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3.3 Solve mixture applications  (Page 3/10)

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In the next example, we’ll show only the completed table—remember the steps we take to fill in the table.

Danny has $2.14 worth of pennies and nickels in his piggy bank. The number of nickels is two more than ten times the number of pennies. How many nickels and how many pennies does Danny have?

Solution

Step 1. Read the problem.
Determine the types of coins involved. pennies and nickels
Create a table.
Write in the value of each type of coin. Pennies are worth $0.01.
Nickels are worth $0.05.
Step 2. Identify what we are looking for. the number of pennies and nickels
Step 3. Name. Represent the number of each type of coin using variables.
The number of nickels is defined in terms of the number of pennies, so start with pennies. Let p = number of pennies.
The number of nickels is two more than ten times the number of pennies. 10 p + 2 = number of nickels.
Multiply the number and the value to get the total value of each type of coin.
This table has three rows and four columns with an extra cell at the bottom of the fourth column. The top row is a header row that reads from left to right Type, Number, Value ($), and Total Value ($). The second row reads pennies, p, 0.01, and 0.01p. The third row reads nickels, 10p plus 2, 0.05, and 0.05 times the quantity (10p plus 2). The extra cell reads $2.14.
Step 4. Translate. Write the equation by adding the total value of all the types of coins. .
Step 5. Solve the equation. .
How many nickels? .
Step 6. Check the answer in the problem and make sure it makes sense
Danny has four pennies and 42 nickels.
Is the total value $2.14?
4 ( 0.01 ) + 42 ( 0.05 ) ≟ 2.14 2.14 = 2.14 ✓
Step 7. Answer the question. Danny has four pennies and 42 nickels.
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Jesse has $6.55 worth of quarters and nickels in his pocket. The number of nickels is five more than two times the number of quarters. How many nickels and how many quarters does Jesse have?

41 nickels, 18 quarters

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Elane has $7.00 total in dimes and nickels in her coin jar. The number of dimes that Elane has is seven less than three times the number of nickels. How many of each coin does Elane have?

22 nickels, 59 dimes

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Solve ticket and stamp word problems

Problems involving tickets or stamps are very much like coin problems. Each type of ticket and stamp has a value, just like each type of coin does. So to solve these problems, we will follow the same steps we used to solve coin problems.

At a school concert, the total value of tickets sold was $1,506. Student tickets sold for $6 each and adult tickets sold for $9 each. The number of adult tickets sold was five less than three times the number of student tickets sold. How many student tickets and how many adult tickets were sold?

Solution

Step 1. Read the problem.

  • Determine the types of tickets involved. There are student tickets and adult tickets.
  • Create a table to organize the information.


This table has three rows and four columns with an extra cell at the bottom of the fourth column. The top row is a header row that reads from left to right Type, Number, Value ($), and Total Value ($). The second row reads Student, blank, 6, and blank. The third row reads Adult, blank, 9, and blank. The extra cell reads 1506.

Step 2. Identify what we are looking for.

  • We are looking for the number of student and adult tickets.

Step 3. Name. Represent the number of each type of ticket using variables.

We know the number of adult tickets sold was five less than three times the number of student tickets sold.

  • Let s be the number of student tickets.
  • Then 3 s − 5 is the number of adult tickets

Multiply the number times the value to get the total value of each type of ticket.

This table has three rows and four columns with an extra cell at the bottom of the fourth column. The top row is a header row that reads from left to right Type, Number, Value ($), and Total Value ($). The second row reads Student, s, 6, and 6s. The third row reads Adult, 3s minus 5, 9, and 9 times the quantity (3s minus 5). The extra cell reads 1506.

Step 4. Translate. Write the equation by adding the total values of each type of ticket.

6 s + 9 ( 3 s − 5 ) = 1506

Step 5. Solve the equation.

6 s + 27 s − 45 = 1506 33 s − 45 = 1506 33 s = 1551 s = 47 student tickets
3 s − 5 3 ( 47 ) − 5
136 adult tickets

Step 6. Check the answer.

There were 47 student tickets at $6 each and 136 adult tickets at $9 each. Is the total value $1,506? We find the total value of each type of ticket by multiplying the number of tickets times its value then add to get the total value of all the tickets sold.

47 · 6 = 282 136 · 9 = 1,224 _____ 1,506 ✓

Step 7. Answer the question. They sold 47 student tickets and 136 adult tickets.

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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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