Exercises: Solve Mixture Applications with Systems of Equations (4.5)
Exercises: Solve Mixture Applications
Instructions: For questions 1-24, translate to a system of equations and solve.
1. Tickets to a Broadway show cost [latex]$35[/latex] for adults and [latex]$15[/latex] for children. The total receipts for [latex]1650[/latex] tickets at one performance were [latex]$47\text{,}150[/latex]. How many adult and how many child tickets were sold?
Solution
There [latex]1120[/latex] adult tickets and [latex]530[/latex] child tickets sold.
2. Tickets for a show are [latex]$70[/latex] for adults and [latex]$50[/latex] for children. One evening performance had a total of [latex]300[/latex] tickets sold and the receipts totaled [latex]$17\text{,}200[/latex]. How many adult and how many child tickets were sold?
3. Tickets for a train cost [latex]$10[/latex] for children and [latex]$22[/latex] for adults. Josie paid [latex]$1\text{,}200[/latex] for a total of [latex]72[/latex] tickets. How many children’s tickets and how many adult tickets did Josie buy?
Solution
Josie bought [latex]40[/latex] adult tickets and [latex]32[/latex] children tickets.
4. Tickets for a baseball game are [latex]$69[/latex] for Main Level seats and [latex]$39[/latex] for Terrace Level seats. A group of sixteen friends went to the game and spent a total of [latex]$804[/latex] for the tickets. How many of Main Level and how many Terrace Level tickets did they buy?
5. Tickets for a dance recital cost [latex]$15[/latex] for adults and [latex]$7[/latex] for children. The dance company sold [latex]253[/latex] tickets and the total receipts were [latex]$2\text{,}771[/latex]. How many adult tickets and how many child tickets were sold?
Solution
There were [latex]125[/latex] adult tickets and [latex]128[/latex] children tickets sold.
6. Tickets for the community fair cost [latex]$12[/latex] for adults and [latex]$5[/latex] dollars for children. On the first day of the fair, [latex]312[/latex] tickets were sold for a total of [latex]$2\text{,}204[/latex]. How many adult tickets and how many child tickets were sold?
7. Brandon has a cup of quarters and dimes with a total value of [latex]$3.80[/latex]. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have?
Solution
Brandon has [latex]12[/latex] quarters and [latex]8[/latex] dimes.
8. Sherri saves nickels and dimes in a coin purse for her daughter. The total value of the coins in the purse is [latex]$0.95[/latex]. 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?
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 [latex]$13.10[/latex]. The number of quarters was fifteen more than three times the number of dimes. How many quarters and how many dimes did Peter have?
Solution
Peter had [latex]11[/latex] dimes and [latex]48[/latex] quarters.
10. Lucinda had a pocketful of dimes and quarters with a value of ? [latex]$6.20[/latex]. The number of dimes is eighteen more than three times the number of quarters. How many dimes and how many quarters does Lucinda have?
11. A cashier has [latex]30[/latex] bills, all of which are [latex]$10[/latex] or [latex]$20[/latex] bills. The total value of the money is [latex]$460[/latex]. How many of each type of bill does the cashier have?
Solution
The cashier has fourteen [latex]$10[/latex] bills and sixteen [latex]$20[/latex] bills.
12. A cashier has [latex]54[/latex] bills, all of which are [latex]$10[/latex] or [latex]$20[/latex] bills. The total value of the money is [latex]$910[/latex]. How many of each type of bill does the cashier have?
13. Marissa wants to blend candy selling for [latex]$1.80[/latex] per pound with candy costing [latex]$1.20[/latex] per pound to get a mixture that costs her [latex]$1.40[/latex] per pound to make. She wants to make [latex]90[/latex] pounds of the candy blend. How many pounds of each type of candy should she use?
Solution
Marissa should use [latex]60[/latex] pounds of the [latex]$1.20\text{/lb}[/latex] candy and [latex]30[/latex] pounds of the [latex]$1.80\text{/lb}[/latex] candy.
14. How many pounds of nuts selling for [latex]$6[/latex] per pound and raisins selling for [latex]$3[/latex] per pound should Kurt combine to obtain [latex]120[/latex] pounds of trail mix that cost him [latex]$5[/latex] per pound?
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 [latex]$1.79[/latex] per gallon and the cost of the fruit drink is [latex]$2.49[/latex] per gallon. Hannah’s budget requires that the punch cost [latex]$2.21[/latex] per gallon. How many gallons of soda and how many gallons of fruit drink does she need?
Solution
Hannah needs [latex]10[/latex] gallons of soda and [latex]15[/latex] gallons of fruit drink.
16. Joseph would like to make [latex]12[/latex] pounds of a coffee blend at a cost of [latex]$6.25[/latex] per pound. He blends Ground Chicory at [latex]$4.40[/latex] a pound with Jamaican Blue Mountain at [latex]$8.84[/latex] per pound. How much of each type of coffee should he use?
17. Julia and her husband own a coffee shop. They experimented with mixing a City Roast Columbian coffee that cost [latex]$7.80[/latex] per pound with French Roast Columbian coffee that cost [latex]$8.10[/latex] per pound to make a [latex]20[/latex] pound blend. Their blend should cost them [latex]$7.92[/latex] per pound. How much of each type of coffee should they buy?
Solution
Julia and her husband should buy [latex]12[/latex] pounds of City Roast Columbian coffee and [latex]8[/latex] pounds of French Roast Columbian coffee.
18. Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost [latex]$4.89[/latex] per bag with peanut butter pieces that cost [latex]$3.79[/latex] per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her [latex]$4.23[/latex] a bag to make. How many bags of chocolate pieces and how many bags of peanut butter pieces should she use?
19. Jotham needs [latex]70[/latex] liters of a [latex]50\%[/latex] alcohol solution. He has a [latex]30\%[/latex] and an [latex]80\%[/latex] solution available. How many liters of the [latex]30\%[/latex] and how many liters of the [latex]80\%[/latex] solutions should he mix to make the [latex]50\%[/latex] solution?
Solution
Jotham should mix [latex]2[/latex] liters of the [latex]30\%[/latex] solution and [latex]28[/latex] liters of the [latex]80\%[/latex] solution.
20. Joy is preparing [latex]15[/latex] liters of a [latex]25\%[/latex] saline solution. She only has [latex]40\%[/latex] and [latex]10\%[/latex] solution in her lab. How many liters of the [latex]40\%[/latex] and how many liters of the [latex]10\%[/latex] should she mix to make the [latex]25\%[/latex] solution?
21. A scientist needs [latex]65[/latex] liters of a [latex]15\%[/latex] alcohol solution. She has available a [latex]25\%[/latex] and a [latex]12\%[/latex] solution. How many liters of the [latex]25\%[/latex] and how many liters of the [latex]12\%[/latex] solutions should she mix to make the [latex]15\%[/latex] solution?
Solution
The scientist should mix [latex]15[/latex] liters of the [latex]25\%[/latex] solution and [latex]50[/latex] liters of the [latex]12\%[/latex] solution.
22. A scientist needs [latex]120[/latex] liters of a [latex]20\%[/latex] acid solution for an experiment. The lab has available a [latex]25\%[/latex] and a [latex]10\%[/latex] solution. How many liters of the [latex]25\%[/latex] and how many liters of the [latex]10\%[/latex] solutions should the scientist mix to make the [latex]20\%[/latex] solution?
23. A [latex]40\%[/latex] antifreeze solution is to be mixed with a [latex]70\%[/latex] antifreeze solution to get [latex]240[/latex] liters of a [latex]50\%[/latex] solution. How many liters of the [latex]40\%[/latex] and how many liters of the [latex]70\%[/latex] solutions will be used?
Solution
[latex]160[/latex] liters of the [latex]40\%[/latex] solution and [latex]80[/latex] liters of the [latex]70\%[/latex] solution will be used.
24. A [latex]90\%[/latex] antifreeze solution is to be mixed with a [latex]75\%[/latex] antifreeze solution to get [latex]360[/latex] liters of a [latex]85\%[/latex] solution. How many liters of the [latex]90\%[/latex] and how many liters of the [latex]75\%[/latex] solutions will be used?
Exercises: Solve Interest Applications
Instructions: For questions 25-32, translate to a system of equations and solve.
25. Hattie had [latex]$3\text{,}000[/latex] to invest and wants to earn [latex]10.6\%[/latex] interest per year. She will put some of the money into an account that earns [latex]12\%[/latex] per year and the rest into an account that earns [latex]10\%[/latex] per year. How much money should she put into each account?
Solution
Hattie should invest [latex]$900[/latex] at [latex]12\%[/latex] and [latex]$2\text{,}100[/latex] at [latex]10\%[/latex].
26. Carol invested [latex]$2\text{,}560[/latex] into two accounts. One account paid [latex]8\%[/latex] interest and the other paid [latex]6\%[/latex] interest. She earned [latex]7.25\%[/latex] interest on the total investment. How much money did she put in each account?
27. Sam invested [latex]$48\text{,}000[/latex], some at [latex]6\%[/latex] interest and the rest at [latex]10\%[/latex]. How much did he invest at each rate if he received [latex]$4\text{,}000[/latex] in interest in one year?
Solution
Sam invested [latex]$28\text{,}000[/latex] at [latex]10\%[/latex] and [latex]$20\text{,}000[/latex] at [latex]6\%[/latex].
28. Arnold invested [latex]$64\text{,}000[/latex], some at [latex]5.5\%[/latex] interest and the rest at [latex]9\%[/latex]. How much did he invest at each rate if he received [latex]$4\text{,}500[/latex] in interest in one year?
29. After four years in college, Josie owes [latex]$65\text{,}800[/latex] in student loans. The interest rate on the federal loans is [latex]4.5\%[/latex] and the rate on the private bank loans is [latex]2\%[/latex]. The total interest she owed for one year was [latex]$2\text{,}878.50[/latex]. What is the amount of each loan?
Solution
The federal loan is [latex]$62\text{,}500[/latex] and the bank loan is [latex]$3\text{,}300[/latex].
30. Mark wants to invest [latex]$10\text{,}000[/latex] to pay for his daughter’s wedding next year. He will invest some of the money in a short term CD that pays [latex]12\%[/latex] interest and the rest in a money market savings account that pays [latex]5\%[/latex] interest. How much should he invest at each rate if he wants to earn [latex]$1\text{,}095[/latex] in interest in one year?
31. A trust fund worth [latex]$25\text{,}000[/latex] is invested in two different portfolios. This year, one portfolio is expected to earn [latex]5.25\%[/latex] interest and the other is expected to earn [latex]4\%[/latex]. Plans are for the total interest on the fund to be [latex]$1\text{,}150[/latex] in one year. How much money should be invested at each rate?
Solution
[latex]$12\text{,}000[/latex] should be invested at [latex]5.25\%[/latex] and [latex]$13\text{,}000[/latex] should be invested at [latex]4\%[/latex].
32. A business has two loans totaling [latex]$85\text{,}000[/latex]. One loan has a rate of [latex]6\%[/latex] and the other has a rate of [latex]4.5\%[/latex]. This year, the business expects to pay [latex]$4\text{,}650[/latex] in interest on the two loans. How much is each loan?
Exercises: Everyday Math
Instructions: For questions 33-34, translate to a system of equations and solve.
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 [latex]$15[/latex] full-year registration fee and how many had paid the [latex]$10[/latex] 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 [latex]$250[/latex] was collected for all the registrations, how many boys had paid the full-year fee and how many had paid the partial-year fee?
Solution
[latex]14[/latex] boys paid the full-year fee. [latex]4[/latex] boys paid the partial-year fee,
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 [latex]$75[/latex] and each adult paid [latex]$30[/latex]. The total amount of money collected for camp was [latex]$765[/latex]. If the number of girls is three times the number of adults, how many girls and how many adults paid for camp?
Exercises: Writing Exercises
Instructions: For questions 35-36, answer the given writing exercises.
34. Take a handful of two types of coins, and write a problem similar to (Example 4.5.2) relating the total number of coins and their total value. Set up a system of equations to describe your situation and then solve it.
Solution
Answers will vary.
35. In (Example 4.5.6) we solved the system of equations
[latex]\left\{\begin{array}{rcl}b+f&=&21\text{,}540\\0.105b+0.059f&=&1669.68\end{array}\right.[/latex]
by substitution. Would you have used substitution or elimination to solve this system? Why?
