Question

You decide to put $175 in a savings account to save for a $3,000 down payment on a new car. If the account has an interest rate of 3% per year and is compounded monthly, how long does it take you to earn $3,000 without depositing any additional funds?.
8.0111 years.
96.1332 years.
93.2914 years.
94.8377 years

Answer 1

3 % = 0.003 : 12 = 0.0025
175 · ( 1 + 0.0025 )^(12 x) = 3,000
175 · (1.0025)^(12 x) = 3,000
1.0025^(12 x ) = 17.142857
$12 x = log _{1.0025} 17.142857$
12 x = 1138.0524
x = 1138.0524 : 12
x = 94.8377
Answer: D ) 94.8377 years

Answer 2

The correct answer is:

94.8377 years

Explanation:

The formula for compound interest is

$A=p(1+\frac{r}{n})^{nt}$, where A is the total amount, p is the amount of principal deposited, r is the interest rate as a decimal number, n is the number of times per year the interest is compounded, and t is the number of years.

For our problem, A = 3000; p = 175; r = 3% = 3/100 = 0.03; n = 12; and t is unknown:

$3000=175(1+\frac{0.03}{12})^{12t} \\3000=175(1+0.0025)^{12t} \\3000=175(1.0025)^{12t}$

Divide both sides by 175:

$\frac{3000}{175}=\frac{175(1.0025)^{12t}}{175} \\ \\\frac{120}{7}=1.0025^{12t}$

We will use logarithms to solve this. The base of the exponent is 1.0025, so this will be the base of the log:

$\log_{1.0025}(\frac{120}{7})=12t$

Divide both sides by 12:

$\frac{\log_{1.0025}(\frac{120}{7})}{12}=\frac{12t}{12} \\ \\94.8377$