Question

Sarah and Azhar both invest $10,000. Sarah invests her $10,000 at a rate of x% compound interest per year. Azhar invests his $10,000 in a bank that pays 2% simple interest per year. After 7 Years, their investments are worth the same amount. Calculate the value of x.

Answer 1

The firts thing we are going to do here is use the simple interest formula:
`A=P(1+rt)`
where

`A` is the final amount after
`t` years

`P` is the initial investment

`r` is the interest rate in decimal form

`t` is the number of years
With this formula we will find the final amount Azhar's investment after 7 years. We know from our problem that
`P=10000`,
`r= (2)/(100)=0.02`, and
`t=7`. Lets replace those values in our formula to find
`A`:

`A=10000(1+(0.02)(7))`

`A=10000(1.14)`

`A=11400`

Now, since Sarah is investing in a compound interest account, we are going to use the compound interest formula:
`A=P(1+ (r)/(n))^(nt)`
where

`A` is the final amount after
`t` years

`P` is the initial investment

`r` is the interest rate in decimal form

`n` is the number of times the interest is compounded per year

`t` is the number of years
Notice that we know from our problem that after 7 years their investments are worth the same amount, so
`A=11400`. We also know that
`P=10000`,
`r= (x)/(100) =0.01x`, and
`t=7`. Since the interest are compounded per year,
`n=1`. Lets replace all the vales in our compound interest formula and solve for
`x` to find our rate:

`11400=10000(1+ (0.01x)/(1) )^((1)(7))`

`(11400)/(10000) =(1+0.01x) ^(7)`

`(1+0.01x) ^(7) =1.14`

`1+0.01x= \sqrt[7]{1.14}`

`0.01x= \sqrt[7]{1.14} -1`

`x= \frac{ \sqrt[7]{1.14}-1 }{0.01}`

`x=1.89`

We can conclude that the interest rate of Sarah's investment is approximately 1.89%, so x=1.89%.