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.