4.3 Solve Mixture Applications with Systems of Equations

Learning objectives

Learning objectives

• Solve mixture applications

• Solve interest applications

• Solve applications of cost and revenue functions

Solve Mixture Applications

Mixture application involve combining two or more quantities. When we solved mixture applications with coins and tickets earlier, we started by creating a table so we could organize the information. For a coin example with nickels and dimes, the table looked like this:

Using one variable meant that we had to relate the number of nickels and the number of dimes. We had to decide if we were going to let n be the number of nickels and then write the number of dimes in terms of n, or if we would let d be the number of dimes and write the number of nickels in terms of d.

Now that we know how to solve systems of equations with two variables, we’ll just let n be the number of nickels and d be the number of dimes. We’ll write one equation based on the total value column, like we did before, and the other equation will come from the number column.

For the first example, we’ll do a ticket problem where the ticket prices are in whole dollars, so we won’t need to use decimals just yet.

Example 4.24

Solution

In the next example, we’ll solve a coin problem. Now that we know how to work with systems of two variables, naming the variables in the ‘number’ column will be easy.

Example 4.25

Solution

Some mixture applications involve combining foods or drinks. Example situations might include combining raisins and nuts to make a trail mix or using two types of coffee beans to make a blend.

Example 4.26

Solution

Another application of mixture problems relates to concentrated cleaning supplies, other chemicals, and mixed drinks. The concentration is given as a percent. For example, a 20% concentrated household cleanser means that 20% of the total amount is cleanser, and the rest is water. To make 35 ounces of a 20% concentration, you mix 7 ounces (20% of 35) of the cleanser with 28 ounces of water.

For these kinds of mixture problems, we’ll use “percent” instead of “value” for one of the columns in our table.

Example 4.27

Solution

Solve Interest Applications

The formula to model simple interest applications is $I=Prt.$ Interest, I, is the product of the principal, P, the rate, r, and the time, t. In our work here, we will calculate the interest earned in one year, so t will be 1.

We modify the column titles in the mixture table to show the formula for interest, as you’ll see in the next example.

Example 4.28

Solution

Did you notice that the Principal column represents the total amount of money invested while the Interest column represents only the interest earned? Likewise, the first equation in our system, $s+b=40,000,$ represents the total amount of money invested and the second equation, $0.08s+0.03b=0.071(40,000),$ represents the interest earned.

The next example requires that we find the principal given the amount of interest earned.

Example 4.29

Solution

Solve applications of cost and revenue functions

Suppose a company makes and sells x units of a product. The cost to the company is the total costs to produce x units. This is the cost to manufacture for each unit times x, the number of units manufactured, plus the fixed costs.

The revenue is the money the company brings in as a result of selling x units. This is the selling price of each unit times the number of units sold.

When the costs equal the revenue we say the business has reached the break-even point.

Example 4.30

Solution

ⓐ The manufacturer has $7,000 of fixed costs no matter how many weight training benches it produces. In addition to the fixed costs, the manufacturer also spends$105 to produce each bench. Suppose x benches are sold.

Write the general Cost function formula. · $C(x)=(\mathrm{cost per unit})\cdot{}x+\mathrm{fixed costs}$ Substitute in the cost values. · $C(x)=105x+7000$

ⓑ The manufacturer sells each weight training bench for $245. We get the total revenue by multiplying the revenue per unit times the number of units sold.

Write the general Revenue function. · $R(x)=(\mathrm{selling}\,\mathrm{price}\,\mathrm{per}\,\mathrm{unit})\cdot{}x$ Substitute in the revenue per unit. · $R(x)=245x$

ⓒ Essentially we have a system of linear equations. We will show the graph of the system as this helps make the idea of a break-even point more visual.

ⓓ To find the actual value, we remember the break-even point occurs when costs equal revenue.

Write the break-even formula. · $\begin{array}{rll} C(x) & = & R(x) \\ 105x+7000 & = & 245x \\ \end{array}$ Solve. · $\,\begin{array}{rll} 7000 & = & 140x \\ 50 & = & x \\ \end{array}$

When 50 benches are sold, the costs equal the revenue.

When 50 benches are sold, the revenue and costs are both $12,250. Notice this corresponds to the ordered pair$(50,12,250).$

Section 4.3 Exercises

Section 4.3 Exercises

Solve Mixture Applications

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

Solve Interest Applications

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

Solve Applications of Cost and Revenue Functions

Writing Exercises

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

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

1. Translate to a system of equations and solve:

The ticket office at the zoo sold 553 tickets one day. The receipts totaled $3,936. How many$9 adult tickets and how many $6 child tickets were sold?

2. Translate to a system of equations and solve:

The box office at a movie theater sold 147 tickets for the evening show, and receipts totaled $1,302. How many$11 adult and how many $8 child tickets were sold?

1. Translate to a system of equations and solve:

Matilda has a handful of quarters and dimes, with a total value of $8.55. The number of quarters is 3 more than twice the number of dimes. How many dimes and how many quarters does she have?

2. Translate to a system of equations and solve:

Priam has a collection of nickels and quarters, with a total value of $7.30. The number of nickels is six less than three times the number of quarters. How many nickels and how many quarters does he have?

1. Translate to a system of equations and solve:

Greta wants to make 5 pounds of a nut mix using peanuts and cashews. Her budget requires the mixture to cost her $6 per pound. Peanuts are$4 per pound and cashews are $9 per pound. How many pounds of peanuts and how many pounds of cashews should she use?

2. Translate to a system of equations and solve:

Sammy has most of the ingredients he needs to make a large batch of chili. The only items he lacks are beans and ground beef. He needs a total of 20 pounds combined of beans and ground beef and has a budget of $3 per pound. The price of beans is$1 per pound and the price of ground beef is $5 per pound. How many pounds of beans and how many pounds of ground beef should he purchase?

1. Translate to a system of equations and solve:

LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?

2. Translate to a system of equations and solve:

Anatole needs to make 250 milliliters of a 25% solution of hydrochloric acid for a lab experiment. The lab only has a 10% solution and a 40% solution in the storeroom. How much of the 10% and how much of the 40% solutions should he mix to make the 25% solution?

1. Translate to a system of equations and solve:

Leon had $50,000 to invest and hopes to earn$6.2\%$ interest per year. He will put some of the money into a stock fund that earns 7% per year and the rest in to a savings account that earns 2% per year. How much money should he put into each fund?

2. Translate to a system of equations and solve:

Julius invested $7000 into two stock investments. One stock paid 11$12.5\%$ interest on the total investment. How much money did he put in each stock?

1. Translate to a system of equations and solve:

Laura owes $18,000 on her student loans. The interest rate on the bank loan is 2.5$1,066. What was the principal for each loan?

2. Translate to a system of equations and solve:

Jill’s Sandwich Shoppe owes $65,200 on two business loans, one at 4.5$3,582. What was the principal for each loan?

1. The manufacturer of a weight training bench spends $15 to build each bench and sells them for$32. The manufacturer also has fixed costs each month of $25,500.

ⓐ Find the cost function C when x benches are manufactured.

ⓑ Find the revenue function R when x benches are sold.

ⓒ Show the break-even point by graphing both the Revenue and Cost functions on the same grid.

ⓓ Find the break-even point. Interpret what the break-even point means.

2. The manufacturer of a weight training bench spends $120 to build each bench and sells them for$170. The manufacturer also has fixed costs each month of $150,000.

ⓐ Find the cost function C when x benches are manufactured.

ⓑ Find the revenue function R when x benches are sold.

ⓒ Show the break-even point by graphing both the Revenue and Cost functions on the same grid.

ⓓ Find the break-even point. Interpret what the break-even point means.

1. Tickets for the Cirque du Soleil 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?

2. Tickets for a Minnesota Twins 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?

1. 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 $2204. How many adult tickets and how many child tickets were sold?

2. 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?

1. 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?

2. 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?

1. 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?

2. Joseph would like to make twelve pounds of a coffee blend at a cost of $6 per pound. He blends Ground Chicory at$5 a pound with Jamaican Blue Mountain at $9 per pound. How much of each type of coffee should he use?

1. Twelve-year old Melody wants to sell bags of mixed candy at her lemonade stand. She will mix M&M’s that cost $4.89 per bag and Reese’s 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 M&M’s and how many bags of Reese’s Pieces should she use?

2. 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?

1. A scientist needs 120 milliliters 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?

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

1. Carol invested $2560 into two accounts. One account paid 8$7.25\%$ interest on the total investment. How much money did she put in each account?

2. Arnold invested $64,000, some at$5.5\%$interest and the rest at 9$4500 in interest in one year?

1. 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$1095 in interest in one year?

2. A business has two loans totaling $85,000. One loan has a rate of 6$4,650 in interest on the two loans. How much is each loan?

1. The manufacturer of a water bottle spends $5 to build each bottle and sells them for$10. The manufacturer also has fixed costs each month of $6500. ⓐ Find the cost function C when x bottles are manufactured. ⓑ Find the revenue function R when x bottles are sold. ⓒ Show the break-even point by graphing both the Revenue and Cost functions on the same grid. ⓓ Find the break-even point. Interpret what the break-even point means.

2. In Example 4.28, we used elimination to solve the system of equations$\mathrm{\{}\begin{array}{l} s+b=40,000 \\ 0.08s+0.03b=0.071(40,000) \\ \end{array}.$

Could you have used substitution or elimination to solve this system? Why?