Given A=P(1+ (r)/(n))^(nt) what interest rate (to the nearest percent) is needed to grow $5000 to $8000 in 5 years if interest is compounded annually?

Question

Given
`A=P(1+ (r)/(n))^(nt)` what interest rate (to the nearest percent) is needed to grow $5000 to $8000 in 5 years if interest is compounded annually?

Answer 1

The interest rate (to the nearest percent) is 10%.

Explanation:

The formula to calculate compound interest is:

`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 is applied per year.
• t = Time (in years).

Given values:

• A = $8000
• P = $5000
• n = 1 (annually)
• t = 5 years

Substitute the values into the formula and solve for r:

`\implies 8000=5000\left(1+(r)/(1)\right)^(1 \cdot 5)`

`\implies 8000=5000\left(1+r\right)^(5)`

`\implies (8000)/(5000)=\left(1+r\right)^(5)`

`\implies1.6=\left(1+r\right)^(5)`

`\implies \sqrt[5]{1.6}=\sqrt[5]{\left(1+r\right)^(5)}`

`\implies \sqrt[5]{1.6}=1+r`

`\implies r=\sqrt[5]{1.6}-1`

`\implies r=0.09856054...`

`\implies r=9.856054...\%`

`\implies r \approx 10\%`

Therefore, the interest rate (to the nearest percent) is 10%.