### All GRE Math Resources

## Example Questions

### Example Question #1 : How To Find Rate

Two cars begin 500 miles apart and begin driving directly toward each other. One car proceeds at a rate of 50 miles per hour, while the other proceeds at a rate of 40 miles per hour. Rounding down, many minutes will it take for the two drivers to be 150 miles apart?

**Possible Answers:**

233

333

250

300

**Correct answer:**

233

You know that the distance between these cars is defined by the following equation: Answer "250" = 500 – 90t. This is because the cars get 90 miles closer every *t* hours. You want to solve for t when the distance is 150: 150 = 500 – 90t; –350 = –90t; t = ^{35}/_{9}. Recall, however, that the question asked for the rounded-down number of minutes; therefore, multiply your answer by 60: 35 * ^{60}/_{9} = 233^{1}/_{3}

Rounding down, you get 233.

### Example Question #2 : How To Find Rate

If Mary has $17 and gains a $2 weekly allowance, while Todd has $4 and gains a $3 weekly allowance, what is the least number of weeks that will pass before Todd has more money than Mary?

**Possible Answers:**

15

13

16

12

14

**Correct answer:**

14

First we set up two equations, one for Mary and one for Todd. Mary’s money growth is represented by y = 2x + 17 (she starts with $17 so this is our y-intercept and she gains $2 weekly so this is our slope).

Todd’s money growth is represented by y = 3x + 4 (he starts with $4 so this is our y-intercept and he gains $3 weekly so this is our slope). Set these two equal and solve for x. We find that after 13 weeks they have the same amount of money. But this is not what the question asked for. They want to know how many weeks it will take before Todd has MORE than Mary. Thus the answer must be 14 weeks.

### Example Question #23 : Proportion / Ratio / Rate

At a widget factory, 60 workers produce 1,000 widgets per week using power from internal generators. If (f) cubic meters of fuel are required by (g) number of generators every day to power the factory, how long will (t) cubic meters of fuel last in days?

**Possible Answers:**

f/tg

t/fg

tg/f

tf/g

g/tf

**Correct answer:**

t/fg

This is a plug and chug sort of problem, where choosing values is arbitrary. Suppose the generators consume 5 cubic meters of fuel per day and there are 10 generators. Then the number of days that 100 cubic meters of fuel will last is expressed as 100/5*10. Switching back to variables, that comes out to t/fg.

### Example Question #3 : How To Find Rate

Beverly just filled up her gas tank, which has enough gas to last her, at her usual driving rate, about 45 days. However, Beverly becomes extra busy and begins driving 66.6% more than she usually does. How many days does the tank of gas last Beverly at her new rate?

**Possible Answers:**

30 days

10 days

27 days

33 days

15 days

**Correct answer:**

27 days

Answer: 27 days

Explanation: Recall that the rate of gas consumption is INVERSELY related to the time it takes to consume the gas. Thus, if the rate of gas consumption increases to 5/3 of it's original rate (a 66.6% increase), then the time it will take to consume all of the gas decreases to the inverse, or 3/5 of the original. The answer is thus 27 (3/5 of 45).

### Example Question #4 : How To Find Rate

A Super Sweet Candy Puff Roll has 1450 calories per roll. A man eats one roll in 10 minutes. During the work day, the man eats a roll at the start of the shift and then eats another a roll every two hours after finishing the last one. Since he is watching his health, he eats only until 3 PM but will not start eating another one at any time after 3 PM. If his shift begins at 8 AM and ends at 5 PM, how many calories per minute does he consume in Super Sweet Candy Puff Rolls® during the whole work day?

**Possible Answers:**

9.21

14.27

5800

13.43

10.74

**Correct answer:**

10.74

It is probably easiest just to write out the eating schedule:

Roll 1: 8:00 AM - 8:10 AM

Roll 2: 10:10 AM - 10:20 AM

Roll 3: 12:20 PM - 12:30 PM

Roll 4: 2:40 PM - 2:50 PM

Therefore, he eats 4 rolls, or 1450 * 4 = 5800 calories. To get the rate, this must be divided across the whole day's minutes: 9 work hours * 60 = 540 minutes. The average calories per minute = 10.74.

### Example Question #5 : How To Find Rate

The cold-water faucet can fill a bucket in 30 minutes, and the hot-water faucet can fill a bucket in 60 minutes. How long will it take to fill a bucket when the two faucets are running together?

**Possible Answers:**

30 minutes

45 minutes

25 minutes

20 minutes

60 minutes

**Correct answer:**

20 minutes

The cold-water faucet fills the bucket in 30 minutes, so in 1 minute it fills 1/30 of the bucket. The hot-water faucet fills the bucket in 60 minutes, so in 1 minute it fills 1/60 of the bucket. Then, when they're both running together they fill 1/30 + 1/60 of the bucket in 1 minute.

1/30 + 1/60 = 2/60 + 1/60 = 3/60 = 1/20, so they fill the whole bucket in 20 minutes.

### Example Question #6 : How To Find Rate

Quantitative Comparison

Alice has a puppy and a kitten. The puppy weighs 4 pounds and grows at a rate of 1 pound per month. The kitten weighs 2 pounds and grows at a rate of 2 pounds per month.

Quantity A: Weight of the puppy after 8 months

Quantity B: Weight of the kitten after 7 months

**Possible Answers:**

The relationship cannot be determined from the information given.

The two quantities are equal.

Quantity A is greater.

Quantity B is greater.

**Correct answer:**

Quantity B is greater.

The puppy starts at 4 pounds and gains 1 pound per month for 8 months, so he weighs 4 + 8 = 12 pounds at the end of 8 months. The kitten starts at 2 pounds and gains 2 pounds per month for 7 months, so he weighs 2 + 14 = 16 pounds at the end of 7 months. Therefore Quantity B is greater.

### Example Question #7 : How To Find Rate

Kathy travels 40 miles in 1 hour and then 60 miles in 3 hours. What is her average speed in miles per hour?

**Possible Answers:**

35 mi/hr

25 mi/hr

18 mi/hr

20 mi/hr

30 mi/hr

**Correct answer:**

25 mi/hr

It's tempting to take the two rates and average them. This is wrong but the GRE will *always* give you this as a trap answer choice. In this problem, looking at the two rates separately would give us 40 mi/hr and 20 mi/hr, making the trap answer here 30 mi/hr.

The key is that Kathy goes 40 mi/hr for 1 hour and then 20 mi/hr for 3 hours, so the speed should be less than the middle number of 30. 18 mi/hr is lower than her lowest speed so that doesn't make sense, and 20 mi/hr is too low because the 1 hour at 40 mi/hr should bring the average up a little bit. Therefore the answer must be 25 mi/hr.

We can also do the computations to find the answer. The correct formula is average miles per hour = total miles / total hours. Plugging in our values, average speed = (40 + 60) miles / (1 + 3) hours = 25 mi/hr.

### Example Question #8 : How To Find Rate

A 10,000 gallon shark tank is filled by two hoses. Hose A fills at a rate of 1,000 gallons per hour. When Hose A and Hose B are both on, they fill the shark tank in 4 hours. At what rate does Hose B fill the tank?

**Possible Answers:**

3000 gallons/hour

1500 gallons/hour

2500 gallons/hour

2000 gallons/hour

1000 gallons/hour

**Correct answer:**

1500 gallons/hour

This problem gives the rate of Hose A instead of the time it takes to fill the shark tank. Let's convert the rest of the problem information into rates as well. The tank is 10,000 gallons, and it takes the two hoses together 4 hours to fill the tank. Therefore, the combined rate is 10,000/4 = 2,500 gallons per hour.

Now we know that Hose A can deliver 1,000 gallons in an hour, and together Hose A and Hose B can deliver 2,500 gallons in an hour. So 1,000 gallons + Hose B gallons = 2,500 gallons. Then Hose B fills the tank at 2500 – 1000 = 1500 gallons/hour.

### Example Question #9 : How To Find Rate

At 9 AM, the temperature is 65 degrees. At 2 PM, the temperature has risen to 100 degrees. What is the rate of temperature change in degrees per hour?

**Possible Answers:**

10 degrees/hour

9 degrees/hour

7 degrees/hour

35 degrees/hour

5 degrees/hour

**Correct answer:**

7 degrees/hour

The change in temperature is 100 – 65 = 35 degrees. The change in time is 9 AM to 2 PM, or 5 hours. So the rate of change is 35 degrees / 5 hours = 7 degrees / hour.