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Mathematics LibreTexts

5.5E: Exercises

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    30415
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    Practice Makes Perfect

    Solve Mixture Applications

    In the following exercises, translate to a system of equations and solve.

    Exercise \(\PageIndex{1}\)

    Tickets to a Broadway show cost $35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were $47,150. How many adult and how many child tickets were sold?

    Answer

    There 1120 adult tickets and 530 child tickets sold.

    Exercise \(\PageIndex{2}\)

    Tickets for a show are $70 for adults and $50 for children. One evening performance had a total of 300 tickets sold and the receipts totaled $17,200. How many adult and how many child tickets were sold?

    Exercise \(\PageIndex{3}\)

    Tickets for a train cost $10 for children and $22 for adults. Josie paid $1,200 for a total of 72 tickets. How many children’s tickets and how many adult tickets did Josie buy?

    Answer

    Josie bought 40 adult tickets and 32 children tickets.

    Exercise \(\PageIndex{4}\)

    Tickets for a baseball game are $69 for Main Level seats and $39 for Terrace Level seats. A group of sixteen friends went to the game and spent a total of $804 for the tickets. How many of Main Level and how many Terrace Level tickets did they buy?

    Exercise \(\PageIndex{5}\)

    Tickets for a dance recital cost $15 for adults and $7 for children. The dance company sold 253 tickets and the total receipts were $2,771. How many adult tickets and how many child tickets were sold?

    Answer

    There were 125 adult tickets and 128 children tickets sold.

    Exercise \(\PageIndex{6}\)

    Tickets for the community fair cost $12 for adults and $5 dollars for children. On the first day of the fair, 312 tickets were sold for a total of $2,204. How many adult tickets and how many child tickets were sold?

    Exercise \(\PageIndex{7}\)

    Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have?

    Answer

    Brandon has 12 quarters and 8 dimes.

    Exercise \(\PageIndex{8}\)

    Sherri saves nickels and dimes in a coin purse for her daughter. The total value of the coins in the purse is $0.95. The number of nickels is two less than five times the number of dimes. How many nickels and how many dimes are in the coin purse?

    Exercise \(\PageIndex{9}\)

    Peter has been saving his loose change for several days. When he counted his quarters and dimes, he found they had a total value $13.10. The number of quarters was fifteen more than three times the number of dimes. How many quarters and how many dimes did Peter have?

    Answer

    Peter had 11 dimes and 48 quarters.

    Exercise \(\PageIndex{10}\)

    Lucinda had a pocketful of dimes and quarters with a value of $ $6.20. The number of dimes is eighteen more than three times the number of quarters. How many dimes and how many quarters does Lucinda have?

    Exercise \(\PageIndex{11}\)

    A cashier has 30 bills, all of which are $10 or $20 bills. The total value of the money is $460. How many of each type of bill does the cashier have?

    Answer

    The cashier has fourteen $10 bills and sixteen $20 bills.

    Exercise \(\PageIndex{12}\)

    A cashier has 54 bills, all of which are $10 or $20 bills. The total value of the money is $910. How many of each type of bill does the cashier have?

    Exercise \(\PageIndex{13}\)

    Marissa wants to blend candy selling for $1.80 per pound with candy costing $1.20 per pound to get a mixture that costs her $1.40 per pound to make. She wants to make 90 pounds of the candy blend. How many pounds of each type of candy should she use?

    Answer

    Marissa should use 60 pounds of the $1.20/lb candy and 30 pounds of the $1.80/lb candy.

    Exercise \(\PageIndex{14}\)

    How many pounds of nuts selling for $6 per pound and raisins selling for $3 per pound should Kurt combine to obtain 120 pounds of trail mix that cost him $5 per pound?

    Exercise \(\PageIndex{15}\)

    Hannah has to make twenty-five gallons of punch for a potluck. The punch is made of soda and fruit drink. The cost of the soda is $1.79 per gallon and the cost of the fruit drink is $2.49 per gallon. Hannah’s budget requires that the punch cost $2.21 per gallon. How many gallons of soda and how many gallons of fruit drink does she need?

    Answer

    Hannah needs 10 gallons of soda and 15 gallons of fruit drink.

    Exercise \(\PageIndex{16}\)

    Joseph would like to make 12 pounds of a coffee blend at a cost of $6.25 per pound. He blends Ground Chicory at $4.40 a pound with Jamaican Blue Mountain at $8.84 per pound. How much of each type of coffee should he use?

    Exercise \(\PageIndex{17}\)

    Julia and her husband own a coffee shop. They experimented with mixing a City Roast Columbian coffee that cost $7.80 per pound with French Roast Columbian coffee that cost $8.10 per pound to make a 20 pound blend. Their blend should cost them $7.92 per pound. How much of each type of coffee should they buy?

    Answer

    Julia and her husband should buy 12 pounds of City Roast Columbian coffee and 8 pounds of French Roast Columbian coffee.

    Exercise \(\PageIndex{18}\)

    Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $4.89 per bag with peanut butter pieces that cost $3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use?

    Exercise \(\PageIndex{19}\)

    Jotham needs 70 liters of a 50% alcohol solution. He has a 30% and an 80% solution available. How many liters of the 30% and how many liters of the 80% solutions should he mix to make the 50% solution?

    Answer

    Jotham should mix 42 liters of the 30% solution and 28 liters of the 80% solution.

    Exercise \(\PageIndex{20}\)

    Joy is preparing 15 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution?

    Exercise \(\PageIndex{21}\)

    A scientist needs 65 liters of a 15% alcohol solution. She has available a 25% and a 12% solution. How many liters of the 25% and how many liters of the 12% solutions should she mix to make the 15% solution?

    Answer

    The scientist should mix 15 liters of the 25% solution and 50 liters of the 12% solution.

    Exercise \(\PageIndex{22}\)

    A scientist needs 120 liters of a 20% acid solution for an experiment. The lab has available a 25% and a 10% solution. How many liters of the 25% and how many liters of the 10% solutions should the scientist mix to make the 20% solution?

    Exercise \(\PageIndex{23}\)

    A 40% antifreeze solution is to be mixed with a 70% antifreeze solution to get 240 liters of a 50% solution. How many liters of the 40% and how many liters of the 70% solutions will be used?

    Answer

    160 liters of the 40% solution and 80 liters of the 70% solution will be used.

    Exercise \(\PageIndex{24}\)

    A 90% antifreeze solution is to be mixed with a 75% antifreeze solution to get 360 liters of a 85% solution. How many liters of the 90% and how many liters of the 75% solutions will be used?

    Solve Interest Applications

    In the following exercises, translate to a system of equations and solve.

    Exercise \(\PageIndex{25}\)

    Hattie had $3,000 to invest and wants to earn 10.6% interest per year. She will put some of the money into an account that earns 12% per year and the rest into an account that earns 10% per year. How much money should she put into each account?

    Answer

    Hattie should invest $900 at 12% and $2,100 at 10%.

    Exercise \(\PageIndex{26}\)

    Carol invested $2,560 into two accounts. One account paid 8% interest and the other paid 6% interest. She earned 7.25% interest on the total investment. How much money did she put in each account?

    Exercise \(\PageIndex{27}\)

    Sam invested $48,000, some at 6% interest and the rest at 10%. How much did he invest at each rate if he received $4,000 in interest in one year?

    Answer

    Sam invested $28,000 at 10% and $20,000 at 6%.

    Exercise \(\PageIndex{28}\)

    Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year?

    Exercise \(\PageIndex{29}\)

    After four years in college, Josie owes $65,800 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owed for one year was $2,878.50. What is the amount of each loan?

    Answer

    The federal loan is $62,500 and the bank loan is $3,300.

    Exercise \(\PageIndex{30}\)

    Mark wants to invest $10,000 to pay for his daughter’s wedding next year. He will invest some of the money in a short term CD that pays 12% interest and the rest in a money market savings account that pays 5% interest. How much should he invest at each rate if he wants to earn $1,095 in interest in one year?

    Exercise \(\PageIndex{31}\)

    A trust fund worth $25,000 is invested in two different portfolios. This year, one portfolio is expected to earn 5.25% interest and the other is expected to earn 4%. Plans are for the total interest on the fund to be $1150 in one year. How much money should be invested at each rate?

    Answer

    $12,000 should be invested at 5.25% and $13,000 should be invested at 4%.

    Exercise \(\PageIndex{32}\)

    A business has two loans totaling $85,000. One loan has a rate of 6% and the other has a rate of 4.5%. This year, the business expects to pay $4650 in interest on the two loans. How much is each loan?

    Everyday Math

    In the following exercises, translate to a system of equations and solve.

    Exercise \(\PageIndex{33}\)

    Laurie was completing the treasurer’s report for her son’s Boy Scout troop at the end of the school year. She didn’t remember how many boys had paid the $15 full-year registration fee and how many had paid the $10 partial-year fee. She knew that the number of boys who paid for a full-year was ten more than the number who paid for a partial-year. If $250 was collected for all the registrations, how many boys had paid the full-year fee and how many had paid the partial-year fee?

    Answer

    14 boys paid the full-year fee. 4 boys paid the partial-year fee

    Exercise \(\PageIndex{34}\)

    As the treasurer of her daughter’s Girl Scout troop, Laney collected money for some girls and adults to go to a three-day camp. Each girl paid $75 and each adult paid $30. The total amount of money collected for camp was $765. If the number of girls is three times the number of adults, how many girls and how many adults paid for camp?

    Writing Exercises

    Exercise \(\PageIndex{35}\)

    Take a handful of two types of coins, and write a problem similar to Example relating the total number of coins and their total value. Set up a system of equations to describe your situation and then solve it.

    Answer

    Answers will vary.

    Exercise \(\PageIndex{36}\)

    In Example we solved the system of equations \(\left\{\begin{array}{l}{b+f=21,540} \\ {0.105 b+0.059 f=1669.68}\end{array}\right.\) by substitution. Would you have used substitution or elimination to solve this system? Why?

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This figure shows a table with four rows and four columns. The columns are labeled, “I can…,” “Confidently.” “With some help.” and “No - I don’t get it.” The only column with filled in cells below it is labeled “I can…” It reads, “solve mixture applications.” “solve interest applications.”

    b. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?