Question

Ella is planning to buy a home in six years. She'll need to make a down payment of $20,000. She has invested $12,000 in an account earning 5% interest, compounded monthly. Ella have enough money saved in six years to buy her home. It will take just over years for Ella's account balance to exceed $20,000.

Answer 1

Answer:

Ella will not enough money saved in 6 years.

It will take 11 years for Ella's account balance to exceed $20,000.

Step-by-step explanation:

We have been given that Ella i s planning to buy a home in six years. She'll need to make a down payment of $20,000. She has invested $12,000 in an account earning 5% interest, compounded monthly. We are supposed to find whether Ella will have enough money saved in six years to buy her home.

We will use compound interest formula to solve our given problem.

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

A = Final amount after t years,

P = Principal amount,

r = Annual interest rate in decimal form,

n = Number of times interest is compounded per year.

$5\%=\frac{5}{100}=0.05$

Upon substituting our given values in compound interest formula, we will get:

$20000=12000(1+\frac{0.05}{12})^{12t}$

Let us solve for t.

$\frac{20000}{12000}=\frac{12000(1+\frac{0.05}{12})^{12t}}{12000}$

$1.666666=(1+0.004166666)^{12t}$

$1.666666=(1.004166666)^{12t}$

Now, we will take natural log of both sides:

$\text{ln}(1.666666)=\text{ln}((1.004166666)^{12t})$

$\text{ln}(1.666666)=12t\cdot \text{ln}(1.004166666)$

$0.5108256237659907=12t\cdot 0.0041580094847633$

$0.5108256237659907=0.0498961138171596t$

$t=\frac{0.5108256237659907}{0.0498961138171596}$

$t=10.23778376$

Since it will take approximately 10.23 years to have an amount of $20,000, therefore, Ella will not enough money saved in 6 years and it will take 11 years for Ella's account balance to exceed $20,000.