a bank loaned out 60000, part of it at the rate 14% per year and the rest at a rate of 8% per year. if the interest received was 6780, how much was loaned at 14%?
P2+14p=
x = amount at 14%
y = amount at 8%
x + y =60000
.14x + .08y = 6780
Most people will multiply the second equation by 100 to get rid of the decimals.
x + y = 60000
14x + 8y = 67800
if you eliminate the y's you will be left with x = the value invested at 14%
Multiply the top equation by -8 and add it to the second equation.
This is called the addition or elimination method.
To solve this problem, we can set up a system of equations based on the information given. Let's call the amount loaned at 14% x, and the amount loaned at 8% y.
From the problem, we have two pieces of information:
1) The total loan amount is $60,000. So we have the equation: x + y = 60000.
2) The interest received from these loans is $6,780. To calculate the interest, we need to multiply the loan amounts by their respective interest rates and then add them together. So we have another equation: 0.14x + 0.08y = 6780.
Now, we can solve this system of equations to find the values of x and y.
First, let's solve the first equation for x in terms of y:
x = 60000 - y.
Now, substitute this value of x in the second equation:
0.14(60000 - y) + 0.08y = 6780.
Distribute the 0.14:
8400 - 0.14y + 0.08y = 6780.
Combine like terms:
-0.06y = 6780 - 8400.
Simplify the right side:
-0.06y = -1620.
Divide both sides by -0.06 to isolate y:
y = (-1620) / (-0.06).
Simplify:
y = 27000.
Now, substitute this value of y back into the first equation to find x:
x + 27000 = 60000.
x = 60000 - 27000.
x = 33000.
Therefore, $33,000 was loaned at a rate of 14%.