Answer:
18,000 is invested at 8%.
12,000 is invested at 6%.
20,000 is invested at 9%.
Explanation:
x = amount invested at 8%.
y = amount invested at 6%.
z = amount invested at 9%.
x + y + z = 50,000
.08x + .06y + .09z = 3960
interest on the first investment is equal to 2 times interest on the second investment.
.08x = 2 * .06y
solve for x to get x = .12y/.08 = 1.5y
since .08x = .12y, you can replace .08x with .12y.
since x = 1.5y, you can replace x with 1.5y
your two equations of x + y + z = 50,000 and .08x + .06y + .09z = 3960 become:
1.5y + y + z = 50,000
.12y + .06y + .09z = 3960
combine like terms in both equations to get: 2.5y + z = 50,000
.18y + .09z = 3960
you now have 2 equations with 2 unknown variables which can be solved simultaneously for a unique solution.
we will eliminate the z variable from the 2 equations in the following manner.
multiply both sides of the first eqution by 9 and multiply both sides of the second equation by 100.
you will get:
2.5y + z = 50,000 becomes 22.5y + 9z = 450,000.
.18y + .09z = 3960 becomes 18y + 9z = 396,000.
the 2 equations are now:
22.5y + 9z = 450,000
18y + 9z = 396,000
subtract the second equation from the first to get:
4.5y = 54,000
divide both sides of this equation by 4.5 to get:
y = 12,000
since x = 1.5y, then x must be equal to 18,000
since x + y + z = 50,000, then z must be equal to 20,000
you have:
x = 18,000
y = 12,000
z = 20,000
the interest on each of these investments is:
.08 * 18,000 = 1,440
.06 * 12,000 = 720
.09 * 20,000 = 1,800
add up the investments and you get 50,000
add up the interest and you get 3,960
the interest on the first investment is 2 times the interest on the second investment because 1,440 = 2 * 720.
all results are confirmed as good.
your solution is:
18,000 is invested at 8%.
12,000 is invested at 6%.
20,000 is invested at 9%.
