Answers: 2700 and 5300 in that order
Katie invested <u>$2700</u> at a rate of 3% and <u>$5300</u> at a rate of 4%
===============================================================
Explanation:
- x = amount invested at a rate of 3%
- y = amount invested at a rate of 4%
Those two amounts must add to 8000 because this is how much she invests total. So x+y = 8000 is the first equation to set up.
The formula we'll be using is the simple interest formula, which is,
i = p*r*t
where p is the amount deposited or principal, r is the interest rate, and t is the number of years. Luckily, we don't have to worry about the time value t because t = 1 so it effectively goes away. We just need to worry about p and r.
The variable p will be tied directly to the values of x and y, as these are deposit amounts. The r values are decimal forms of the percentages (eg: 3% converts to 0.03)
In short, Katie will earn
- 0.03x dollars from the first account (3% account)
- 0.04y dollars from the second account (4% account)
In total, she earns 0.03x+0.04y dollars combined. This is set equal to 293 as this is the stated interest she earns over the entire year.
So another equation to set up is 0.03x+0.04y = 293
This only works for simple interest. Compound interest may likely be a different story.
----------------------------------------------------
To summarize so far, we found these two equations:
- x+y = 8000
- 0.03x+0.04y = 293
We have a number of options on how to solve this system. I'll use substitution.
Solve the first equation for y
x+y = 8000
y = 8000-x
Then we'll plug this into the other equation to solve for x like so,
0.03x+0.04y = 293
0.03x+0.04( y ) = 293
0.03x+0.04( 8000-x ) = 293
0.03x+0.04(8000)+0.04(-x) = 293
0.03x+320-0.04x = 293
-0.01x+320 = 293
-0.01x = 293-320
-0.01x = -27
x = -27/(-0.01)
x = 2700
This tells us that Katie invested $<u>2700</u> at 3%
Use this x value to find y
y = 8000-x
y = 8000-2700
y = 5300
She also invested $<u>5300</u> at 4%
----------------------------
As a quick check,
so that works out
Also,
- 0.03*x = 0.03*2700 = 81
- 0.04*y = 0.04*5300 = 212
- 0.03x+0.04y = 81+212 = 293
which also works out. The answers are fully confirmed.
