Question

Consider an investment of $6000 that earns 4.5% interest.

How long would it take for the investment
to reach $15,000 if the interest is
compounded monthly? Round your
answer to the nearest tenth.

Answer 1

20.4 years (nearest tenth)

Explanation:

Compound Interest Formula

`\large \text{$ \sf A=P\left(1+(r)/(n)\right)^(nt) $}`

where:

• A = final amount
• P = principal amount
• r = interest rate (in decimal form)
• n = number of times interest applied per time period
• t = number of time periods elapsed

Given:

• A = $15,000
• P = $6,000
• r = 4.5% = 0.045
• n = 12 (monthly)

Substitute the given values into the formula and solve for t:

`\implies \sf 15000=6000\left(1+(0.045)/(12)\right)^(12t)`

`\implies \sf (15000)/(6000)=\left(1.00375\right)^(12t)`

`\implies \sf 2.5=\left(1.00375\right)^(12t)`

`\implies \sf \ln (2.5)=\ln \left(1.00375\right)^(12t)`

`\implies \sf \ln (2.5)=12t \ln \left(1.00375\right)`

`\implies \sf t=(\ln (2.5))/(12 \ln (1.00375))`

`\implies \sf t=20.40017123`

Therefore, it would take 20.4 years (nearest tenth) for the investment to reach $15,000.