Interest Problems

Ann invested \$12,000 in two bank accounts.  One of the accounts pays 6% annual interest, and the other account pays 5% annual interest.  If the combined interest earned in both accounts after a year was \$700, how much money was invested in each account?

What are we trying to find in this problem?

We want to know the amount of money invested in each account-- in other words, we want to know the amount invested in the 6% account and the amount invested in the 5% account.  Each of the things we are trying to find will be represented by a variable:

x = amount invested at 6%
y = amount invested at 5%

Since we have two variables to solve for, we will need to find a system of two equations to solve.

How do we find the two equations we need?

We are given two numbers in the problem:

\$12,000 = total money invested in both accounts
\$700 = total interest earned in both accounts

Let's start with the \$12,000.  Ann wants to split this money into two parts.  We have chosen to call the two parts x and y.  Since these two parts must total to \$12,000, this gives us our first equation:

x + y = 12,000

Now let's look at the \$700, the interest earned on the two accounts together.  Let's think about the formula for calculating simple interest :

Interest = (Principle)(Rate)(Time)

Since the time period in this problem is one year, our simple interest equation becomes:

Interest = (Principle)(Rate)(1)
or
Interest = (Principle)(Rate)

Each account has a different amount of money invested in it (either x dollars or y dollars), and each account has a different interest rate (either 6% or 5%). This gives us the following:

Interest earned on x dollars = (x)(6%) = .06x

and

Interest earned on y dollars = (y)(5%) = .05y

The total interest earned in both accounts is \$700, so our second equation is:

Interest earned on x dollars + interest earned on y dollars = total interest
.06x + .05y = 700

If we multiply both sides of this equation by 100 to clear the decimals, it becomes:
6x + 5y = 70,000

Now we'll solve the system of equations:

x + y = 12,000
6x + 5y = 70,000

Multiply the first equation by -5, then add the equations:

-5x - 5y = -60,000
6x + 5y = 70,000
x = 10,000

Ann invested \$10,000 in the account that pays 6% interest.

To find the amount invested in the other account, substitute 10,000 for x in either of our equations.  We'll choose the easier equation:

x + y = 12,000
10,000 + y = 12,000
y = 2,000

Ann invested \$2,000 in the account that pays 5% interest.