GRE Math : Linear / Rational / Variable Equations

Study concepts, example questions & explanations for GRE Math

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Example Questions

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 : Gre Quantitative Reasoning

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.

Example Question #141 : Algebra

How much pure water must be added to 2 gallons of 90% pure cleaning solution to yield a 30% pure cleaning solution?

Possible Answers:

Correct answer:

Explanation:

Let pure water be 0% and pure solution be 100%.

So the general equation to solve is:

V_{1}P_{1} + V_{2}P_{2} = V_{f}P_{f} where  is the volume and the  is percent solution.

So the equation to solve becomes

Solving shows that we need to add 4 gallons of pure water to 2 gallons of 90% pure cleaning solution to get a 30% pure solution.

Example Question #141 : Algebra

Susan got a new piggy bank and counted the change she put into it.  She had one more nickel than dimes and two fewer quarters than nickles.  The value of her change was $1.40.  How many total coins did she have?

Possible Answers:

Correct answer:

Explanation:

Let  = number of dimes,  = number of nickels, and 

= number of quarters.

The general equation to use is:

V_{1}N_{1} + V_{2}N_{2} + V_{3}N_{3} = V_{f} where  is the money value and  is the number of coins

So the equation to solve becomes

Thus, solving the equation shows that she had five nickels, four dimes, and three quarters giving a total of 12 coins.

Example Question #42 : Equations / Inequalities

How much pure water should be added to 1\ L of 80% cleaning solution to dilute it to 25% cleaning solution.

Possible Answers:

2.2\ L

1.5\ L

3.0\ L

4.1\ L

2.6\ L

Correct answer:

2.2\ L

Explanation:

Pure water is 0% and pure solution is 100%

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

So the equation to solve becomes x(0)+1(0.80)=(1+x)(0.25)

So we need to add 2.2\ L pure water to 1\ L of 80% cleaning solution to yield 25% cleaning solution.

Example Question #41 : Linear / Rational / Variable Equations

Luke purchased a tractor for $1200. The value of the tractor decreases by 25 percent each year. The value, , in dollars, of the tractor at  years from the date of purchase is given by the function .

In how many years from the date of purchase will the value of the tractor be $675?

Possible Answers:

1

3

5

2

4

Correct answer:

2

Explanation:

We are looking for the value of t  that gives $675 as the result when plugged in V (t ). While there are many ways to do this, one of the fastest is to plug in the answer choices as values of t .

When we plug = 1 into V (t ), we get V (1) = 1200(0.75)1 = 1000(0.75) = $900, which is incorrect.

When we plug = 2 into V (t ), we get V (2) = 1200(0.75)2 = $675, so this is our solution.

The value of the tractor will be $675 after 2 years.

Finally, we can see that if = 3, 4, or 5, the resulting values of the V (t ) are all incorrect.

Example Question #151 : Algebra

Solve for :

Possible Answers:

Correct answer:

Explanation:

First combine like terms. In this case, 4x and 9x can be added together:

13x + 13 = 0

Subtract 13 from both sides:

13x = -13

Divide both sides by 13 to isolate x:

x = -13/13

x = -1

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