MATH 105 (COLLEGE LEVEL) finical algebra compounding computations

Question

Suppose that $2500 is placed in a savings account at an annual rate of 3.3%, compounded quarterly. Assuming that no withdrawals are made, how long will it take for the account to grow to $4500? Do not round any intermediate computations, and round your answer to the nearest hundredth.

Find ___years

AND

To help with a down payment on a home, Tammy is going to invest. Assuming an interest rate of 1.77% compounded monthly, how much would she have to invest to have $27,300 after 9 years?

find $___

Answer 1

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

where r = rate of interest=3.3%= 0.033

n= number of compounding periods per year=4

t= number of years

P=starting amount=$2500

A=ending amount=$4500

4500=2500(1+.033/4)^4t

45/25 = (1.00825)^4t

1.8 = 1.00825^4t

4t >64

t> 16 years

4t= log to the base 1.00825 of 1.8 = log_1.008251.8

t = (log_1.008251.8)/4 = about 72/4 = 18 years

t = slightly less than 18 years to turn $2500 into $4500

4t = ln1.8/log1.00825 = .58779/.008216= 71.54 years

t = 17.885 = about 17.89 years = about 17 years 10.5 months

if it had been continuous compounding, then

4500=2500e^.033t

ln9/5 = .033t

t = (ln1.8)/.033 = 17.81 years

with compounding only quarterly, it takes slightly longer, another 0.08 years more

27300 = P(1+0.0177/12)^12(9)

P = 27,300/(1.001475)^108

use a calculator

P=about $23,282.55 rounded to nearest cent

if it had been continuous compounding

27300 = Pe^.0177(9)

P= 27300/1,1727=$23,279.61