Question

At what interest rate compounded quarterly must you invest $7800 to have$22797.37 in 11 years? First round the interest rate r to 4 decimal places andthen re-write it as a percentage with 2 decimal places.

Answer 1

The rate in 4 decimal place = 0.0988

The rate in percentage in 2 decimal place = 9.88%

Step-by-step explanation:

Principal = P = $7800

Future avalue = FV = $22797.37

Time = t = 11 years

n = number of times compounded = quarterly

n = 4

r = rate = ?

Using the compound interest formula:

`FV\text{ = P(1 +}(r)/(n))^(nt)`
`\begin{gathered} \text{Inserting the values into the formula:} \\ 22797.37\text{ = 7800(1 + }(r)/(4))^((4*11)) \\ 22797.37\text{ = 7800(1 + }(r)/(4))^(44) \end{gathered}`
`\begin{gathered} \text{divide through by 7800:} \\ (22797.37)/(7800)=\text{ (1 + }(r)/(4))^(44) \\ 2.9227\text{ = (1 + }(r)/(4))^(44) \\ find\text{ 44th root of both sides:} \\ \sqrt[44\text{ }]{2.9227\text{ }}\text{ = }\sqrt[44\text{ }]{\text{ (1 + }(r)/(4))^(44)} \end{gathered}`
`\begin{gathered} 1.0247\text{ = (1 + }(r)/(4)) \\ 1.0247\text{ = 1 + }(r)/(4) \\ \text{subtract 1 from both sides:} \\ 1.0247\text{ -1 = 1 - 1 + }(r)/(4) \\ 0.0247\text{ = }(r)/(4) \end{gathered}`
`\begin{gathered} \text{cross multiply:} \\ 0.0247(4)\text{ = r} \\ r\text{ = 0.0988} \end{gathered}`

The rate in 4 decimal place = 0.0988

The rate in percentage in 2 decimal place = 0.0988 ×100 = 9.88%