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Answer :
To solve the question of how much money will be in Josiah's account after a certain number of years, we need to understand how interest compounds annually. Here’s a step-by-step guide to solving this problem:
1. Understanding the Initial Information:
- Josiah invests $360 in an account.
- The account accrues interest at a rate of 3% per year.
2. Converting Percentage to Decimal:
- The interest rate of 3% can be written as a decimal by dividing by 100:
[tex]\[
3\% = 0.03
\][/tex]
3. Formulating the Compound Interest Formula:
- Compound interest grows the principal amount by a certain percentage every year. The formula for the amount of money in an account with 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 (in decimal),
- [tex]\(x\)[/tex] is the number of years.
4. Applying the Values to the Formula:
- For Josiah:
- The principal [tex]\(P = 360\)[/tex],
- The interest rate [tex]\(r = 0.03\)[/tex],
- Plug these values into the formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
- Simplifying, we get:
[tex]\[
y = 360(1.03)^x
\][/tex]
5. Verifying the Correct Equation:
- Among the options given, the equation that correctly represents the amount of money in Josiah’s account after [tex]\(x\)[/tex] years is:
[tex]\[
y = 360(1.03)^x
\][/tex]
This is how we determine the correct equation that represents how Josiah’s investment will grow with annual compounding interest.
1. Understanding the Initial Information:
- Josiah invests $360 in an account.
- The account accrues interest at a rate of 3% per year.
2. Converting Percentage to Decimal:
- The interest rate of 3% can be written as a decimal by dividing by 100:
[tex]\[
3\% = 0.03
\][/tex]
3. Formulating the Compound Interest Formula:
- Compound interest grows the principal amount by a certain percentage every year. The formula for the amount of money in an account with 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 (in decimal),
- [tex]\(x\)[/tex] is the number of years.
4. Applying the Values to the Formula:
- For Josiah:
- The principal [tex]\(P = 360\)[/tex],
- The interest rate [tex]\(r = 0.03\)[/tex],
- Plug these values into the formula:
[tex]\[
y = 360(1 + 0.03)^x
\][/tex]
- Simplifying, we get:
[tex]\[
y = 360(1.03)^x
\][/tex]
5. Verifying the Correct Equation:
- Among the options given, the equation that correctly represents the amount of money in Josiah’s account after [tex]\(x\)[/tex] years is:
[tex]\[
y = 360(1.03)^x
\][/tex]
This is how we determine the correct equation that represents how Josiah’s investment will grow with annual compounding interest.
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