Answer:
$13500 was loaned out at 5% annual interest.
$20000 was loaned out at 4% annual interest.
Explanation:
We can determine the amount loaned at each rate using a system of equations where:
- x represents the amount loaned at 5% annual interest,
- and y represents the amount loaned at 4% annual interest.
First equation:
We know that the sum of the amounts loaned at 5% and 4% equals the total amount loaned out:
amount loaned at 5% + amount loaned at 4% = total amount loaned
Since the bank loaned out $33500, our first equation is given by:
x + y = 33500
Second equation:
We also know that the sum of the interests earned at 5% and 4% equals the total interest earned:
(5% interest rate * loan amount) + (4% interest rate * loan amount) = total interest
- Note that for our equation, we convert 5% and 4% to a decimal (i.e., 0.05 and 0.04).
Since the total interest earned was $1475, our second equation is given by:
0.05x + 0.04y = 1475
Method to solve: Elimination:
Solving for x (the amount loaned at 5%):
First we need to multiply the first equation by -0.04, which will eventually allow us to eliminate y since -0.04y + 0.04y = 0:
-0.04(x + y = 33500)
-0.04x - 0.04y = -1340
Now we can add the two equations to eliminate y and solve for x:
-0.04x - 0.04y = -1340
+
0.05x + 0.04y = 1475
----------------------------------------------------------------------------------------------------------(-0.04x + 0.05x) + (-0.04y + 0.04y) = (-1340 + 1475)
(0.01x = 135) / 0.01
x = 13500
Thus, $13500 was loaned out at 5% annual interest.
Solving for y (the amount loaned at 4%):
Now we can solve for y by plugging in 13500 for y in the first equation (x + y = 33500):
(13500 + y = 33500) - 13500
y = 20000
Thus, $20000 was loaned out at 4% annual interest.