Question

Find the interest rate for the given deposit and compound amount.

$4000 accumulating to ​$​5994.75, compounded monthly for 7 years.

Answer 1

The interest rate is 5.79% (to the nearest hundredth).

Explanation:

To find the interest rate for the given deposit of $4000 accumulating to ​$5994.75, compounded monthly for 7 years, use the compound interest formula.

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

Given values:

• P = $4,000
• A = $5,994.75
• n = 12 (monthly)
• t = 7 years

Substitute the given values into the formula and solve for r.

`\implies \sf 5994.75=4000\left(1+(r)/(12)\right)^(12 * 7)`

`\implies \sf 5994.75=4000\left(1+(r)/(12)\right)^(84)`

`\implies \sf (5994.75)/(4000)=\left(1+(r)/(12)\right)^(84)`

`\implies \sf \sqrt[\sf 84]{\sf (5994.75)/(4000)}=1+(r)/(12)`

`\implies \sf \sqrt[\sf 84]{\sf (5994.75)/(4000)}-1=(r)/(12)`

`\implies \sf r=12\left(\sqrt[\sf 84]{\sf (5994.75)/(4000)}-1\right)`

`\implies \sf r=0.057937950...`

`\implies \sf r=5.7937950...\%`

Therefore, the interest rate is 5.79% (to the nearest hundredth).