GRE Math : Equations / Inequalities

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #81 : Algebra

Abby works at a car dealership and receives a commission "c" which is a percent of the profit the dealership makes from the sale, which is the difference between the price "p" of the car and the value "v" of the car. How much, in dollars, does the dealership earn per transaction?

Possible Answers:

(p – v)(0.01c)

pv(0.01c)

(p – v)(1 – c)

(p – v)(1 – 0.01c)

p(v – 0.01c)

Correct answer:

(p – v)(1 – 0.01c)

Explanation:

To show that c is of the profit of the transaction, we must represent the profit as the difference between the price and the value of the car, or "(p – v)"

To show that Abby's commission in dollars is a percentage of the profit, we use 0.01 * c to convert the commission she earns to a percent.

To shift the earnings from Abby to the dealership (which is what the question requires of us), we must take 1 – 0.01c since this will accommodate for the remaining percentage. For example, it shifts 75% (0.75) to 25% (1 – 0.75 or 0.25).

Putting this all together, we get a final expression of:

(p – v)(1 – 0.01c) = dealership earnings

Check answer with arbitrary values: letting p = 300, v = 200, and c = 20, we get a value of 80 which makes sense as the $100 profit must be distributed evenly between Abby ($20) and the dealership ($80).

Example Question #81 : Gre Quantitative Reasoning

Sally is 2 years younger than Abby

Daisy is 5 years older than Tracy

Abby is 6 years older than Tracy

A

---

Sally's age

 

B

---

Daisy's age

Possible Answers:

The relationship cannot be determined

The two quantities are equal

Quantity B is greater

Quantity A is greater

Correct answer:

Quantity B is greater

Explanation:

To simplify the word problem, express the ages in terms of variables in a system of equations. Note that we want to compare S with D:

S = A – 2

D = T + 5

A = T + 6

By substituting A for T in the first equation, we can get S in terms of T, which will let us directly compare the values of S and D.

S = (T + 6) – 2 = T + 4

If D = T + 5, and S = T + 4, D must be the greater value and Daisy is one year older than Sally.  B is the correct answer.

Example Question #41 : Equations / Inequalities

Quantitative Comparison

3x + 4y = 5

x – y = 6

Quantity A: x

Quantity B: y

Possible Answers:

Quantity B is greater.

Quantity A is greater.

The relationship cannot be determined from the information given.

The two quantities are equal.

Correct answer:

Quantity A is greater.

Explanation:

First let's solve for y using the second equation, x – y = 6.

x = 6 + y.  Then plug this in to the other equation.

3 (6 + y) + 4y = 5

18 + 3y + 4y = 5

18 + 7y = 5

7y = –13

y = –13/7. Now plug this value back into x = 6 + y.

x = 6 – 13/7 = 29/7.  x is positive and y is negative, so clearly x is larger, so Quantity A is bigger.

Example Question #42 : Equations / Inequalities

x + y = 12 and 2xy = 6

Quantity A: x

Quantity B: y

Possible Answers:

The relationship cannot be determined from the information given.

Quantity B is greater.

The two quantities are equal.

Quantity A is greater.

Correct answer:

The two quantities are equal.

Explanation:

Because there are two different equations for the two variables (x and y), you are able to solve for the value of each. You can rearrange the first equation to show that

y = 12 – x by subtracting x from both sides.  

Then you can substitute this equation into the second equation to give you

2x – (12 – x) = 6.  

This new equation can be simplified to

2x – 12 + x = 6

3x =18

x = 6

Filling this back into the first equation, we get 6 + y = 12 which means y must also equal 6.  Because x and y are equal, we choose the option both quantities are equal.

Example Question #21 : How To Find The Solution To An Equation

A theme park charges $10 for adults and $5 for kids. How many kids tickets were sold if a total of 548 tickets were sold for a total of $3750?

Possible Answers:

346

157

269

431

248

Correct answer:

346

Explanation:

Let c = number of kids tickets sold. Then (548 – c) adult tickets were sold. The revenue from kids tickets is $5c, and the total revenue from adult tickets is $10(548 – c). Then,

5c + 10(548 – c) = 3750

5c + 5480 – 10c = 3750

5c = 1730

c = 346. 

We can check to make sure that this number is correct:

$5 * 346 tickets + $10 * (548 – 346) tickets = $3750 total revenue

Example Question #44 : Equations / Inequalities

Two palm trees grow next to each other in Luke's backyard. One of the trees gets sick, so Luke cuts off the top 3 feet. The other tree, however, is healthy and grows 2 feet. How tall are the two trees if the healthy tree is now 4 feet taller than the sick tree, and together they are 28 feet tall?

Possible Answers:

11 and 17 feet

cannot be determined

12 and 16 feet

14 and 14 feet

8 and 20 feet

Correct answer:

12 and 16 feet

Explanation:

Let s stand for the sick tree and h for the healthy tree. The beginning information about cutting the sick tree and the healthy tree growing is actually not needed to solve this problem! We know that the healthy tree is 4 feet taller than the sick tree, so h = s + 4.

We also know that the two trees are 28 feet tall together, so s + h = 28. Now we can solve for the two tree heights.

Plug h = s + 4 into the second equation: (s + 4) + s = 28. Simplify and solve for h: 2s = 24 so s = 12. Then solve for h: h = s + 4 = 12 + 4 = 16.

Example Question #41 : Linear / Rational / Variable Equations

Solve for z

3(z + 4)3 – 7 = 17

Possible Answers:

4

2

–8

–2

8

Correct answer:

–2

Explanation:

1. Add 7 to both sides

3(z + 4)3 – 7 + 7= 17 + 7

3(z + 4)3 = 24

2. Divide both sides by 3

(z + 4)3 = 8

3. Take the cube root of both sides

z + 4 = 2

4. Subtract 4 from both sides

z = –2

Example Question #71 : Algebra

Jen and Karen are travelling for the weekend. They both leave from Jen's house and meet at their destination. Jen drives 45mph the whole way. Karen drives 60mph but leaves a half hour after Jen. How long after Jen leaves does Karen catch up with her? 

Possible Answers:

She can't catch up. 

\dpi{100} \small 1.5 \ hours

\dpi{100} \small 1 \ hour

\dpi{100} \small 3 \ hours

\dpi{100} \small 2 \ hours

Correct answer:

\dpi{100} \small 2 \ hours

Explanation:

For this type of problem, we use the formula:

\dpi{100} \small distance = rate\times time

When Karen catches up with Jen, their distances are equivalent. Thus,

\dpi{100} \small rate (Jen) \times time (Jen)=rate(Karen)\times time(Karen)

We then make a variable for Jen's time, \dpi{100} \small t. Thus we know that Karen's time is \dpi{100} \small t-.5 (since we are working in hours).

Thus,

\dpi{100} \small 45t=60(t-.5))

\dpi{100} \small 45t=60t-30

\dpi{100} \small 30=15t 

\dpi{100} \small t=2 \ hours

Example Question #81 : Algebra

Bill and Bob are working to build toys. Bill can build \dpi{100} \small k toys in 6 hours. Bob can build \dpi{100} \small k toys in 3 hours. How long would it take Bob and Bill to build \dpi{100} \small 4k toys working together? 

Possible Answers:

\dpi{100} \small 4\ hours

\dpi{100} \small 2\ hours

\dpi{100} \small 8\ hours

\dpi{100} \small 9\ hours

\dpi{100} \small 12\ hours

Correct answer:

\dpi{100} \small 8\ hours

Explanation:

Bill builds \dpi{100} \small \frac{k}{6} toys an hour. Bob builds \dpi{100} \small \frac{k}{3} toys an hour. Together, their rate of building is \dpi{100} \small \frac{k}{6}+\frac{k}{3}=\frac{k}{2}. Together they can build \dpi{100} \small k toys in 2 hours. They would be able to build \dpi{100} \small 4k toys in 8 hours. 

Example Question #361 : Algebra

A hybrid car gets 40 miles per gallon. Gasoline costs $3.52 per gallon. What is the cost of the gasoline needed for the car to travel 120 miles? 

Possible Answers:

\dpi{100} \small \$ 10.36

\dpi{100} \small \$ 12.53

\dpi{100} \small \$ 9.54

\dpi{100} \small \$ 14.08

\dpi{100} \small \$ 10.56

Correct answer:

\dpi{100} \small \$ 10.56

Explanation:

The car will be using \dpi{100} \small \frac{120\ miles}{40\ mpg}=3\ gallons of gas during this trip. Thus, the total cost would be \dpi{100} \small 3\times \$ 3.52=\$ 10.56

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