if $9,000 is to be invested, part at 13% and the rest at 8% simple interest , how much should be invested at each rate so that the total annual return will be the same as $9,000 invested at 9% set up as a system of linear equation

Question

if $9,000 is to be invested, part at 13% and the rest at 8% simple interest , how much should be invested at each rate so that the total annual return will be the same as $9,000 invested at 9% set up as a system of linear equation

Answer 1

It's a linear partition problem, since both investments are paid simple interest.

Amount invested at 8%
=$9000*(13-9)/(13-8)
=$7200
Amount invested at 13%
=$9000*(9-8)/(13-8)
=$1800

Alternatively, using algebra,
Let x=amount invested at 8%,
Then (9000-x)=amount invested at 13%
and the interest obtained should be the same as $9000 invested at 9%:
x*(8/100)+(9000-x)*(13/100)=9000*(9/100)
Isolate x,
x((13-8)/100)=9000(13-9)/100
x=9000*(13-9)/(13-8)=7200 as before.

Answer 2

To solve this problem, we can set up a system of linear equations using the principle of simple interest.

Let's assume the amount invested at 13% is x, and the amount invested at 8% is y.

The total amount invested is $9,000, so we have:

x + y = 9,000 ---(Equation 1)

The total annual return from the amount invested at 13% is obtained by multiplying the amount invested at 13% by the interest rate of 13%, while the total annual return from the amount invested at 8% is obtained by multiplying the amount invested at 8% by the interest rate of 8%.

This can be expressed as:

0.13x + 0.08y = 0.09(9,000) ---(Equation 2)

Let's simplify Equation 2:

0.13x + 0.08y = 810 ---(Equation 2)

Now we have a system of linear equations:

x + y = 9,000 ---(Equation 1)
0.13x + 0.08y = 810 ---(Equation 2)

To solve this system of equations, we can use any method, such as substitution or elimination.

Let's solve by elimination:

Multiply Equation 1 by 0.08 and Equation 2 by 100 to make the coefficients of y the same:

0.08x + 0.08y = 720 ---(Equation 1, multiplied by 0.08)
13x + 8y = 81000 ---(Equation 2, multiplied by 100)

Now, subtract Equation 1 from Equation 2:

13x + 8y - (0.08x + 0.08y) = 81000 - 720
12.92x + 7.92y = 80280

Simplify and convert to whole numbers:

1292x + 792y = 802800 ---(Equation 3)

Now we have the system of equations:

x + y = 9,000 ---(Equation 1)
1292x + 792y = 802800 ---(Equation 3)

You can now solve this system of equations using various methods such as substitution or elimination to find the values of x and y, which will give you the amount that should be invested at each rate to obtain a total annual return equal to $9,000 invested at 9%.