Sally invests some money at 6%/a compounded annually. After 5 years, she takes the principal and interest and reinvests it all at a 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?

I tried solving this and got 7276.81. am i correct?

Hmm, let me see if I can calculate this for you.

To find out how much Sally originally invested, let's break it down into two parts. First, we'll calculate the amount she had after 5 years of compounding annually, and then we'll determine the principal she reinvested for the next 6 years compounded quarterly.

For the first part, we can use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal, r is the interest rate, n is the number of times compounded per year, and t is the number of years.

Using the given information: A = $14,784.56, r = 6%, t = 5, and n = 1, we can rearrange the formula to solve for P.

14,784.56 = P(1 + 0.06/1)^(1*5)
14,784.56 = P(1.06)^5
14,784.56 = P(1.338225)
P = 14,784.56 / 1.338225 ≈ $11,049.20

So, Sally initially invested approximately $11,049.20.

Hmm, it seems your answer is a bit off. Maybe next time you'll come closer to this amusingly precise amount!

To solve this problem, we can use the formula for compound interest:

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

Where:
A = the final amount
P = the principal (initial investment)
r = the interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years

Let's break the problem down into two parts:

Part 1: Investment after 5 years at 6% compounded annually
Using the formula for compound interest, and plugging in the given values:
14,784.56 = P(1 + 0.06/1)^(1*5)

Dividing both sides of the equation by (1 + 0.06)^5, we get:
P = 14,784.56 / (1.06)^5

Calculating P using a calculator, we find:
P ≈ 10,000

So, Sally originally invested approximately $10,000.

Therefore, your initial calculation of $7,276.81 is incorrect. The correct amount is $10,000.

To find the original amount Sally invested, we need to use the compound interest formula.

Let's break down the problem into two parts:

First, let's calculate the amount after 5 years of the initial investment:
A = P(1 + r/n)^(nt)
Where:
A = Final amount
P = Principal amount (initial investment)
r = Annual interest rate (6% = 0.06 in decimal form)
n = Number of times compounded per year (annual compounding = 1)
t = Number of years (5)

Plugging in the values, we have:
A1 = P(1 + 0.06/1)^(1*5)
A1 = P(1.06)^5

Next, let's calculate the amount after 6 years of reinvesting the principal and interest, at a different interest rate compounded quarterly:
A2 = A1(1 + r/n)^(nt)
Where:
A2 = Final amount after 6 more years
A1 = Amount after the initial 5 years
r = Annual interest rate (7.2% = 0.072 in decimal form)
n = Number of times compounded per year (quarterly compounding = 4)
t = Number of years (6)

Plugging in the values, we have:
A2 = A1(1 + 0.072/4)^(4*6)

We are given that the final amount at the end of this time is $14,784.56:
A2 = $14,784.56

Combining the two equations, we have:
P(1.06)^5 * (1 + 0.072/4)^(4*6) = $14,784.56

To solve for P, we rearrange the equation:
P = $14,784.56 / [(1.06)^5 * (1 + 0.072/4)^(4*6)]

Evaluating this expression, we find:
P ≈ $6,517.85

So, the original amount Sally invested was approximately $6,517.85, not $7,276.81 as you calculated.

Note: It is crucial to use the correct values and parentheses when calculating compound interest. Double-check your calculations to determine any potential errors made while solving the problem.