SOLUTION: Part of $20,000 is invested at 5% and the rest at 3%. How much is invested at each rate if the total income is 4.5% of the total investment?

Question 111380: Part of $20,000 is invested at 5% and the rest at 3%. How much is invested at each rate if the total income is 4.5% of the total investment?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Let x=amount invested for 5%, y=amount for 3%

So we know the sum of the parts is the total $20,000. So this means

Also we know that 5% of x amount of dollars plus 3% of y amount of dollars is 4.5% of $20,000 (which is $900)

So

Multiply both sides by 100 to make every number a whole number

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition

Lets start with the given system of linear equations

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 5 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 5 is 5, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -1 like this:

Multiply the top equation (both sides) by 5
Multiply the bottom equation (both sides) by -1

So after multiplying we get this:

Notice how 5 and -5 add to zero (ie )

Now add the equations together. In order to add 2 equations, group like terms and combine them

Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.

So after adding and canceling out the x terms we're left with:

Divide both sides by to solve for y

Reduce

Now plug this answer into the top equation to solve for x

Plug in

Multiply

Subtract from both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out on the left side.

Multiply the terms on the right side

So our answer is

,

which also looks like

(, )

Notice if we graph the equations (if you need help with graphing, check out this solver)

we get

graph of (red) (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).

and we can see that the two equations intersect at (,). This verifies our answer.

note: ignore the graph. It didn't turn out right

So $15,000 was invested at 5% and $5,000 was invested at 3%
RELATED QUESTIONS
part of 100,000.00 invested at 5% and the rest at 6%. if the income from both invesment... (answered by macston)

A man invested P140,000. Part of his capital was invested at 11% and the rest at 9%. If... (answered by Theo,greenestamps)

A total of $12,000 was invested. Part of the money was invested at 4% and the rest... (answered by checkley79)

A total of $9000 was invested, part at 6% and the rest at 3% per year. If the annual... (answered by ikleyn)

Part of a $8, 000 inheritance was invested at 10% interest and the rest was in- vested at (answered by stanbon)

Sergio invests part of his 90,000 at 4% and the rest at 3%. How much is invested at each... (answered by lynnlo,mananth)

Sergio invests part of his 90,000 PESOS at 4% and the rest at 3%. How much money is... (answered by mananth)

Mrs. Jones invested a total of $12,000, part at 10% and the rest at 5%. If the income... (answered by scott8148)

Nestor invested 5000 pesos, part of it at 3% and the rest at 5% simple interest. If the... (answered by mananth)