compound interest

Question

Quan has 20,000 to invest over the next three years. He has the choice of investing his money at 6.25% simple interest or 6% compound interest 

A) calculate the interest that quan will earn if he selects the simple interest option.

B)calculate the amount of interest that wuan will earn if he selects the simple interest option is compounded i) annually ii) six monthly iii)quarterly 

C) which of these three options will esra quan the most interest 

D) with this option, how long will it take quan to double his money? 

E) quan also has $17918 in another account that grew from 11,000 in the last ten years. Find, to the nearest percent, the interest rate for the account if it's compounded annually 

 

Please show working out and answers. Thankyou

Answer 1

Note that when using the interest formulas to change the interest rate to a decimal.

6.25% = 6.25/100 = .0625      6% = 6/100 = .06

 

A) Simple interest:  I = Prt

    I = 20000(.0625)(3) = $3,750

 

B) Interest earned if we use the compound interest...  A = P(1+r/n)^nt

    A = future amount

    P = Principal (initial investment)

    r = interest rate

    n = number of times compounded each year

    t = time in yers

 

    1.  Annually ... n=1

        A = 20000(1+.06/1)¹⁽³⁾ = 2000(1.06)³ = $23820.32

        Interest earned: A-P = 23820.32 - 20000 = $3,820.32

 

    2.  Six monthly (semi-annually) ...  n=2

        A = 20000(1+.06/2)²⁽³⁾ = 20000(1.03)⁶ = $23,881.05

        Interest earned: A-P = 23881.05-20000 = $3,881.05

 

    3.  Quarterly ... n=4

        A = 20000(1+.06/4)⁴⁽³⁾ = $23,912.36

        Interest earned:  A-P = 23912.36-20000 = $3,912.36

 

C)  As can be seen from part B the highest interest earned was $3,912.36

     Esra will earn the most if he invests the money at compound interest 6% compounded quarterly

 

D)  If we invest at 6% compounded quarterly we want to know how long it will take to double

     the principal.  This means we want to know for what value of t will A=2P

 

    2(20000) = 20000(1+.06/4)^(4)t

    Divide both sides by 20000

    2 = (1.15)^4t

    Take the log of both sides

    log 2 = log(1.15)^4t

    log 2 = 4t[log(1.15)]

 

   t = (log 2)/[4log(1.15)]

   t = 1.24 years

 

   (.24 years)(12 mos/yr) = 2.88 months... so round to 3 months

 

the money will be doubled in 1 year 3 months

 

E)  $17,918 grew from $11,000 in 10 years.  Find the annual interest rate to nearest percent

 

     A = 17918,   P = 11000,   n=1, r = ??,  t=10

 

    17918 = 11000(1+r/1)¹⁽¹⁰⁾

    17918/11000 = (1+r)¹⁰

    Take the 10th root of both sides

    ¹⁰√(17918/11000) = 1+r

     r = ¹⁰√(17918/11000) -1

     r = .05 = 5/100 = 5%

 

    The annual interest rate was 5%