19. Given a 10 percent interest rate, compute the year 9 future value if deposits of $10,000 and $20,000 are made in years 1 and 5 respectively, and a withdrawal of $5,000 is made in year 7.

Well, investing money is like feeding a hungry clown. It's all about making deposits and withdrawals. So, let's do some circus math!

In year 1, you make a deposit of $10,000. The clown jumps with joy!

In year 5, another deposit of $20,000 is made. The clown now does a somersault!

But in year 7, there's a withdrawal of $5,000. The clown looks sad.

Now, let's calculate the future value in year 9. We'll need to consider the interest rate too.

First, let's find out how much the clown has after year 1. He already had $10,000, and now with a 10% interest rate, he gets an additional $1,000. So, after year 1, the clown has $11,000.

In year 5, the clown receives another deposit of $20,000. With an interest rate of 10%, he gets $2,000 as interest. So, after year 5, the clown has $33,000.

Now, in year 7, there's a withdrawal of $5,000. Poor clown, he now has $28,000 left.

Lastly, in year 9, the clown receives more interest. With a 10% interest rate, he gains $2,800. So, in year 9, the future value will be $30,800.

That's the circus math for you! Remember, investing is all about keeping the clown happy with deposits and minimizing withdrawals.

To compute the future value in year 9, we need to calculate the accumulated value of the deposits and withdrawals over the period.

First, let's calculate the future value of the $10,000 deposit made in year 1. We can use the formula for compound interest:

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

Where:
A = Accumulated or future value
P = Principal (initial deposit)
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years

In this case, the principal is $10,000, the interest rate is 10% (0.10 as a decimal), the compounding is annual (n = 1), and the time is 9 years (t = 9).

Using the formula, we can calculate the accumulated value of the $10,000 deposit in year 9:

A1 = 10000(1 + 0.10/1)^(1 * 9)
= 10000(1 + 0.10)^9

Now let's calculate the future value of the $20,000 deposit made in year 5. By year 9, these deposits will have accumulated interest for 4 years.

Using the same formula, we can calculate the accumulated value of the $20,000 deposit in year 9:

A2 = 20000(1 + 0.10/1)^(1 * 4)
= 20000(1 + 0.10)^4

Finally, we need to account for the withdrawal of $5,000 made in year 7. By year 9, this withdrawal will have lost two years of accumulated interest.

To calculate the adjusted value of the withdrawal, we can use the same formula but with negative values for the withdrawal amount and interest rate:

Adjusted withdrawal = -5000(1 + 0.10/1)^(1 * 2)
= -5000(1 + 0.10)^2

Now, we can compute the total future value in year 9 by summing the accumulated values of the deposits and adjusting for the withdrawal:

Future value in year 9 = A1 + A2 + Adjusted withdrawal

Note: The adjusted withdrawal is subtracted because it represents a decrease in the accumulated value.

Once you substitute the individual values into the equations above, you can calculate the future value.