Question 473718: An investment of $59,000 was made by a business club. The investment was split into three parts and lasted for one year. The first part of the investment earned 8% interest, the second 6%, the third 9%. Total interest from the investment was $4710. The interest from the first investment was 6 times the interest from the second.
What are the amounts from the three parts of the investments?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x + y + z = 59000
.08*x + .06*y + .09*z = 4710
since the interest from the first investment is equal to 6 times the interest from the second investment, then we get:
.08*x = 6 * .06*y (additional requirement)
this makes x equal to (6*.06*y)/.08 which equals .36*y/.08 which simplifies to:
x = 4.5y
x is the first investment
y is the second investment
z is the third investment
since we have x equal to 4.5y, then we can substitute for x in both equations which allows us to make this problem 2 equations in 2 unknowns rather than 2 equations in 3 unknowns.
2 equations in 2 unknowns can be solved for a single solution.
2 equations in 3 unknowns can't.
our 2 equations that we have to work with are:
x + y + z = 59000 (first equation)
.08x + .06y + .09z = 4710 (second equation)
after we substitute for x in each equation, we get:
4.5y + y + z = 59000
.08*(4.5y) + .06y + .09z = 4710
these 2 equations simplify to:
5.5y + z = 59000 (third equation)
.42y + .09z = 4710 (fourth equation)
we can substitute for z in the fourth equation as follows:
from the third equation, we solve for z to get z = 59000 - 5.5y
in the fourth equation, we substitute for z to get:
.42y + .09*(59000-5.5y) = 4710 (fifth equation)
we can now solve for y as follows:
.42y + .09*59000 - .09*5.5y = 4710
this simplifies to:
.42y + 5310 - .495y = 4710 which further simplifies to:
-.075y + 5310 = 4710
subtract 5310 from both sides of this equation to get:
-.075y = -600
divide both sides by -.075 to get:
y = -600/-.075 = 8000.
substitute for y in the third equation of 5.5y + z = 59000 to get:
5.5*8000 + z = 59000
simplify this to get:
44000 + z = 59000
subtract 44000 from both sides of this equation to get:
z = 59000 - 44000 = 15000
so far we have:
y = 8000
z = 15000
since x + y + z = 59000 (first equation), this forces x to be equal to 36000.
we now have:
x = 36000
y = 8000
z = 15000
our first equation is satisfied because:
x + y + z = 36000 + 8000 + 15000 = 59000.
our second equation is satisfied because:
.08*36000 + .06*8000 + .09*15000 = 2880 + 480 + 1350 = 4710
our additional requirement is satisfied because:
2880 is equal to 6 * 480.
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