Question

How much money will you need to invest initially to have $750.00 in 10 years and 8 months if the money is compounded daily at an annual rate of 2 1/2%?A.$585.90B. $574.45C. $576.33D. $574.60

Answer 1

Given:

The amount after 10 years and 8 months, A=$750.00.

The rate of interest, r =2 1/2 %.

The period of time, t =10 years and 8 months.

The interest is compounded daily

Required:

We need to find the intial investment amount.

Step-by-step explanation:

Conver the period of time to years.

`1\text{ year =12 months.}`
`(8)/(12)\text{ year =8 months.}`

`10\text{ +}(8)/(12)\text{ years =10 years and 8 months.}`

`10\text{ +}(8)/(12)\text{ years =10 years and 8 months.}`

`10\text{ }*(12)/(12)\text{+}(8)/(12)\text{ years =10 years and 8 months.}`

`(120)/(12)\text{+}(8)/(12)\text{ years =10 years and 8 months.}`

`(120+8)/(12)\text{ years =10 years and 8 months.}`

`(128)/(12)\text{ years =10 years and 8 months.}`

We get t =128/12.

The annual interest rate is

`r=2.5\text{ \%.}`
`r=0.025.`

The number of days in a year = 365 days.

The money is compounded daily, n=365.

Consider the formula to find the amount in compound interest.

`A=P(1+(r)/(n))^(nt)`

Substitute A =750, r=0.025, n=365 and t =128/12 in the formula.

`750=P(1+(0.025)/(365))^{365*(128)/(12)}`

`750=P(1+(0.025)/(365))^(3893.333)`

`P=(750)/((1+(0.025)/(365))^(3893.333))`
`P=574.4516`

Final answer:

The initial amount is $ 574.45.