Let’s use the problem solving process outlined in Module 1 to help us work through a solution to the problem.

Read and Understand: We are looking for a new amount – in this case a volume –  based on the words “how much”.  We know two starting  concentrations and the final concentration, as well as one volume.

Define and Translate: Solution 1 is the 70 mL of 50% methane and solution 2 is the unknown amount with 80% methane.  We can call our unknown amount x.

Write and Solve:  Set up the mixture table. Remember that concentrations are written as decimals before we can perform mathematical operations on them.

Multiply amount by concentration to get total, be sure to distribute on the last row: [latex]\left(70 + x\right)0.6[/latex]Add the entries in the amount column to get final amount. The concentration for this amount is 0.6 because we want the final solution to be 60% methane.

Add the total mass for solution 1 and solution 2 to get the total mass for the 60% solution. This is our equation for finding the unknown volume.

[latex]35+0.8x=42+0.6x[/latex]

[latex]\begin{array}{c}35+0.8x=42+0.6x\\\underline{-0.6x}\,\,\,\,\,\,\,\underline{-0.6x}\\35+0.2x=42\\\end{array}[/latex]

Subtract 35 from both sides

[latex]\begin{array}{c}35+0.2x=42\\\underline{-35}\,\,\,\,\,\,\,\underline{-35}\\0.2x=7\end{array}[/latex]

Divide both sides by 0.2

[latex]\begin{array}{c}0.2x=7\\\frac{0.2x}{0.2}=\frac{7}{0.2}\end{array}[/latex]
x=35

Answer

35mL must be added to the original 70 mL to gain a solution with a concentration of 60%

Read and Understand: We are asked to find two starting volumes of milk whose concentrations of butterfat are both known. We also know the final volume is 42 gallons. There are two unknowns in this problem.

Define and Translate: We will call the unknown volume of the  24% solution x, and the unknown volume of the 18% solution y.

Write and Solve: Fill in the table with the information we know.

Find the total mass by multiplying the amount of each solution by the concentration. The total mass of the final solution comes from

When you sum the amount column you get one equation: x+ y = 42
When you sum the total column you get a second equation: 0.24x + 0.18y = 8.4

Use elimination to find a value for x, and y.

Multiply the first equation by -0.18

-0.18(x+y) = (42)(-0.18)

-0.18x – -0.18y = -7.56

Now our system of equations looks like this:

-0.18x – -0.18y = -7.56

0.24x + 0.18y = 8.4

Adding the two equations together to eliminate the y terms gives this equation:

0.06x = 8.4

Divide by 0.06 on each side:

x = 14

Now substitute the value for x into one of the equations in order to solve for y.

(14) + y = 42

y = 28

Answer

This can be interpreted as 14 gallons of 24% butterfat milk added to 28 gallons of 18% butterfat milk will give 42 gallons of 20% butterfat milk.

Read and Understand: We want to find the number of child and adult tickets, we know the total number of tickets sold, the total revenue and the cost of a child and adult ticket.

Define and Translate: Let c = the number of children and a = the number of adults in attendance.  Revenue comes from number of tickets sold multiplied by the price of the ticket.  We will get revenue for adults by multiplying $50.00 times a.  $25.00 times c will give the revenue from the number of child tickets sold.

Write and Solve: We can use a table as we did in the mixture problems section to organize the information we have.  Although a table is not necessary, it can help you get started.  For this problem, we labeled columns as amount, value, and total revenue because that is the information we are given.

The total number of people is [latex]2,000[/latex].

The total revenue is $70,000. We can use this and the revenue from child and adult tickets to write an equation for the revenue.[latex]25c+50a=70,000[/latex]

The number of people at the game that day is the total number of child tickets sold plus the total number of adult tickets, [latex]c+a=2,000[/latex]

We now have a system of linear equations in two variables.[latex]\begin{array}{r}c+a=2,000\,\,\,\\ 25c+50a=70,000\end{array}[/latex].

We can use any method of solving systems of equations to solve this system for a and c.  Substitution looks easiest because we can  solve the first equation for either [latex]c[/latex] or [latex]a[/latex]. We will solve for [latex]a[/latex].

[latex]\begin{array}{c}c+a=2,000\\ a=2,000-c\end{array}[/latex]

Substitute the expression [latex]2,000-c[/latex] in the second equation for [latex]a[/latex] and solve for [latex]c[/latex].

[latex]\begin{array}{r} 25c+50\left(2,000-c\right)=70,000\,\,\,\, \\ 25c+100,000 - 50c=70,000\,\,\,\, \\ -25c=-30,000 \\ c=1,200\,\,\,\,\,\,\, \end{array}[/latex]

Substitute [latex]c=1,200[/latex] into the first equation to solve for [latex]a[/latex].

[latex]\begin{array}{r}1,200+a=2,000 \\ a=800\,\,\,\,\,\, \end{array}[/latex]

Answer

We find that [latex]1,200[/latex] children and [latex]800[/latex] adults bought tickets to the game that day. The marketing group may want to focus their advertising toward attracting young people.

Read and Understand: We want to find the number of quarters and the number of dimes in the jar.  We know that dimes are $0.10 and quarters are $0.25, and the total number of coins is 11.

Define and Translate: We will call the number of quarters q and the number of dimes d. The part of the total $1.85 that comes from quarters will be determined by how many quarters and the fact that each one is worth $0.25, so $0.25q represents the amount of $1.85 that is quarters.  The same idea can be used for dimes, so $0.10d represents the amount of $1.85 that is dimes.

Write and Solve: We can label a new table with the information we are given.

We can write our two equations, remember that we need two to solve for two unknowns.

 [latex]\begin{array}{r}q+d=11\,\,\,\\0.25q+0.10d=1.85\end{array}[/latex]

Substitution looks like the easiest path to a solution, solve for q.

[latex]\begin{array}{c}q+d=11\\ q=11-d\end{array}[/latex]

Substitute this into the other equation, and solve for d.

[latex]\begin{array}{r}0.25\left(11-d\right)+0.10d=1.85\,\,\,\, \\2.75-0.25d+0.10d=1.85\,\,\,\,\\ 2.75-0.15d=1.85\\-0.15d=-0.9\\\,\,\,\,\,\,\, d=6\end{array}[/latex]

Substitute [latex]d=6[/latex] into the first equation to solve for [latex]q[/latex].

[latex]\begin{array}{r}q+6=11 \\q=5\,\,\,\,\,\, \end{array}[/latex]

Answer

We have 6 dimes and 5 quarters.

Cost: [latex]y=0.85x+35,000[/latex]

Revenue:[latex]y=1.55x[/latex]

The cost equation represents money leaving the company, namely how much it costs to produce a given number of bike frames. If we use the skateboard example as a model, x would represent the number of frames produced (instead of skateboards) and y would represent the amount of money it would cost to produce them (the same as the skateboard problem).

The revenue equation represents money coming into the company, so in this context x still represents the number of bike frames manufactured, and y now represents the amount of money made from selling them.  Let’s organize this information in a table:

Read and Understand: We want the break even point for this system that represents cost and revenue.  This means we want to find where the two lines cross, and we have learned a few different methods for doing this because this is the solution to the system of equations!  Substitution looks like the easiest method since the revenue equation is already solved for y, [latex]y=1.55x[/latex].

Define and Translate: Write the system of equations.

[latex]\begin{array}{l}\\ y=0.85x+35,000\hfill \\ y=1.55x\hfill \end{array}\\[/latex]

Write and Solve: The equations are already written for us, so we just need to solve the system using substitution.

Substitute the expression [latex]0.85x+35,000[/latex] from the first equation into the second equation and solve for [latex]x[/latex].
[latex]\begin{array}{r}0.85x+35,000=1.55x\\ 35,000=0.7x\,\,\,\\ 50,000=x\,\,\,\,\,\,\,\,\,\,\end{array}\\[/latex]
Then, we substitute [latex]x=50,000[/latex] into either the cost equation or the revenue equation.
[latex]1.55\left(50,000\right)=77,500\\[/latex]
The break-even point is [latex]\left(50,000,77,500\right)[/latex].

Check and Interpret:

The solution to this system is[latex]\left(50,000,77,500\right)[/latex], but what does that mean? Think of a point as  [latex]\left(x,y)[/latex], where in this case x is the quantity of bikes manufactured and y is an amount of money. For our system y represents two different things and x represents one thing.  Refer to tehe table we made in the first example, shown below:

Let’s interpret the solution with respect to the Cost equation first. x = 50,000 and y = 77,500.  Using our table, we can translate this as “the cost for producing 50,000 bike frames is $75,500”.

In the same way, the Revenue equation can be interpreted as “the amount of money the company makes from selling 50,000 bike frames is $77,500”.

Read and Understand: We know that graphically,  solutions to linear inequalities are entire regions, and we learned how to graph systems of linear inequalities earlier in this module. Based on the graph below and the equations that define cost and revenue, we can use inequalities to define the region for which the bike manufacturer will make a profit.

Let’s start with the revenue equation.  We know that the break even point is at (50,000, 77,500) and the profit region is the blue area.  If we choose a point in the region and test it like we did for finding solution regions to inequalities, we will know which kind of inequality sign to use.

Let’s test the point [latex]\left(65,00,100,000\right)[/latex] in both equations to determine which inequality sign to use.

Cost:

[latex]\begin{array}{l}y=0.85x+{35,000}\\{100,000}\text{ ? }0.85\left(65,000\right)+35,000\\100,000\text{ ? }90,250\end{array}[/latex]

We need to use > because 100,000 is greater than 90,250

The cost inequality that will ensure the company makes profit – not just break even – is [latex]y>0.85x+35,000[/latex]

Now test the point in the revenue equation:

Revenue:

[latex]\begin{array}{l}y=1.55x\\100,000\text{ ? }1.55\left(65,000\right)\\100,000\text{ ? }100,750\end{array}[/latex]

We need to use < because 100,000 is less than 100,750

The revenue inequality that will ensure the company makes profit – not just break even – is [latex]y<1.55x[/latex] The systems of inequalities that defines the profit region for the bike manufacturer:

[latex]\begin{array}{l}y>0.85x+35,000\\y<1.55x\end{array}[/latex]

Answer

The cost to produce 50,000 units is $77,500, and the revenue from the sales of 50,000 units is also $77,500. To make a profit, the business must produce and sell more than 50,000 units. The system of linear inequalities that represents the number of units that the company must produce in order to earn a profit is:

[latex]\begin{array}{l}y>0.85x+35,000\\y<1.55x\end{array}[/latex]