Sally invests some money at 65/a compounded annually. After 5 years . she takes the principal and interest and reinvests it all at 7.2%/a compounded quarterly for 6 more years. At the end of this time, her investment is worth $ 14 784.56 . How much did Sally originally invest?

65/a ??

check typing

oh sorry 6%/a

To find out the original amount Sally invested, we can use the formula for compound interest:

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

where:
A is the final amount,
P is the principal amount (the initial investment),
r is the annual interest rate (as a decimal),
n is the number of times that interest is compounded per year, and
t is the number of years.

Let's break down the information given in the problem step by step:

Step 1: Calculate the first investment amount after 5 years:
Let's assume the principal amount Sally initially invested is P.

A1 = P(1 + r/1)^(1*5)
A1 = P(1 + 0.65/1)^5
A1 = P(1.65)^5
A1 = 2.88825P

Step 2: Calculate the second investment amount after 6 more years:
Now, Sally takes the principal and interest from the first investment and reinvests it at a new interest rate.

A2 = A1(1 + r/4)^(4*6)
A2 = 2.88825P(1 + 0.072/4)^(4*6)
A2 = 2.88825P(1.018)^24
A2 = 2.88825P(1.310796)

Step 3: Set up an equation to find the original investment amount:
At the end of the second investment period, the total value is given as $14,784.56.

A2 = 14,784.56
2.88825P(1.310796) = 14,784.56

Now, we can solve this equation to find the value of P.

2.88825P(1.310796) = 14,784.56
3.78325974P = 14,784.56
P = 14,784.56 / 3.78325974
P ≈ $3,905.43

Therefore, Sally originally invested approximately $3,905.43.