Question 351114: PLS. HELP ME SOLVE THIS PROBLEM. TNX.
1. A man divides P100,000 among his three investments, at 4%, 5%, and 6% per annum, respectively. The total income is P4,900 per year. His income from the 5% and 6% investments exceeds the income at 4% by P 2,500. Find how much is invested at each rate.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Total investment is 100,000
x amount of investment is invested at 4% per year.
y amount of investment is invested at 5% per year.
z amount of investment is invested at 6% per year.
income from y and z exceeds income from x by 2,500.
first equation states that the total investment is composed of investment x, y, and z, and totals 100,000.
x + y + z = 100,000 (first equation)
second equation states that income from all of these investments equals 4,900 per year.
.04*x + .05*y + .06*z = 4,900 (second equation)
you have 2 equations in 3 unknowns.
we'll try to reduce the number of unknowns by establishing some equivalencies.
you are also given that the income from investment y and investment z exceeds the income from investment x by 2,500.
the equation to express that relationship is shown below:
.05*y + .06*z = .04*x + 2,500
we can use this equation to reduce the number of unknowns in the first 2 equations.
we'll solve for .04*x to get:
.04*x = .05*y + .06*z - 2,500
we'll solve for x to get:
x = (.05*y + .06*z - 2,500) / .04
we now have values for x and .04*x in terms of y and z.
we'll take the first equation and the second equation and substitute for x and .04*x in them as follows:
the equations are:
x + y + z = 100,000 (first equation)
.04*x + .05*y + .06*z = 4,900 (second equation)
substituting for x and .04*x in these equations gets us:
((.05*y + .06*z - 2,500) / .04) + y + z = 100,000
and:
(.05*y + .06*z - 2,500) + .05*y + .06*z = 4,900
if we solve these 2 equations in 2 unknowns simultaneously, we should be able to get our answer.
first step is to simplify these equations.
modified first equation is:
((.05*y + .06*z - 2,500) / .04) + y + z = 100,000
modified first equation simplifies to:
1.25*y + 1.5*z - 62,500 + y + z = 100,000
combine like terms to get:
2.25*y + 2.5*z - 62,500 = 100,000
add 62,500 to both sides of the equation to get:
2.25*y + 2.5*z = 162,500 (modified first equation)
modified second equation is:
(.05*y + .06*z - 2,500) + .05*y + .06*z = 4,900
modified second equation simplifies to:
.05*y + .06*z - 2,500 + .05*y + .06*z = 4,900
combine like terms to get:
.10*y + .12*z - 2,500 = 4,900
add 2,500 to both sides of the equation to get:
.10*y + .12*z = 7,400 (modified second equation).
your 2 equations that now need to be solved simultaneously are:
2.25*y + 2.5*z = 162,500 (modified first equation)
and:
.10*y + .12*z = 7,400 (modified second equation).
Multiply both sides of the second equation by 22.5 to get:
2.25*y + 2.5*z = 162,500 (modified first equation)
and:
2.25*y + 2.7*z = 166,500 (modified second equation).
Subtract the first equation from the second equation to get:
.2*z = 4,000
Divide both sides of this equation by .2 to get:
z = 4,000 / .2 = 20,000
use this value of z to solve for y in either one of the modified first equation or modified second equation (the original modified before we started multiplying them out.
those 2 original modified equations are:
2.25*y + 2.5*z = 162,500 (modified first equation)
and:
.10*y + .12*z = 7,400 (modified second equation).
In the first of these equation, let z = 20,000 to get:
2.25*y + 2.5*(20,000) = 162,500
Simplify to get:
2.25*y + 50,000 = 162,500
Subtract 50,000 from both sides of this equation to get:
2.25*y = 162,500 - 50,000
Simplify further to get:
2.25*y = 112,500
Divide both sides of this equation by 2.25 to get:
y = 50,000
If you solved for y in the second equation, you would have gotten the same answer, as you should. I checked it out and it's good.
You now have 2 values.
y = 50,000, and z = 20,000
Since x + y + z = 100,000, that must mean that x = 30,000
Your answer should be:
x = 30,000
y = 50,000
z = 20,000
It remains only to check your answers to confirm that they are good.
Your original first equation is:
x + y + z = 100,000
That's clearly true.
Your original second equation is:
.04*x + .05*y + .06*z = 4,900
Substituting for x,y,z that we solved for, this becomes:
.04*30,000 + .05*50,000 + .06*20,000 = 4,900
This simplifies to:
1,200 + 2,500 + 1,200 = 4,900
This simplifies further to:
4,900 = 4,900 which is also true.
The third equation that we used to reduce the number of unknowns to 2 was:
.05*y + .06*z = .04*x + 2,500
Substituting for x, y, and z in this equation, we get:
.05*50,000 + .06*20,000 = .04*30,000 + 2,500
This equation simplifies to:
2,500 + 1,200 = 1,200 + 2,500
This equation simplifies to:
3,700 = 3,700 which is also true.
Our answers for x,y,z are good.
x = investment at 4% = 30,000
y = investment at 5% = 50,000
z = investment at 6% = 20,000
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