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Orlando is mixing nuts and cereal squares to make a party mix. Nuts sell for $7 a pound and cereal squares sell for $4 a pound. Orlando wants to make 30 pounds of party mix at a cost of $6.50 a pound, how many pounds of nuts and how many pounds of cereal squares should he use?

5 pounds cereal squares, 25 pounds nuts

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Becca wants to mix fruit juice and soda to make a punch. She can buy fruit juice for $3 a gallon and soda for $4 a gallon. If she wants to make 28 gallons of punch at a cost of $3.25 a gallon, how many gallons of fruit juice and how many gallons of soda should she buy?

21 gallons of fruit punch, 7 gallons of soda

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

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.

Let x = amount invested at 3%. 20,000 x = amount invested at 5%

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.

This table has four rows and four columns. The top row is a header row that reads from left to right Type, Amount invested, Rate, and Interest. The second row reads 3%, x, 0.03, and 0.03x. The third row reads 5%, 20,000 minus x, 0.05, and 0.05 times the quantity (20,000 minus x). The fourth row reads 4.5%, 20,000, 0.045, and 0.045 times 20,000.


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 ( 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.
0.03 x + 0.05 ( 20,000 x ) = 0.045 ( 20,000 ) 0.03 x + 1,000 0.05 x = 900 0.02 x + 1,000 = 900 0.02 x = 100 x = 5,000
amount invested at 3%
Find the amount invested at 5%. .
.
.
Check.
0.03 x + 0.05 ( 15,000 + x ) 0.045 ( 20,000 ) 150 + 750 900 900 = 900
Stacey should invest $5,000 in the account that
earns 3% and $15,000 in the account that earns 5%.

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Remy has $14,000 to invest in two mutual funds. One fund pays interest at 4% per year and the other fund pays interest at 7% per year. How much should she invest in each fund if she wants to earn 6.1% interest on the total amount?

$4,200 at 4%, $9,800 at 7%

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Marco has $8,000 to save for his daughter’s college education. He wants to divide it between one account that pays 3.2% interest per year and another account that pays 8% interest per year. How much should he invest in each account if he wants the interest on the total investment to be 6.5%?

$2,500 at 3.2%, $5,500 at 8%

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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.
    n u m b e r · v a l u e = t o t a l v a l u e 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.

Questions & Answers

Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Katherine Reply
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
Marisol Reply
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid
I'm mathematics teacher from highly recognized university.
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is anyone else having issues with the links not doing anything?
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Yes
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chapter 1 foundations 1.2 exercises variables and algebraic symbols
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i need help how to do this is confusing
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Practice Key Terms 1

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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