It takes 10.155 years until you have $3,000 ⇒ 3rd
Step-by-step explanation:
The formula for compound interest, including principal sum is:
, 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}](https://tex.z-dn.net/?f=3000%3D2000%281%2B%5Cfrac%7B0.04%7D%7B12%7D%29%5E%7B12t%7D)
- Divide both sides by 2000
∴ ![1.5=(1+\frac{1}{300})^{12t}](https://tex.z-dn.net/?f=1.5%3D%281%2B%5Cfrac%7B1%7D%7B300%7D%29%5E%7B12t%7D)
∴ ![1.5=(1\frac{1}{300})^{12t}](https://tex.z-dn.net/?f=1.5%3D%281%5Cfrac%7B1%7D%7B300%7D%29%5E%7B12t%7D)
- Change the mixed number to an improper fraction
∴ ![(1.5)=(\frac{301}{300})^{12t}](https://tex.z-dn.net/?f=%281.5%29%3D%28%5Cfrac%7B301%7D%7B300%7D%29%5E%7B12t%7D)
- Insert ㏒ for both sides
∴ ![log(1.5)=log(\frac{301}{300})^{12t}](https://tex.z-dn.net/?f=log%281.5%29%3Dlog%28%5Cfrac%7B301%7D%7B300%7D%29%5E%7B12t%7D)
- Remember ![log(a)^{n}=nlog(a)](https://tex.z-dn.net/?f=log%28a%29%5E%7Bn%7D%3Dnlog%28a%29)
∴ ![log(1.5)=(12t)log(\frac{301}{300})](https://tex.z-dn.net/?f=log%281.5%29%3D%2812t%29log%28%5Cfrac%7B301%7D%7B300%7D%29)
- Divide both sides by ![log(\frac{301}{300})](https://tex.z-dn.net/?f=log%28%5Cfrac%7B301%7D%7B300%7D%29)
∴ 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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