Investing at Two Different Interest Rates

Date: 11/24/2003 at 00:47:23
From: Ryan
Subject: (no subject)

I am having problems with simple interest.

If Pedro invested $5000 for one year, part at 12% annual interest and 
the balance at 10% annual interest.  His total interest for the year 
was $568.  How much money did he invest at each rate?

I know the formula i = prt but I'm lost.

Date: 11/24/2003 at 05:21:59
From: Doctor Jeremiah
Subject: Re: (no subject)

Hi Ryan,

Your formula i = prt is correct, but in this question there are two
values of p because some money was done at 10% and some was done at
12%, so we need to treat them as completely separate.  That also means
that there are two values of r and i (one for each value of p).  So we
can write four equations that represent the problem:

  i1 + i2 = 568      (total interest is $568)
  p1 + p2 = 5000     (total principal is $5000)
  i1 = p1 * r1 * t   (interest earned at rate 1)
  i2 = p2 * r2 * t   (interest earned at rate 2)

In your specific problem, t = 1, r1 = 0.10 and r2 = 0.12 so:

  i1 + i2 = 568
  p1 + p2 = 5000
  i1 = p1(0.10)(1)
  i2 = p2(0.12)(1)

Now we have four equations and four unknowns.  The last two equations
tell us what i1 and i2 are equal to, so let's replace the i1 and i2 in
the first equation with those two expressions:

  i1 + i2 = 568   where   i1 = p1(0.10)(1) and i2 = p2(0.12)(1)
  p1(0.10)(1) + p2(0.12)(1) = 568

Now we have these two equations:

  p1(0.10) + p2(0.12) = 568
  p1 + p2 = 5000

If you know how to solve a system of two linear equation, you can
solve these two for p1 and p2, which are the amounts that he invested
at each rate.  

If you aren't sure how to solve a system of equations, and this method
seems a little complicated, here's another way that uses only one
variable, even though the math behind it is the same as what we just did.

Let x = the amount Pedro invested at 12% interest.  Since his total
investment was $5000, he must have 5000 - x left to invest at 10%
interest.  Do you see why?  That's the key step to this method!

Now we can make a chart using i = prt and filling in what we know for
each of the two investments:

          Principal    Rate    Time    Interest
          ---------    ----    ----    -------------------
  At 12% |    x        .12       1     (x)(.12)(1)        |
  At 10% |5000 - x     .10       1     (5000 - x)(.10)(1) |

Since we know that the total interest he earned was $568, we can write
an equation showing that adding the two interest amounts from the
chart must make 568:

  (x)(.12)(1) + (5000 - x)(.10)(1) = 568

Distributing we get:

  .12x + 500 - .10x = 568

You can solve this equation for x and then use x to calculate the
principals, or how much he invested at each rate, in the chart above.

Let me know if you get stuck or if you have other questions.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/