A bank has loaned out $750,000, part of it at 5% per annum and the rest of it at 8% per annum. If the bank receives $52,500 in interest each year, how much was loaned at 5%?
$500,000 was loaned at 5% per annum.
Let's assume that the amount loaned at 5% per annum is x dollars.
Therefore, the amount loaned at 8% per annum is (750,000 - x) dollars.
Now, let's calculate the interest earned from each rate. The interest from the amount loaned at 5% is:
Interest1 = x * 0.05
And the interest from the amount loaned at 8% is:
Interest2 = (750,000 - x) * 0.08
The total interest earned from both loans is given as $52,500:
Interest1 + Interest2 = 52,500
Substituting the values of Interest1 and Interest2:
x * 0.05 + (750,000 - x) * 0.08 = 52,500
Simplifying the equation:
0.05x + 0.08(750,000 - x) = 52,500
0.05x + 60,000 - 0.08x = 52,500
-0.03x = -7,500
Now, let's solve for x:
x = (-7,500) / (-0.03)
x = 250,000
Therefore, $250,000 was loaned at 5%.
To solve this problem, we can set up a system of equations. Let's call the amount loaned at 5% x and the amount loaned at 8% y.
x + y = $750,000 (since the sum of the loan amounts equals $750,000)
The interest earned from the amount loaned at 5% (x) can be calculated as 0.05x, and the interest earned from the amount loaned at 8% (y) is 0.08y.
Since the bank receives a total interest of $52,500, we can write the equation:
0.05x + 0.08y = $52,500
Now we can solve the system of equations.
First, let's solve the first equation for x by subtracting y from both sides:
x = $750,000 - y
Substituting this expression for x into the second equation, we have:
0.05($750,000 - y) + 0.08y = $52,500
Now, distribute 0.05:
($37,500 - 0.05y) + 0.08y = $52,500
Simplify the equation:
$37,500 + 0.03y = $52,500
To isolate y, subtract $37,500 from both sides:
0.03y = $52,500 - $37,500
0.03y = $15,000
Now, divide both sides by 0.03 to solve for y:
y = $15,000 / 0.03
y = $500,000
We now know that the amount loaned at 8% is $500,000. To find the amount loaned at 5%, we can substitute this value back into the first equation:
x + $500,000 = $750,000
Now, subtract $500,000 from both sides:
x = $750,000 - $500,000
x = $250,000
Therefore, $250,000 was loaned at 5%.