Objectives 

• Identifying the Relevant Variables

• Constructing Equations 

• Solving Equations and Checking Solutions

Examples

• Example 1

• Example 2

• Example 3

• Example 4

A contestant wins $4000 on the "Wheel of Wealth" game show. She invests part of her winnings in an account which earns 8% annual interest. She places the rest of the money in an account which earns 6% annual interest. After one year the total interest earned from both accounts is $300. How much money was in each account originally?

Hint:  Let x = amount invested at 8 %

Solution
    Our goal is to construct a one-variable equation which 
    matches the given information.

    Step 1: Identify the desired variable.

        The hint tells us to use the variable "x" to represent the
        amount invested at 8 % interest.

        That is,

x = amount invested at 8 %

    Step 2: Identify the known relationships in the problem.

        The problem statement tells us that the total money
        invested is $4000.

     Since we are representing the amount in the 8 % account
        by the variable "x", we can represent the amount of money
        in the 6 % account by the expression "4000 - x."

        That is,

4000 - x = amount invested at 6 %

       We will also need the formula:

(money invested)(annual interest rate)(time) = interest earned


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Vocabulary & notation: 

        "Principal" =money invested = P

         annual interest rate = r

         time = t

         interest earned = I

Note:  You MUST write the interest rate as a DECIMAL!

8 % = 8/100 = 0.08

6% = 6/100 =0 .06

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        We can construct a table to help us organize the given
        information.
 

	P	r	t	I
8 % account	x	0.08	1	(0.08)(x)
6 % account	4000 - x	0.06	1	(0.06)(4000 - x)

Step 3: Construct a one variable equation which fits the known
              relationships in the problem.

        The problem statement tells us that the total interest earned from the two accounts is $300.

        Using the entires from the "interest earned" column in our
        table, we have the equation:

Total Interest earned = $300

(0.08)(x) + (0.06)(4000 - x)  = 300

Step 4: Solve the equation.

         Solving for x, we get

0.08x + 240 - 0.06x  = 300

0.02x = 60

x = (60)/(0.02) = 3000

Step 5: Use the solution from Step 4 to answer the question(s)
              posed in the original problem. 

Plugging the value x = 3000 into our table, we see that the contestant invested $3000 at 8% and $1000 at 6 %.


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Special Note: Be sure to check that your final answers "make sense" in the context of the original problem. 

In this problem, for example, our equation needed to yield a non-negative number for the value of "x". 

That is, it wouldn't "make sense" for the contestant to have invested a negative amount of money.