Algebra problem, figuring rate of return.?

Question

Suppose that you invested $16,000 in a high-risk growth fund that after 2 years was worth $25,000. Your broker used the following equation to find the average annual rate of return, (r)

16,000(1 + r)2 = 25,000
What is the annual rate of return?

I don't understand the equation.

Answer 1

Imagine you have $16000. You put it somewhere for a year, and at the end of it, you get $16000 back. You have the same amount of money at the end as you started with, so you would multiply it by 1.

16,000(1) = 16,000

Now imagine you put it somewhere for a year and get 17,600 back. 17,600 is the initial 16,000 plus an additional 1,600 (10% or 0.1). This 0.1 is r

16,000(1 + r) = 16,000(1 + 0.1) = 16,000(1.1) = 17,600

Now imagine you put that 17,600 back in the fund for another year. You would get 10% of that 17,600 back.
i.e. 17,600(1 + r) = 17,600(1 + 0.1) = 17,600(1.1) = 19,360

So in two years you went from 16,000 to 19,360. This was calculated by:

16,000(1 + r) to get 17,600
17,600(1 + r) to get 19,360

This can be simplified to 16,000(1 + r)(1 + r) or 16,000(1 + r)² where the square (²) shows that the initial amount was put into the fund for 2 years.

So the formula for this example would be 16,000(1 + r)² = 19,360

Hopefully the above explained how the formula was derived, so I will use the above figures to walk you through a similar question which you can use as a model for your own calculations.

16,000(1 + r)² = 19,360

We want to work out what r is, so we need to leave r on its own.

The first thing to do is to get rid of the 16,000 on the left. So we need to divide both sides by 16,000

(1 + r)² = 19,360/16,000
(1 + r)² = 1.21

We now need to square root both sides to get rid of the square

1 + r = √1.21
1 + r = 1.1

Then just subtract 1 from each side

r = 1.1 - 1
r = 0.1
r = 10% (rate of return is usually given as a percentage)
which is correct, as can be seen from my initial explanation.

IMPORTANT NOTE: THIS IS NOT THE ANSWER TO YOUR QUESTION. THIS IS SIMPLY A WORKED THROUGH ANSWER OF A SIMILAR QUESTION TO SHOW HOW YOU CAN WORK IT OUT YOURSELF.

Answer 2

This is a compound interest problem

A = P(1+r)^n

where P is the amount invested, r is the rate of return of the investment and A is the amount after 2 years

16,000(1 + r)² = 25,000

(1 + r)² = 25000/16000

(1 + r)² = 1.5625

1 + r = 1.25

r = 25%

.

Answer 3

Assume that the interest is compounded once a year.
You'll have to solve for r.
16000(1+r)(1+r)=25000
(1+r)(1+r)=25000/16000=25/16
1+r+r+r^2=25/16
1+2r+r^2=25/16
multiply both sides by 16
16+32r+16r^2=25
16r^2+32r-9=0
you'll have to solve this quadratic equation
ar^2+br+c=0
a=16 b=32 c=-9
r=[-b+-sqrt(b^2-4ac)]/2a
solving r=0.25 or -2.25
r is the rate of return and cannot be negative
so r=0.25 or 1/4 or 4 % per year.
I hope you're familiar with this.

Answer 4

Let r = annual rate of interest, $16,000 and $25,000 are represented by 16 and 25, respectively.

Finding the rate of interest:
16(1 + r)^2 = 25
4(1 + r) = 5
1 + r = 1.25
r = 0.25

Answer: rate of interest per year is 25%

Proof (determine balances per year after interests are added):
1st year:
= $16,000 + 0.25($16,000)
= $16,000 + $4,000
= $20,000

2nd year:
= $20,000 + 0.25($20,000)
= $20,000 + $5,000
= $25,000

Answer 5

16000*(1+r)^2 =25000
(1+r)^2=25000/16000

dividing numerator & denominator in 1000
(1+r)^2=25/16

taking squareroot on both sides
(1+r) = 5/4
r =5/4 -1
r= 1/4
r=.25

r =25%