suppose you invest $15,000 in two funds paying 11 1/2% and 14% simple interest.
the total annual interest is $1950. how much is invested in each fund?
Let we invest $X in 1st fund, then
0.115X+0.14(15000-X)=1950
0.115X+2100-0.14X=1950
150=0.025X
X=6000
To solve this problem, we can set up a system of linear equations.
Let's assume that the amount invested in the fund paying 11 1/2% simple interest is x dollars, and the amount invested in the fund paying 14% simple interest is y dollars.
According to the given information, the total amount invested is $15,000, so we have the equation:
x + y = 15000 ---(Equation 1)
The total annual interest earned is $1950. We can calculate the interest earned on each investment using the formula:
Interest = Principal * Rate of Interest
For the first investment, the interest is x * (11 1/2%) = x * 11.5/100
For the second investment, the interest is y * (14%) = y * 14/100
The sum of the interests should be equal to $1950, so we have the equation:
x * 11.5/100 + y * 14/100 = 1950 ---(Equation 2)
Now we can solve this system of equations (Equation 1 and Equation 2) to find the values of x and y.
To make the calculations simpler, we can multiply Equation 1 by 100 to eliminate the decimal:
100x + 100y = 150000
Now we can rewrite Equation 2 in terms of x:
11.5x + 14y = 195000
We can use the method of substitution or elimination to solve this system of equations. Let's use the elimination method:
Multiply Equation 1 by 11.5 and Equation 2 by 100 to eliminate x:
1150x + 1150y = 1725000 ---(Equation 3)
115x + 140y = 195000 ---(Equation 4)
Now, subtract Equation 4 from Equation 3:
1150x + 1150y - (115x + 140y) = 1725000 - 195000
1035x + 1010y = 1530000
Now, we have a new equation:
1035x + 1010y = 1530000 ---(Equation 5)
Now, we can divide Equation 5 by 5 to simplify it:
207x + 202y = 306000 ---(Equation 6)
Now we have two equations:
100x + 100y = 150000 ---(Equation 3)
207x + 202y = 306000 ---(Equation 6)
Now we can solve this system of equations either by substitution or elimination.
Using the elimination method, multiply Equation 3 by 2 and subtract it from Equation 6:
2(100x + 100y) = 2 * 150000
207x + 202y - 200x - 200y = 306000 - 300000
2007y = 6000
Divide both sides by 2007 to solve for y:
y = 6000 / 2007
Simplify further:
y ≈ 2.98853
Now substitute this value of y into Equation 3 to solve for x:
100x + 100(2.98853) = 150000
100x + 298.853 = 150000
100x = 150000 - 298.853
100x = 149701.147
Divide both sides by 100 to solve for x:
x = 149701.147 / 100
x ≈ 1497.01147
So, approximately $1,497.01 is invested in the fund paying 11 1/2% interest, and approximately $2,988.53 is invested in the fund paying 14% interest.