Questions: If 1000 is invested in an account earning 3% compounded monthly, how long will it take the account to grow in value to 1500? Round to the nearest month.
years, months
Transcript text: If $\$ 1000$ is invested in an account earning $3 \%$ compounded monthly, how long will it take the account to grow in value to $\$ 1500$ ? Round to the nearest month.
$\square$ years, $\square$ months
Solution
Solution Steps
To solve this problem, we need to use the formula for compound interest to find the time it takes for the investment to grow to a certain amount. The formula for compound interest is:
A=P(1+nr)nt
where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
r is the annual interest rate (decimal).
n is the number of times that interest is compounded per year.
t is the time the money is invested for in years.
We need to solve for t when A=1500, P=1000, r=0.03, and n=12.
Solution Approach
Use the compound interest formula to set up the equation.
Solve for t by isolating it on one side of the equation.
Use logarithms to solve for t.
Convert the time in years to months and round to the nearest month.
Step 1: Set Up the Compound Interest Formula
We start with the compound interest formula:
A=P(1+nr)nt
where:
A=1500 (the amount we want to reach),
P=1000 (the initial investment),
r=0.03 (the annual interest rate),
n=12 (the number of times interest is compounded per year).
Step 2: Solve for Time t
Rearranging the formula to solve for t:
t=n⋅log(1+nr)log(PA)
Substituting the known values:
t=12⋅log(1+120.03)log(10001500)
Step 3: Calculate t in Years and Convert to Months
After performing the calculations, we find:
t≈13.5324 years
To convert this to months:
tmonths=t×12≈162.3888 months
Rounding to the nearest month gives:
tmonths=162 months
Step 4: Determine Years and Remaining Months
To express tmonths in terms of years and months:
Years=⌊12162⌋=13Months=162mod12=6
Final Answer
The time it takes for the account to grow from \$1000 to \$1500 is: