3.3 Solve mixture applications  (Page 6/10)

We can also use the mixture model to solve investment problems using simple interest    . We have used the simple interest formula, $I=Prt,$ where $t$ represented the number of years. When we just need to find the interest for one year, $t=1,$ so then $I=Pr.$

Stacey has $20,000 to invest in two different bank accounts. One account pays interest at 3% per year and the other account pays interest at 5% per year. How much should she invest in each account if she wants to earn 4.5% interest per year on the total amount?

Solution

We will fill in a chart to organize our information. We will use the simple interest formula to find the interest earned in the different accounts.

The interest on the mixed investment will come from adding the interest from the account earning 3% and the interest from the account earning 5% to get the total interest on the $20,000.

$\begin{array}{ccc}\hfill \text{Let}x& =\hfill & \text{amount invested at 3\%.}\hfill \\ \hfill \mathrm{20,000}-x& =\hfill & \text{amount invested at 5\%}\hfill \end{array}$

The amount invested is the principal for each account.

We enter the interest rate for each account.

We multiply the amount invested times the rate to get the interest.

Notice that the total amount invested, 20,000, is the sum of the amount invested at 3% and the amount invested at 5%. And the total interest, $0.045(\mathrm{20,000}),$ is the sum of the interest earned in the 3% account and the interest earned in the 5% account.

As with the other mixture applications, the last column in the table gives us the equation to solve.

Write the equation from the interest earned. Solve the equation.	$\begin{array}{ccc}\hfill 0.03x+0.05(\mathrm{20,000}-x)& =\hfill & 0.045\left(\mathrm{20,000}\right)\hfill \\ \\ \\ \hfill 0.03x+\mathrm{1,000}-0.05x& =\hfill & 900\hfill \\ \hfill -0.02x+\mathrm{1,000}& =\hfill & 900\hfill \\ \hfill -0.02x& =\hfill & -100\hfill \\ \hfill x& =\hfill & \mathrm{5,000}\hfill \end{array}$ amount invested at 3%
Find the amount invested at 5%.
Check. $\begin{array}{}\\ \hfill 0.03x+0.05(\mathrm{15,000}+x)& \stackrel{?}{=}\hfill & 0.045\left(\mathrm{20,000}\right)\hfill \\ \hfill 150+750& \stackrel{?}{=}\hfill & 900\hfill \\ \hfill 900& =\hfill & 900✓\hfill \end{array}$
	Stacey should invest $5,000 in the account that earns 3% and $15,000 in the account that earns 5%.

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Key concepts

• Total Value of Coins For the same type of coin, the total value of a number of coins is found by using the model.
  $number·value=totalvalue$ where number is the number of coins and value is the value of each coin; total value is the total value of all the coins
• Problem-Solving Strategy—Coin Word Problems
  1. Read the problem. Make all the words and ideas are understood. Determine the types of coins involved.
     
     • Create a table to organize the information.
     • Label the columns type, number, value, total value.
     • List the types of coins.
     • Write in the value of each type of coin.
     • Write in the total value of all the coins.
  2. Identify what we are looking for.
  3. Name what we are looking for. Choose a variable to represent that quantity.
     Use variable expressions to represent the number of each type of coin and write them in the table.
     Multiply the number times the value to get the total value of each type of coin.
  4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation.
     Write the equation by adding the total values of all the types of coins.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.