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
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I'm sorry, I cannot provide an answer without a specific question. Can you please provide more information?
To find out how much Sally originally invested, we can use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A = the final amount
P = the principal amount
r = the annual interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years
We have two stages of investment, so let's break it down.
Stage 1:
Principal = P
Interest Rate = 65% = 0.65 (in decimal form)
Compounded Annually = 1 time per year
Years = 5
Stage 2:
Principal = Total amount after Stage 1 (P + PA) = P1
Interest Rate = 7.2% = 0.072 (in decimal form)
Compounded Quarterly = 4 times per year
Years = 6
From the given information, we know that at the end of Stage 2, the investment is worth $14,784.56.
Let's solve it step by step:
Step 1: Find the value after Stage 1
A1 = P(1 + r/n)^(n*t)
A1 = P(1 + 0.65/1)^(1*5)
A1 = P(1 + 0.65)^5
A1 = P(1.65)^5
A1 = 1.65^5 * P
Step 2: Find the value after Stage 2
A2 = P1(1 + r/n)^(n*t)
A2 = P1(1 + 0.072/4)^(4*6)
A2 = P1(1 + 0.018)^24
A2 = (1.018)^24 * P1
Step 3: Set up the final equation using the given value at the end of Stage 2
A2 = (1.018)^24 * P1 = $14,784.56
Step 4: Solve for P1 (Total amount after Stage 1)
P1 = $14,784.56 / (1.018)^24
Step 5: Find the original principal amount (P)
P = P1 - A1
Now, let's calculate the values.
P1 = $14,784.56 / (1.018)^24
P1 ≈ $10,140.42 (rounded to two decimal places)
P = $10,140.42 - A1
To find A1, we calculate it based on the given formula:
A1 = 1.65^5 * P
A1 ≈ 3.185625 * P
P = $10,140.42 - (3.185625 * P)
Next, we solve for P:
P + 3.185625 * P = $10,140.42
4.185625 * P = $10,140.42
P ≈ $2,423.74 (rounded to two decimal places)
Therefore, Sally originally invested approximately $2,423.74.