Arithmetic - GRE Quantitative Reasoning
Card 1 of 4976
Which of the following is a prime number?
Which of the following is a prime number?
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a prime number is divisible by itself and 1 only
list the factors of each number:
6: 1,2,3,6
9: 1,3,9
71: 1,71
51: 1, 3,17,51
15: 1,3,5,15
a prime number is divisible by itself and 1 only
list the factors of each number:
6: 1,2,3,6
9: 1,3,9
71: 1,71
51: 1, 3,17,51
15: 1,3,5,15
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It takes Mary 45 minutes to completely frost 100 cupcakes, and it takes Benjamin 80 minutes to completely frost 110 cupcakes. How many cupcakes can they completely frost, working together, in 1 hour?
It takes Mary 45 minutes to completely frost 100 cupcakes, and it takes Benjamin 80 minutes to completely frost 110 cupcakes. How many cupcakes can they completely frost, working together, in 1 hour?
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In this rate word problem, we need to find the rates at which Mary and Bejamin frost their respective cupcakes, and then sum their respective rates per hour. In one hour Mary frosts 133 cupcakes. (Note: the question specifies COMPLETELY frosted cupcakes only, so the fractional results here will need to be rounded down to the nearest integer.) Benjamin frosts 82 cupcakes.
82 + 133=215
In this rate word problem, we need to find the rates at which Mary and Bejamin frost their respective cupcakes, and then sum their respective rates per hour. In one hour Mary frosts 133 cupcakes. (Note: the question specifies COMPLETELY frosted cupcakes only, so the fractional results here will need to be rounded down to the nearest integer.) Benjamin frosts 82 cupcakes.
82 + 133=215
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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?
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?
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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.
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.
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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?
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?
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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).
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).
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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?
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?
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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.
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.
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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?
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?
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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.
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.
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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
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
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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.
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.
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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?
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?
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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.
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.
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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?
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?
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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.
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.
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Ben mows the lawn in 1 hour. Kent mows the lawn in 2 hours. How long will it take them to mow the lawn working together?
Ben mows the lawn in 1 hour. Kent mows the lawn in 2 hours. How long will it take them to mow the lawn working together?
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Ben mows 1 lawn in 1 hour, or 1/60 of the lawn in 1 minute. Ken mows 1 lawn in 2 hours, or 1/120 of the lawn in 1 minute. Then each minute they mow 1/60 + 1/120 = 3/120 = 1/40 of the lawn. That means the entire lawn takes 40 minutes to mow.
Ben mows 1 lawn in 1 hour, or 1/60 of the lawn in 1 minute. Ken mows 1 lawn in 2 hours, or 1/120 of the lawn in 1 minute. Then each minute they mow 1/60 + 1/120 = 3/120 = 1/40 of the lawn. That means the entire lawn takes 40 minutes to mow.
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A train travels at 50 feet per second. If there are 5280 feet in a mile, how many miles will the train travel in an hour?
A train travels at 50 feet per second. If there are 5280 feet in a mile, how many miles will the train travel in an hour?
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First, we must determine how many feet per hour the train travels.
50 feet per second * 60 seconds in a minute * 60 minutes in an hour.
50 * 60 * 60 = 180,000
Then, it's just a matter of converting 180,000 feet to miles. Because there are 5280 feet in a mile, just divide.
180,000 / 5280 = 34.091
First, we must determine how many feet per hour the train travels.
50 feet per second * 60 seconds in a minute * 60 minutes in an hour.
50 * 60 * 60 = 180,000
Then, it's just a matter of converting 180,000 feet to miles. Because there are 5280 feet in a mile, just divide.
180,000 / 5280 = 34.091
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Simplify:
(2_x_ + 4)/(x + 2)
Simplify:
(2_x_ + 4)/(x + 2)
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(2_x_ + 4)/(x + 2)
To simplify you must first factor the top polynomial to 2(x + 2). You may then eliminate the identical (x + 2) from the top and bottom leaving 2.
(2_x_ + 4)/(x + 2)
To simplify you must first factor the top polynomial to 2(x + 2). You may then eliminate the identical (x + 2) from the top and bottom leaving 2.
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If a is the greatest common divisor of 64 and 14 and b is the least common multiple of 16 and 52 then a + b = ?
If a is the greatest common divisor of 64 and 14 and b is the least common multiple of 16 and 52 then a + b = ?
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The greatest common divisor of 64 and 14 is 2, as found by the prime factorization of 64 and 14. The least common multiple of 16 and 52 is 208, which can be found by looking at the decimal when 52 is divided by 16. The remainder is 0.25, or 1/4 so the fourth multiple of 52 is 208, which is also divisible by 16.
The greatest common divisor of 64 and 14 is 2, as found by the prime factorization of 64 and 14. The least common multiple of 16 and 52 is 208, which can be found by looking at the decimal when 52 is divided by 16. The remainder is 0.25, or 1/4 so the fourth multiple of 52 is 208, which is also divisible by 16.
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A prime number is divisible by:
A prime number is divisible by:
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The definition of a prime number is a number that is divisible by only one and itself. A prime number can't be divided by zero, because numbers divided by zero are undefined. The smallest prime number is 2, which is also the only even prime.
The definition of a prime number is a number that is divisible by only one and itself. A prime number can't be divided by zero, because numbers divided by zero are undefined. The smallest prime number is 2, which is also the only even prime.
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If x is a prime number, then 3_x_ is
If x is a prime number, then 3_x_ is
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Pick a prime number to see that 3_x_ is not always even, for example 3 * 3 = 9.
But 2 is a prime number as well, so 3 * 2 = 6 which is even, so we can't say that 3_x_ is either even or odd.
Neither 9 nor 6 in our above example is prime, so 3_x_ is not a prime number.
Lastly, 9 is not divisible by 4, so 3_x_ is not always divisible by 4.
Therefore the answer is "Cannot be determined".
Pick a prime number to see that 3_x_ is not always even, for example 3 * 3 = 9.
But 2 is a prime number as well, so 3 * 2 = 6 which is even, so we can't say that 3_x_ is either even or odd.
Neither 9 nor 6 in our above example is prime, so 3_x_ is not a prime number.
Lastly, 9 is not divisible by 4, so 3_x_ is not always divisible by 4.
Therefore the answer is "Cannot be determined".
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A factory has fixed costs of $25,000 per month. It manufactures widgets at a total manufacturing cost of $45 per widget. They are sold at $60. How many widgets must be sold in any given month in order to break even?
A factory has fixed costs of $25,000 per month. It manufactures widgets at a total manufacturing cost of $45 per widget. They are sold at $60. How many widgets must be sold in any given month in order to break even?
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Let's first represent the total costs per month:
C = 25000 + 45n, where n is the number of widgets manufactured.
The profit can be represented as 60n – C or 60n – 25000 – 45n = 15n – 25000. Now, we merely have to solve this for 0 in order to find the "break even" line.
15n – 25000 = 0 → 15n = 25000 → n = 1666.67. We must sell a whole number of widgets, so it must be 1667.
Let's first represent the total costs per month:
C = 25000 + 45n, where n is the number of widgets manufactured.
The profit can be represented as 60n – C or 60n – 25000 – 45n = 15n – 25000. Now, we merely have to solve this for 0 in order to find the "break even" line.
15n – 25000 = 0 → 15n = 25000 → n = 1666.67. We must sell a whole number of widgets, so it must be 1667.
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John can paint a room in 4 hours, and Susan can paint the same room in 6 hours. How many minutes would it take them to paint the room together?
John can paint a room in 4 hours, and Susan can paint the same room in 6 hours. How many minutes would it take them to paint the room together?
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First find what fraction of the job is completed per hour. If John can paint the room in 4 hours, then he completes 
of it per hour. Similarly, Susan paints 
of it per hour. So together they paint 
of the room per hour.
Add 
by rewriting them with the same common denominator (least common multiple of 4 and 6 is 12, so 12 is the least common denominator):
This means 
of the job is completed per hour. To find the number of hours for the whole job (1 room), divide the whole by the fraction completed per hour:
Convert hours to minutes:
First find what fraction of the job is completed per hour. If John can paint the room in 4 hours, then he completes
Add
This means
Convert hours to minutes:
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Carol ate 3 pancakes in 5 minutes. If she continues to eat at the same rate, how many whole pancakes can she eat in 24 minutes?
Carol ate 3 pancakes in 5 minutes. If she continues to eat at the same rate, how many whole pancakes can she eat in 24 minutes?
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If Carol ate 3 pancakes in 5 minutes, she can eat 
of a pancake every minute. 
.
That means she ate 14 whole pancakes (and an additional 2/5 of another pancake).
If Carol ate 3 pancakes in 5 minutes, she can eat
That means she ate 14 whole pancakes (and an additional 2/5 of another pancake).
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According to the graph above, what was the percent change in the sales of camcorders from 2002-2006?
According to the graph above, what was the percent change in the sales of camcorders from 2002-2006?
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For this problem, pay attention only to the values between 2002 and 2006 - each of which is 17 (thousand) and 22 (thousand). Note that the decrease within this range does not need to be accounted for, as the question asks only for values relative to 2002 and 2006.
To calculate percent change, take it as a fraction: 22/17 = 1.294; this represents approximately 129% of the actual value or a 29% increase in sales across these 4 years.
For this problem, pay attention only to the values between 2002 and 2006 - each of which is 17 (thousand) and 22 (thousand). Note that the decrease within this range does not need to be accounted for, as the question asks only for values relative to 2002 and 2006.
To calculate percent change, take it as a fraction: 22/17 = 1.294; this represents approximately 129% of the actual value or a 29% increase in sales across these 4 years.
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What number is 150% greater than 6?
What number is 150% greater than 6?
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In the problem, 150% of 6 represents the change in the value, not the new value. Percent "greater than" is a key phrase that refers to percent change. Therefore, 9, or 150% of 6, is how much 6 increases. The answer is 6 + 9 = 15.
Formula: Original * (1 + percent increase/100) = New
6(1 + 1.50) = 15
In the problem, 150% of 6 represents the change in the value, not the new value. Percent "greater than" is a key phrase that refers to percent change. Therefore, 9, or 150% of 6, is how much 6 increases. The answer is 6 + 9 = 15.
Formula: Original * (1 + percent increase/100) = New
6(1 + 1.50) = 15
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