# ACT Math : Equations / Inequalities

## Example Questions

### Example Question #71 : How To Find The Solution To An Equation

How much water should be added to  of 90% cleaning solution to yield 50% cleaning solution?

Explanation:

Pure water is 0% and pure solution 100%.  Let  = water to be added.

in general where  is the volume and  is the percent.

So the equation to solve becomes

and

### Example Question #72 : How To Find The Solution To An Equation

Solve  and

Explanation:

This problem is a good example of the substitution method of solving a system of equations.  We start by rewritting the first equation in terms of  to get  and then substutite the  into the second equation to get

Solving this equation gives  and substituting this value into one of the original equations gives , thus the correct answer is .

Joy bought some art supplies.  She bought colored pencils for $1.25 per box and sketch pads for$2.25 each.  Joy bought one more sketch pad than colored pencil boxes and spent $9.25. How many sketch pads did she buy? Possible Answers: Correct answer: Explanation: Let = # of color pencil boxes and = # of sketch pads purchased. So the equation to solve becomes Solving this equations leads to 2 colored pencil boxes and 3 sketch pads. ### Example Question #74 : How To Find The Solution To An Equation Possible Answers: Correct answer: Explanation: This question deals with absolute value equations which will normally gives you two solutions. You need to solve two sets of equations for absolute value problems: and ### Example Question #141 : Algebra Steve sells cars. His monthly salary is$1,000.  He gets a $500 commission for each car he sells. If Steve wants to make$7,500 this month, how many cars would he have to sell?

Explanation:

Let  = money earned and  = number of cars sold

So

and solving shows that he needs to sell 13 cars to make $7,500. ### Example Question #123 : Equations / Inequalities A chemistry student needs to dilute some acid. How much pure water should be added to 2 gallons of 80% acid solution to yield 20% acid solution? Possible Answers: Correct answer: Explanation: Let pure water = 0 % and pure acid = 100% The general equation to use is: where is the volume and is the percent solution. So the equation to solve becomes and gallons of pure water needs to be added to get a 20% acid solution. ### Example Question #141 : Algebra The Widget Company makes widgets. The monthly fixed costs are$750.  It costs $45 to make each widget. The widgets sell for$75 a piece.

What is the monthly break-even point?

Explanation:

The break-even point is where the costs equal revenue.

Let = # of widgets sold.

Costs:

Revenue:

So the equation to solve becomes

So the break-even point occurs when they sell 25 widgets.

### Example Question #81 : How To Find The Solution To An Equation

The Widget Company makes widgets.  The monthly fixed costs are $750. It costs$45 to make each widget.  The widgets sells for $75 a piece. The Widget Company wants to make a profit of$3,000.  How many widgets must be sold?

Explanation:

Profits = Revenues - Costs

Revenue:

Costs:

Profit:

So the equation to solve becomes

So a $3,000 profit occurs when they sell 125 widgets ### Example Question #141 : Algebra Sally sells custom picture frames. Her monthly fixed costs are$350.  It costs $10 to make each frame. Sally sells her picture frames for$35 each.

How many picture frames must Sally sell in order to break even?

Explanation:

The break-even point is where the costs equal the revenues.

Let  = # of frames sold

Costs:

Revenues:

Thus,

So 14 picture frames must be sold each month to break-even.

### Example Question #145 : Algebra

Sally sells custom picture frames.  Her monthly fixed costs are $350. It costs$10 to make each frame.  Sally sells her picture frames for $35 each. To make a profit of$500, how many frames need to be sold?

Explanation:

Let  = # of frames sold

Revenues:

Costs:

Profits =

So the equation to solve becomes

So 34 picture frames must be sold to make a \$500 profit.