$100,000 is divided into two investments with annual returns of 4% and 7.5%. find the amount invested at the lower rate if the total income at the end of the year is $5050?
amount invested at 4% --- x
amount invested at 7.5% -- 100000-x
solve for x:
.04x + .075(100000-x) = 5050
solve for x and express the solution in a+bi form : ix^2-3x+4i=0
You sure this is right?
ix^2 - 3x - 4i = (ix+1)(x-4i)
but that's not what we have.
So, using the good old quadratic formula,
x = (3 ± √5)]/2i
To solve this problem, let's assume that the amount invested at the lower rate is denoted by 'x'.
So, the amount invested at the higher rate will be the remaining amount, which is (100,000 - x).
The total income from the investments is the sum of the incomes from the two investments:
Income from the first investment at 4% = x * 0.04
Income from the second investment at 7.5% = (100,000 - x) * 0.075
According to the problem, the total income at the end of the year is $5050. Therefore, we can set up the equation:
x * 0.04 + (100,000 - x) * 0.075 = 5050
To solve the equation, we can follow these steps:
Step 1: Multiply the decimals with the corresponding variables:
0.04x + (0.075 * 100,000) - (0.075 * x) = 5050
Step 2: Simplify the equation by combining like terms:
0.04x + 7,500 - 0.075x = 5050
Step 3: Combine the x terms:
-0.035x + 7,500 = 5050
Step 4: Move the constants to the other side of the equation:
-0.035x = 5050 - 7,500
Step 5: Simplify the equation further:
-0.035x = -2,450
Step 6: Divide both sides of the equation by -0.035 to solve for x:
x = -2,450 / -0.035
Step 7: Evaluate the expression:
x ≈ 70,000
Therefore, approximately $70,000 is invested at the lower rate.