three times plus the principal, which is incorrect.
800 × 0.06 × 1/4 : This calculates the interest for one quarter only, which is incorrect.
( 800 × 0.06 × 1/4 ) + 800 : This calculates the amount after one quarter, not one year, which is incorrect.
800 × 0.06 × 3 : This calculates simple interest for a portion of the year, which is incorrect.
Further Analysis and Conclusion The correct expression should be 800 ( 1.015 ) 4 , which is not among the options. However, we can approximate ( 1.015 ) 4 :
( 1.015 ) 4 ≈ 1.06136 . Therefore, 800 ( 1.015 ) 4 ≈ 800 × 1.06136 ≈ 849.09 . None of the options directly represent the correct compound interest calculation. However, let's examine the options more closely to see if any can be manipulated to resemble the correct formula.
Since the correct formula is 800 ( 1 + 4 0.06 ) 4 , let's look at the option that involves adding to 800: ( 800 × 0.06 × 1/4 ) + 800 . This represents the amount after the first quarter. It does not account for compounding over the entire year.
Final Answer Based on the compound interest formula, none of the provided options accurately represent the amount of money you will have in one year. The correct expression should be: 800 × ( 1 + 4 0.06 ) 4 = 800 × ( 1.015 ) 4 Since this expression is not available, we must conclude that there might be an error in the provided options.
Examples
Compound interest is a powerful tool for growing wealth over time. For example, if you invest $1000 in a retirement account that earns 7% interest compounded annually, after 30 years, your investment could grow to approximately $7,612. This demonstrates the long-term benefits of compound interest in financial planning and investment strategies.
To find the amount after one year with 6% interest compounded quarterly on $800, we use the formula for compound interest, yielding approximately $849.09. None of the provided answer choices correctly represent the calculation. Hence, the correct expression is not listed in the options.
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