Sergon has $5000 to invest for six months. The rates offered on three-month and six-month
term deposits at his bank are 5.5% and 5.8%, respectively. He is trying to choose between
the six-month term deposit and two consecutive three-month term deposits. What would
the simple interest rate on three-month term deposits have to be, three months from now,
for Sergon to end up in the same financial position with either alternative? Assume that he
would place both the principal and the interest from the first three-month term deposit in
the second three-month term deposit.
To determine the simple interest rate on the three-month term deposits that would result in Sergon being in the same financial position with either alternative, we can set up an equation:
Let's assume the principal amount is $5000.
With the six-month term deposit, the amount after six months would be:
Principal + Simple interest = $5000 + ($5000 * 5.8% * 6/12) = $5000 + $145
With the two consecutive three-month term deposits, the amount after six months would be:
Principal + Simple interest(term1) + Simple interest(term2) = $5000 + ($5000 * X * 3/12) + (($5000 + ($5000 * X * 3/12)) * X * 3/12)
Now we can set up the equation to find the value of X.
$5000 + ($5000 * 5.8% * 6/12) = $5000 + ($5000 * X * 3/12) + (($5000 + ($5000 * X * 3/12)) * X * 3/12)
Simplifying the equation:
$5000 + ($5000 * 0.058 * 0.5) = $5000 + ($5000 * X * 0.25) + (($5000 + ($5000 * X * 0.25)) * X * 0.25)
$5000 + $145 = $5000 + ($5000 * X * 0.25) + (($5000 + ($5000 * X * 0.25)) * X * 0.25)
$5000 + $145 = $5000 + 0.25X * $5000 + (0.25X^2 * $5000) + (0.25X * $5000 * X)
Cancelling out like terms and simplifying:
$145 = 0.25X * $5000 + (0.25X^2 * $5000)
Dividing both sides by $5000:
0.028 = 0.25X + (0.25X^2)
Rearranging the equation:
0.25X^2 + 0.25X - 0.028 = 0
Now, we can solve this quadratic equation to find the value of X using any suitable method like factoring or the quadratic formula.
Once you find the value of X, that would be the simple interest rate on the three-month term deposits three months from now for Sergon to end up in the same financial position with either alternative.