Question

You decide to put $2,000 in a savings account to save for a $3,000 downpayment on a new car. If the account has an interest rate of 4% per year and is compounded monthly, how long does it take until you have $3,000 without depositing any additional funds?
121.862 years
12.1862 years
10.155 years
1.0155 years

Answer 1

It takes 10.155 years until you have $3,000 ⇒ 3rd

Step-by-step explanation:

The formula for compound interest, including principal sum is:

$A=P(1+\frac{r}{n})^{nt}$ , where

• A is the future value of the investment/loan, including interest
• P is the principal investment amount  
• r is the annual interest rate (decimal)
• n is the number of times that interest is compounded per unit t
• t is the time the money is invested or borrowed for

∵ You decide to put $2,000 in a savings account

∴ P = 2000

∵ You want to save for $3,000

∴ A = 3000

∵ The account has an interest rate of 4% per year and is

   compounded monthly

∴ r = 4% = 4 ÷ 100 = 0.04

∴ n = 12 ⇒ compounded monthly

- Substitute all of these values in the formula above to find t

∵ $3000=2000(1+\frac{0.04}{12})^{12t}$

- Divide both sides by 2000

∴ $1.5=(1+\frac{1}{300})^{12t}$

∴ $1.5=(1\frac{1}{300})^{12t}$

- Change the mixed number to an improper fraction

∴ $(1.5)=(\frac{301}{300})^{12t}$

- Insert ㏒ for both sides

∴ $log(1.5)=log(\frac{301}{300})^{12t}$

- Remember $log(a)^{n}=nlog(a)$

∴ $log(1.5)=(12t)log(\frac{301}{300})$

- Divide both sides by $log(\frac{301}{300})$

∴ 121.84 = 12 t

- Divide both sides by 12

∴ 10.155 = t

It takes 10.155 years until you have $3,000

Learn more:

You can learn more about the compound interest in brainly.com/question/4361464

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