# 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?

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?

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**

**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**

**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?**