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Answer :
To solve the problem and find the correct equation representing the amount of money in Josiah's account after a certain number of years with an annual 3% interest rate, we need to use the compound interest formula. Here’s a step-by-step reasoning:
1. Understanding Compound Interest:
- The formula for compound interest is:
[tex]\[ y = P(1 + r)^x \][/tex]
- Where:
- [tex]\( y \)[/tex] is the amount of money after [tex]\( x \)[/tex] years,
- [tex]\( P \)[/tex] is the principal amount (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( x \)[/tex] is the number of years the money is invested.
2. Identify the Variables Given in the Problem:
- Principal (initial investment) [tex]\( P = 360 \)[/tex].
- Annual interest rate [tex]\( r = 3\% = 0.03 \)[/tex].
- Number of years is represented by [tex]\( x \)[/tex].
3. Form the Equation:
- Substitute the given values into the compound interest formula:
[tex]\[ y = 360(1 + 0.03)^x \][/tex]
4. Simplify the Formula:
- Calculate [tex]\( 1 + 0.03 = 1.03 \)[/tex].
- The formula becomes:
[tex]\[ y = 360(1.03)^x \][/tex]
5. Choose the Correct Answer:
- Given the options:
- [tex]\( y=360(1.3)^x \)[/tex] – Incorrect, because it uses a larger interest factor.
- [tex]\( y=360(0.3)^x \)[/tex] – Incorrect, because it uses a much smaller factor.
- [tex]\( y=360(0.03)^x \)[/tex] – Incorrect, because the interest should be added to 1.
- [tex]\( y=360(1.03)^x \)[/tex] – This option is correct.
Thus, the correct equation that represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years is:
[tex]\[ y = 360(1.03)^x \][/tex]
1. Understanding Compound Interest:
- The formula for compound interest is:
[tex]\[ y = P(1 + r)^x \][/tex]
- Where:
- [tex]\( y \)[/tex] is the amount of money after [tex]\( x \)[/tex] years,
- [tex]\( P \)[/tex] is the principal amount (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( x \)[/tex] is the number of years the money is invested.
2. Identify the Variables Given in the Problem:
- Principal (initial investment) [tex]\( P = 360 \)[/tex].
- Annual interest rate [tex]\( r = 3\% = 0.03 \)[/tex].
- Number of years is represented by [tex]\( x \)[/tex].
3. Form the Equation:
- Substitute the given values into the compound interest formula:
[tex]\[ y = 360(1 + 0.03)^x \][/tex]
4. Simplify the Formula:
- Calculate [tex]\( 1 + 0.03 = 1.03 \)[/tex].
- The formula becomes:
[tex]\[ y = 360(1.03)^x \][/tex]
5. Choose the Correct Answer:
- Given the options:
- [tex]\( y=360(1.3)^x \)[/tex] – Incorrect, because it uses a larger interest factor.
- [tex]\( y=360(0.3)^x \)[/tex] – Incorrect, because it uses a much smaller factor.
- [tex]\( y=360(0.03)^x \)[/tex] – Incorrect, because the interest should be added to 1.
- [tex]\( y=360(1.03)^x \)[/tex] – This option is correct.
Thus, the correct equation that represents the amount of money in Josiah's account after [tex]\( x \)[/tex] years is:
[tex]\[ y = 360(1.03)^x \][/tex]
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Rewritten by : Barada
