ACT Math : Equations / Inequalities

Study concepts, example questions & explanations for ACT Math

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Example Questions

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

How much water should be added to 2\ L of 90% cleaning solution to yield 50% cleaning solution?

Possible Answers:

1.2\ L

2.4\ L

1.6\ L

0.8\ L

1.5\ L

Correct answer:

1.6\ L

Explanation:

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

V_{1}P_{1} + V_{2}P_{2} = V_{f}P_{f}  in general where V is the volume and P is the percent.

So the equation to solve becomes x(0)+2(0.90)= (x+2)(0.50)

and x=1.6\ L

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

Solve x+2y=14 and 2x+y=13

Possible Answers:

(3,2)

(4,5)

(5,4)

(1,3)

(-4,-5)

Correct answer:

(4,5)

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 x to get x=14-2y and then substutite the x into the second equation to get

2(14-2y)+y=13

Solving this equation gives y=5 and substituting this value into one of the original equations gives x=4, thus the correct answer is (4,5).

Example Question #73 : How To Find The Solution To An Equation

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:

3

1

5

4

2

Correct answer:

3

Explanation:

Let x = # of color pencil boxes and x+1 = # of sketch pads purchased.

So the equation to solve becomes 1.25x+2.25(x+1)=9.25

Solving this equations leads to 2 colored pencil boxes and 3 sketch pads.

Example Question #74 : How To Find The Solution To An Equation

\left | 2x - 3 \right |-x= 5

Possible Answers:

x=-8\ or \ x=-\frac{3}{2}

x=8

x=-\frac{2}{3}

x=8\ or\ x=-\frac{2}{3}

x=-8

Correct answer:

x=8\ or\ x=-\frac{2}{3}

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:

2x-3 = x+5

and

2x-3=-\left ( x+5 \right )

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?

Possible Answers:

Correct answer:

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:

V_{1}P_{1} + V_{2}P_{2} = V_{f}P_{f}  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?

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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.

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