Question

4. If you deposit money today in an account that pays 2.5% annual interest, how long will it take to double your money? Steps (9 points): Answer (1 point): 5. You have $100,000 in a brokerage account, and you plan to deposit an additional $7,500 at the end of every future year until your account totals $1,000,000. You expect to earn 10% annually on the account How many years will it take to reach your goal? Steps (9 points): Answer (1 point): 2 Page

Answer 1

Solution:
4.	Length of time 28.07 years in two decimal
	or 28 years if answer required in whole number
	Hence, it takes 28.07 years to make double at 2.5%
Working Notes:

Notes:	To know the time period required to double money at rate of 2.5% annual interest rate. We will using concept of future value of present value in which present value will be the amount deposited today and which get compounded at given interest rate for time after which we get future value of the deposit which is double in our case.
	using this formula
	FV= PV (1+r)^t
	where

	Future value(FV) = $2	double
	Present value (PV) = $1
	r= rate of interest =2.5%=0.025
	t=time taken to have future value as double= ??
Now
	FV= PV x (1+r)^t
	2=1 x (1.025)^t
	(1.025)^t = 2
	Taking log on both side
	Log(1.025)^t = Log(2)
	t x log(1.025) = log(2)
hence,
	t = log(2)/log(1.025)

	t= 0.30102999566 /0.010723865	Putting log(2) value = 0.30102999566
	t=28.07103555	and log(1.025) = 0.010723865
	t= 28.07 years

5.
	Number of years to reach goal is   19.05 years in two decimal or 19 years in whole number.
Working Notes:
	Amount already in $100,000 & 7500 each year to have 1,000,000 in future at interest rate is 10%
Hence	We get 1,000,000 in future of the sum of future value of 100,000 compounded @10% annual in future plus future value of annuity of 7500 per year.
So
	1,000,000 = Future value of annuity of 7,500 + future value of 100,000
	Future value of annuity = P x ((1+i)^n - 1)/i
	P= Annual deposit = $7500
	i= 10% p.a
	n= no. Of years=??
	Future value of annuity = P x ((1+i)^n - 1)/i
	Future value of annuity = 7500 x ((1+ 10%)^n - 1)/10%
	future value of 100,000
	= initial amount x (1+ i)^n
	=100,000 x (1+ 10%)^n
Now	1,000,000 = Future value of annuity of 7,500 + future value of 100,000
	1,000,000 = [7500 x ((1+ 10%)^n - 1)/10%]+ [ 100,000 x (1+ 10%)^n]
	1,000,000 = [75,000/10% x ((1+ 10%)^n - 1)]+ [ 100,000 x (1+ 10%)^n]
	1,000,000 = [75,000 x ((1+ 10%)^n - 1)]+ [ 100,000 x (1+ 10%)^n]
	1,000,000 = [75,000 x (1+ 10%)^n - 75,000 ]+ [ 100,000 x (1+ 10%)^n]
	1,000,000 = 75,000 x (1+ 10%)^n - 75,000 + 100,000 x (1+ 10%)^n
	1,000,000 = 25,000 x (1+ 10%)^n + 100,000 x (1+ 10%)^n - 75,000
	1,000,000 + 75,000 = 75,000 x (1+ 10%)^n + 100,000 x (1+ 10%)^n
	1,075,000 = 175,000 x (1+ 10%)^n
	6.14285714285 = (1+ 10%)^n
	Taking Log on both side
	Log (6.14285714285) = log(1+ 10%)^n
	using relation loga^b = b x Log a
	Log (6.14285714285) = n x log(1+ 0.10)

	0.788370416 = n x 0.041392685	Log (6.14285714285) = 0.788370416
	n= 0.788370416/ 0.041392685	Log (1.10) = 0.041392685
	n=19.04612895
	n= 19.05 years

lets check it	1,000,000 = Future value of annuity of 5000 + future value of 100,000
	1,000,000 = [7500 x ((1+ 10%)^19.04612895 - 1)/10%]+ [ 100,000 x (1+ 10%)^19.04612895]
	1,000,000 = 385714.2895 + 614285.7193
	1,000,000 = 1,000,000.009
	1,000,000 = 1,000,000
	Hence our computation is correct.

Please feel free to ask if anything about above solution in comment section of the question.