### All GRE Math Resources

## Example Questions

### Example Question #61 : Algebra

A given university has an average professor pay of $40,000 a year and an average administrator pay of $45,000 per year. If the ratio of professors to administrators is 4 to 3, and the total pay for professors and administrators in a year is $40,415,000, how many professors does the college have?

**Possible Answers:**

411

475

548

375

500

**Correct answer:**

548

Set up a system of linear equations based on our data:

40,000P + 45,000A = 40,415,000

P/A = 4/3

To make things easiest, solve the second equation for A in terms of P:

A = (3/4) P

Replace this value into the first equation:

40,000P + 45,000 * (3/4)P = 40,415,000

Simplify:

40,000P + 33,750P = 40,415,000

73,750P = 40,415,000

P = 548 (The number of professors)

### Example Question #61 : How To Find The Solution To An Equation

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:**

pv(0.01c)

(p – v)(1 – 0.01c)

(p – v)(1 – c)

p(v – 0.01c)

(p – v)(0.01c)

**Correct answer:**

(p – v)(1 – 0.01c)

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 #41 : Equations / Inequalities

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

Quantity A is greater

Quantity B is greater

The two quantities are equal

**Correct answer:**

Quantity B is greater

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 #81 : Algebra

Quantitative Comparison

3*x* + 4*y* = 5

*x* – *y* = 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:**

Quantity A is greater.

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*) + 4*y* = 5

18 + 3*y* + 4*y* = 5

18 + 7*y* = 5

7*y* = –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 #85 : Algebra

*x* + *y* = 12 and 2*x* – *y* = 6

Quantity A: *x*

Quantity B: *y*

**Possible Answers:**

The two quantities are equal.

The relationship cannot be determined from the information given.

Quantity A is greater.

Quantity B is greater.

**Correct answer:**

The two quantities are equal.

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

2*x* – (12 – *x*) = 6.

This new equation can be simplified to

2*x* – 12 + *x* = 6

3*x* =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 #71 : 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

269

431

157

248

**Correct answer:**

346

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

5*c* + 10(548 – *c*) = 3750

5*c* + 5480 – 10*c* = 3750

5*c* = 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 #87 : Algebra

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:**

14 and 14 feet

12 and 16 feet

8 and 20 feet

11 and 17 feet

cannot be determined

**Correct answer:**

12 and 16 feet

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*: 2*s* = 24 so *s* = 12. Then solve for *h*: *h* = *s* + 4 = 12 + 4 = 16.

### Example Question #71 : Algebra

Solve for *z*:

3(*z* + 4)^{3} – 7 = 17

**Possible Answers:**

4

8

2

–8

–2

**Correct answer:**

–2

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 #89 : Algebra

Jen and Karen are travelling for the weekend. They both leave from Jen's house and meet at their destination 250 miles away. Jen drives 45mph the whole way. Karen drives 60mph but leaves a half hour after Jen. How long does it take for Karen to catch up with Jen?

**Possible Answers:**

She can't catch up.

**Correct answer:**

For this type of problem, we use the formula:

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

We then make a variable for Jen's time, . Thus we know that Karen's time is (since we are working in hours).

Thus,

There's a logical shortcut you can use on "catching up" distance/rate problems. The difference between the faster (Karen at 60mph) and slower (Jen at 45mph) drivers is 15mph. Which means that every one hour, the faster driver, Karen, gains 15 miles on Jen. We know that Jen gets a 1/2 hour head start, which at 45mph means that she's 22.5 miles ahead when Karen gets started. So we can calculate the number of hours (H) of the 15mph of Karen's "catchup speed" (the difference between their speeds) it will take to make up the 22.5 mile gap:

15H = 22.5

So H = 1.5.

### Example Question #90 : Algebra

Bill and Bob are working to build toys. Bill can build toys in 6 hours. Bob can build toys in 3 hours. How long would it take Bob and Bill to build toys working together?

**Possible Answers:**

**Correct answer:**

Bill builds toys an hour. Bob builds toys an hour. Together, their rate of building is . Together they can build toys in 2 hours. They would be able to build toys in 8 hours.