Arithmetic - GMAT Quantitative
Card 1 of 2420
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number 
were to be placed in its correct region in the diagram, which one would it be - I, II, III, IV, or V?
Statement 1: If 
, then 
would be placed in Region IV.
Statement 2: If 
, then 
would be placed in Region IV.
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number
Statement 1: If
Statement 2: If
Tap to reveal answer
Region IV comprises the rational numbers that are not integers. A number is rational if and only if it can be expressed as the quotient of integers.
From Statement 1 alone, it can be inferred that 
is rational, and that it is not an integer. Since 
, it follows that 
. However, this is not sufficient to narrow it down completely.
For example:
If 
, then 
, a natural number, putting it in Region I.
If 
, then 
, a rational number but not an integer, putting it in Region IV.
From Statement 2 alone, it can be inferred that 
is rational, and that it is not an integer. From 
, it follows that 
. The nonzero rational numbers are closed under division, so 
must be a rational number. However, since 
is not an integer, 
cannot be an integer, since the integers are closed under multiplication. Therefore, Statement 2 alone proves that 
belongs in Region IV.
Region IV comprises the rational numbers that are not integers. A number is rational if and only if it can be expressed as the quotient of integers.
From Statement 1 alone, it can be inferred that
For example:
If
If
From Statement 2 alone, it can be inferred that
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Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number 
were to be placed in its correct region in the diagram, which one would it be - I, II, III, IV, or V?
Statement 1: If 
, then 
would be in Region I.
Statement 2: If 
, then 
would be in Region III.
Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.
If real number
Statement 1: If
Statement 2: If
Tap to reveal answer
Assume Statement 1 alone. It cannot be determined what region 
is in.
For example, suppose 
, which is in Region I (the set of natural numbers, or positive integers). It is possible that 
, putting it in Region I, or 
, putting it in Region III (the set of integers that are not whole numbers - that is, the set of negative integers).
Assume Statement 2 alone. It cannot be determined what region is in.
For example, suppose 
, which is in Region III; then 
, which is also in Region III. But suppose 
; then 
, which, as an irrational number, is in Region V.
Now assume both statements. Then 
has an integer as a square and an integer as a cube. 
must either be an integer or an irrational number. But
, making it the quotient of integers, which is rational. Therefore, 
is an integer. Furthermore, its cube is negative, so 
is negative. The two statements together prove that 
is a negative integer, which belongs in Region III.
Assume Statement 1 alone. It cannot be determined what region
For example, suppose
Assume Statement 2 alone. It cannot be determined what region
For example, suppose
Now assume both statements. Then
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Which, if either, is the greater number: 
or 
?
Statement 1: 
Statement 2: 
Which, if either, is the greater number:
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 alone gives insufficient information. For example, if 
, then:
or 
Since 
, it is unclear which of 
and 
is greater, if either.
Statement 2 gives insufficient information; if 
is positive, 
is negative, and vice versa.
Assume both to be true. The two statements form a system of equations that can be solved using substitution:
Case 1:
Case 2:
This equation has no solution.
Therefore, the only possible solution is 
. Therefore, it can be concluded that 
.
Statement 1 alone gives insufficient information. For example, if
Since
Statement 2 gives insufficient information; if
Assume both to be true. The two statements form a system of equations that can be solved using substitution:
Case 1:
Case 2:
This equation has no solution.
Therefore, the only possible solution is
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and 
are integers. Is 
positive, negative, or zero?
Statement 1: 
is negative.
Statement 2: 
is odd.
Statement 1:
Statement 2:
Tap to reveal answer
A negative integer to an even power is positive:
Example: 
A negative integer to an odd power is negative:
Example: 
A positive integer to an odd power is positive:
Example: 
So, as seen in the first two statements, knowing only that base 
is negative is insufficient to detemine the sign of 
; as seen in the last two statements, knowing only that exponent 
is odd is also insufficient. But by the middle statement, knowing both facts tells us 
is negative.
The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient.
A negative integer to an even power is positive:
Example:
A negative integer to an odd power is negative:
Example:
A positive integer to an odd power is positive:
Example:
So, as seen in the first two statements, knowing only that base
The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient.
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Mrs. Smith purchased groceries whose price, before tax, was $147.64. What was the tax rate on those groceries (nearest hundredth of a percent)?
-
The tax she paid was $9.23.
-
The total amount she paid was $156.87
Mrs. Smith purchased groceries whose price, before tax, was $147.64. What was the tax rate on those groceries (nearest hundredth of a percent)?
-
The tax she paid was $9.23.
-
The total amount she paid was $156.87
Tap to reveal answer
To determine the tax rate, you need to know the purchase price, which is given, and the amount of tax paid. The amount of tax is given in Statement 1. Statement 2 alone, however, also allows you to find the amount of tax; just subtract:
Either way, you can now find the tax rate:
To determine the tax rate, you need to know the purchase price, which is given, and the amount of tax paid. The amount of tax is given in Statement 1. Statement 2 alone, however, also allows you to find the amount of tax; just subtract:
Either way, you can now find the tax rate:
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Data sufficiency question- do not actually solve the question
Find the mean of a set of 5 numbers.
1. The sum of the numbers is 72.
2. The median of the set is 15.
Data sufficiency question- do not actually solve the question
Find the mean of a set of 5 numbers.
1. The sum of the numbers is 72.
2. The median of the set is 15.
Tap to reveal answer
Statement 2 does not provide enough information about the mean as it can vary greatly from the median. Statement 1 is sufficient to calculate the mean, because even though it is impossible to calculate the set of numbers, the mean is calculated by dividing the sum by the total number of incidences in the set.
Statement 2 does not provide enough information about the mean as it can vary greatly from the median. Statement 1 is sufficient to calculate the mean, because even though it is impossible to calculate the set of numbers, the mean is calculated by dividing the sum by the total number of incidences in the set.
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How much greater is the average of the integers from 500 to 700 than the average of the integers from 60 to 90?
How much greater is the average of the integers from 500 to 700 than the average of the integers from 60 to 90?
Tap to reveal answer
In this case, average is also the middle value.
In this case, average is also the middle value.
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Data sufficiency question- do not actually solve the question
A bag of marbles consist of a mixture of black and red marbles. What is the probability of choosing a red marble followed by a black marble?
1. The probability of choosing a black marble first is 
.
2. There are 10 black marbles in the bag.
Data sufficiency question- do not actually solve the question
A bag of marbles consist of a mixture of black and red marbles. What is the probability of choosing a red marble followed by a black marble?
1. The probability of choosing a black marble first is
2. There are 10 black marbles in the bag.
Tap to reveal answer
From statement 1, we know the probabilty of choosing the first marble. However, since the marble is not replaced, it is impossible to calculate the probability of choosing the second marble. By knowing the information in statement 2 combined with statement 1, we can calculate the total number of marbles initially present.
From statement 1, we know the probabilty of choosing the first marble. However, since the marble is not replaced, it is impossible to calculate the probability of choosing the second marble. By knowing the information in statement 2 combined with statement 1, we can calculate the total number of marbles initially present.
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A certain major league baseball player gets on base 25% of the time (once every 4 times at bat).
For any game where he comes to bat 5 times, what is the probability that he will get on base either 3 or 4 times? - Hint – add the probability of 3 to the probability of 4.
A certain major league baseball player gets on base 25% of the time (once every 4 times at bat).
For any game where he comes to bat 5 times, what is the probability that he will get on base either 3 or 4 times? - Hint – add the probability of 3 to the probability of 4.
Tap to reveal answer
Binomial Table
Binomial Table
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Assume that we are the immortal gods of statistics and we know the following population statistics:
-
average driving speed for women=50 mph with a standard deviation of 12
-
average driving speed for men=45 mph with a standard deviation of 11
We look down from our statistical Mount Olympus and notice that the Earth mortals have randomly sampled 60 women and 65 men in an attempt to detect a significant difference in the average driving speed.
What is the probability that the Earth mortals will properly reject the assumption (i.e. the null hypothesis) that there is no significant difference between the average driving speeds. The Earth mortals have decided to use a 2-tailed 95% confidence test.
Assume that we are the immortal gods of statistics and we know the following population statistics:
-
average driving speed for women=50 mph with a standard deviation of 12
-
average driving speed for men=45 mph with a standard deviation of 11
We look down from our statistical Mount Olympus and notice that the Earth mortals have randomly sampled 60 women and 65 men in an attempt to detect a significant difference in the average driving speed.
What is the probability that the Earth mortals will properly reject the assumption (i.e. the null hypothesis) that there is no significant difference between the average driving speeds. The Earth mortals have decided to use a 2-tailed 95% confidence test.
Tap to reveal answer
standard deviation of the difference between the sample means =
at 95% (2-tailed) = 1.96
the sample difference must be 4 or greater
(note: the probability of the sample difference being -4 or less is so small (4.5 standard deviations) that it will be ignored and we will only consider the probability that the difference is 4 or more.)
the probablity of the sample difference being 4 or greater (knowing that the population difference is 5) =
the table shows that .3156 lies below -.48, so, .6844 lies above -.48
In English - there is a .6844 probability that the 2 sample means will yield a sample difference that is 1.96 or more standard deviations above 0.
standard deviation of the difference between the sample means =
the sample difference must be 4 or greater
(note: the probability of the sample difference being -4 or less is so small (4.5 standard deviations) that it will be ignored and we will only consider the probability that the difference is 4 or more.)
the probablity of the sample difference being 4 or greater (knowing that the population difference is 5) =
the table shows that .3156 lies below -.48, so, .6844 lies above -.48
In English - there is a .6844 probability that the 2 sample means will yield a sample difference that is 1.96 or more standard deviations above 0.
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In a popular state lottery game, a player selects 5 numbers (on 1 ticket) out of a possible 39 numbers. There are 575,757 possible 5 number combinations.
So, the odds are 575,757 to 1 against winning.
What are the odds of getting 4 of the 5 numbers correct on 1 ticket?
In a popular state lottery game, a player selects 5 numbers (on 1 ticket) out of a possible 39 numbers. There are 575,757 possible 5 number combinations.
So, the odds are 575,757 to 1 against winning.
What are the odds of getting 4 of the 5 numbers correct on 1 ticket?
Tap to reveal answer
575,757 must be divided by -
575,757 must be divided by -
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One card from another deck is added to a standard deck of fifty-two cards. The cards are shuffled and one card is removed.
A card is then drawn at random. What is the probability that that card is an ace?
Statement 1: The card that was added was a spade.
Statetment 2: The card that was removed was a jack.
One card from another deck is added to a standard deck of fifty-two cards. The cards are shuffled and one card is removed.
A card is then drawn at random. What is the probability that that card is an ace?
Statement 1: The card that was added was a spade.
Statetment 2: The card that was removed was a jack.
Tap to reveal answer
You need to know two things to answer this question - the rank of the added card, and the rank of the removed card. The second statement is useful but not sufficient; the first is irrelevant to the question.
You need to know two things to answer this question - the rank of the added card, and the rank of the removed card. The second statement is useful but not sufficient; the first is irrelevant to the question.
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Your friend at work submits a bold hypothesis. He suggests that the number of sales per day follow the pattern - Monday-10%, Tuesday-10%, Wednesday-10%, Thursday 35% and Friday 35%.
You and he then record the number of sales for the following week: Monday-120, Tuesday-85, Wednesday-105, Thursday-325 and Friday-365.
After viewing the observed data, your friend expresses serious concern regarding his hypothesis.
You can help; you can tell him the probability of the observed data occuring if the hypothesis is true. Hint - Excel ChiTest.
Your friend at work submits a bold hypothesis. He suggests that the number of sales per day follow the pattern - Monday-10%, Tuesday-10%, Wednesday-10%, Thursday 35% and Friday 35%.
You and he then record the number of sales for the following week: Monday-120, Tuesday-85, Wednesday-105, Thursday-325 and Friday-365.
After viewing the observed data, your friend expresses serious concern regarding his hypothesis.
You can help; you can tell him the probability of the observed data occuring if the hypothesis is true. Hint - Excel ChiTest.
Tap to reveal answer
Use Excel ChiTest to get the .063 probability. If you are old-fashioned, you can also obtain the Chi-Squared number (8.928) by using ChiInv; but, it is not needed
Use Excel ChiTest to get the .063 probability. If you are old-fashioned, you can also obtain the Chi-Squared number (8.928) by using ChiInv; but, it is not needed
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A certain tutor boasts that his 2 week training program will increase a student's score on a 2400 point exam by at least 100 points (4.167%). A 10 student 'before-and-after' study was conducted to validate the claim. The following results were obtained - the 3 columns represent the before, afer and increase numbers for each of the 10 students:
1300 1340 40
1670 1790 120
1500 1710 210
1360 1660 300
1580 1730 150
1160 1320 160
1910 2100 190
1410 1490 80
1710 1880 170
1990 2060 70
Assume the null Hypothesis:
'The average increase is less than 100 points'
What is the highest level of significance (p-value) at which the null hypothesis will be rejected?
A certain tutor boasts that his 2 week training program will increase a student's score on a 2400 point exam by at least 100 points (4.167%). A 10 student 'before-and-after' study was conducted to validate the claim. The following results were obtained - the 3 columns represent the before, afer and increase numbers for each of the 10 students:
1300 1340 40
1670 1790 120
1500 1710 210
1360 1660 300
1580 1730 150
1160 1320 160
1910 2100 190
1410 1490 80
1710 1880 170
1990 2060 70
Assume the null Hypothesis:
'The average increase is less than 100 points'
What is the highest level of significance (p-value) at which the null hypothesis will be rejected?
Tap to reveal answer
Using Excel, the average increase (column 3) is 149 and Standard Deviation of the increases is 76.37
Using Excel - a t value of 2.03 for 9 degrees of freedom = .036
Using Excel, the average increase (column 3) is 149 and Standard Deviation of the increases is 76.37
Using Excel - a t value of 2.03 for 9 degrees of freedom = .036
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A test for a new drug was conducted. In the control or placebo group, 7 of 210 participants experienced positive results. The group that took the drug experienced 27 out of 374 positive resluts.
The placebo group had a sucess rate of .0333 and the drug group had a success rate of .0722. The difference is .0389 and the overall percentage (for both groups combined) is .0582
At what level is the difference of .0389 significant? Asked another way - what is the p-value for .0389?
A test for a new drug was conducted. In the control or placebo group, 7 of 210 participants experienced positive results. The group that took the drug experienced 27 out of 374 positive resluts.
The placebo group had a sucess rate of .0333 and the drug group had a success rate of .0722. The difference is .0389 and the overall percentage (for both groups combined) is .0582
At what level is the difference of .0389 significant? Asked another way - what is the p-value for .0389?
Tap to reveal answer
The standard error of the difference (.0389) =
test statistic -
from the table (or excel NormsDist) - Z=1.9245 translates to .9729
The standard error of the difference (.0389) =
test statistic -
from the table (or excel NormsDist) - Z=1.9245 translates to .9729
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A type 1 error (False Alarm or 'Convicting the innocent man') occurs when we incorrectly reject a true null hypothesis.
A type 2 error (failure to detect) occurs when we fail to reject a false null hypothesis.
Which one of the following 5 statements is false?
Note - only 1 of the statements is false.
A) For a given sample size (n=100), decreasing the significane level (from .05 to .01) will decrease the chance of a type 1 error.
B) For a given sample size (n=100), increasing the significane level (from .01 to .05) will decrease the chance of a type 2 error.
C) The ability to correctly detect a false null hypothesis is called the 'Power' of a test.
D) Increasing sample size (from 100 to 120) will always decrease the chance of both a type 1 error and a type 2 error.
E) None of the above statements are true.
A type 1 error (False Alarm or 'Convicting the innocent man') occurs when we incorrectly reject a true null hypothesis.
A type 2 error (failure to detect) occurs when we fail to reject a false null hypothesis.
Which one of the following 5 statements is false?
Note - only 1 of the statements is false.
A) For a given sample size (n=100), decreasing the significane level (from .05 to .01) will decrease the chance of a type 1 error.
B) For a given sample size (n=100), increasing the significane level (from .01 to .05) will decrease the chance of a type 2 error.
C) The ability to correctly detect a false null hypothesis is called the 'Power' of a test.
D) Increasing sample size (from 100 to 120) will always decrease the chance of both a type 1 error and a type 2 error.
E) None of the above statements are true.
Tap to reveal answer
Statements A, B, C, and D are all true - so -
The only false statement is E (the statement that declares that A and B and C and D are all false)
Statements A, B, C, and D are all true - so -
The only false statement is E (the statement that declares that A and B and C and D are all false)
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Four coins - a fair penny, a loaded penny, a fair nickel, and a loaded nickel - are tossed. What is the probability that all four will come up heads?
Statement 1: Yesterday, out of 100 tosses, the loaded penny came up heads 70 times.
Statement 2: Yesterday, out of 100 tosses, the loaded nickel came up heads 40 times.
Four coins - a fair penny, a loaded penny, a fair nickel, and a loaded nickel - are tossed. What is the probability that all four will come up heads?
Statement 1: Yesterday, out of 100 tosses, the loaded penny came up heads 70 times.
Statement 2: Yesterday, out of 100 tosses, the loaded nickel came up heads 40 times.
Tap to reveal answer
While experiments such as repeated tossings can give an idea of the probability that a coin will come up heads or tails, they do not provide a definitive answer, so neither statement is helpful here.
While experiments such as repeated tossings can give an idea of the probability that a coin will come up heads or tails, they do not provide a definitive answer, so neither statement is helpful here.
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Jerry is a Cardinal fan and he and his family live on a street with 9 other families that are all Cardinal fans. One block to the north, there are 11 families and they are all Cub fans. These 21 households all buy their lawn fertilizer from Ben's Lawn and Garden Shop. Jerry suspects that Ben (who is originally from Chicago) is a Cub fan and that he provides better fertilizer to the Cub fans than to the Cardinal fans, while charging the same price for all.
Last Saturday everyone in town mowed their lawn. At 2:00 AM Sunday morning, Jerry snuck around town and weighed all of the grass clippings for the 21 households in question.
The weights (in lbs) of the grass clippings for the 10 Cardinal homes were:
82, 85, 90, 74, 80, 89, 75, 81, 93, 75
The weights (in lbs) of the grass clippings for the 11 Cub homes were:
90, 87, 93, 75, 88, 96, 90, 82, 95, 97, 78
The Cardinal average was 82.4; the Cub average was 88.27.
At what level is the 5.87 lb difference significant? - Asked another way - what is the p value for the 5.87 lb difference.
Jerry is a Cardinal fan and he and his family live on a street with 9 other families that are all Cardinal fans. One block to the north, there are 11 families and they are all Cub fans. These 21 households all buy their lawn fertilizer from Ben's Lawn and Garden Shop. Jerry suspects that Ben (who is originally from Chicago) is a Cub fan and that he provides better fertilizer to the Cub fans than to the Cardinal fans, while charging the same price for all.
Last Saturday everyone in town mowed their lawn. At 2:00 AM Sunday morning, Jerry snuck around town and weighed all of the grass clippings for the 21 households in question.
The weights (in lbs) of the grass clippings for the 10 Cardinal homes were:
82, 85, 90, 74, 80, 89, 75, 81, 93, 75
The weights (in lbs) of the grass clippings for the 11 Cub homes were:
90, 87, 93, 75, 88, 96, 90, 82, 95, 97, 78
The Cardinal average was 82.4; the Cub average was 88.27.
At what level is the 5.87 lb difference significant? - Asked another way - what is the p value for the 5.87 lb difference.
Tap to reveal answer
Cub variance = 53.218; Cardinal variance = 45.378.
The standard deviation of the difference (5.87) is:
TDIST of 1.918 with 19 Degrees of Freedom = .035
So, we would reject the null hypothesis (the hypothesis that claims that the means are equal) at 95% confidence (p=.05) and not reject at 99% (p=.01)
Cub variance = 53.218; Cardinal variance = 45.378.
The standard deviation of the difference (5.87) is:
TDIST of 1.918 with 19 Degrees of Freedom = .035
So, we would reject the null hypothesis (the hypothesis that claims that the means are equal) at 95% confidence (p=.05) and not reject at 99% (p=.01)
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A marble is selected at random from a box of red, yellow, and blue marbles. What is the probability that the marble is yellow?
-
There are ten blue marbles in the box.
-
There are eight red marbles in the box.
A marble is selected at random from a box of red, yellow, and blue marbles. What is the probability that the marble is yellow?
-
There are ten blue marbles in the box.
-
There are eight red marbles in the box.
Tap to reveal answer
To determine the probability that the marble is yellow we need to know two things: the number of yellow marbles, and the number of marbles total. The first quantity divided by the last quantity is our probablility.
But the two given statements together only tell us that eighteen marbles are not yellow. This is not enough information. For example, if there are two yellow marbles, the probability of drawing a yellow marble is 
. But if there are twenty-two yellow marbles, the probability of drawing a yellow marble is 
Therefore, the answer is that both statements together are insufficient to answer the question.
To determine the probability that the marble is yellow we need to know two things: the number of yellow marbles, and the number of marbles total. The first quantity divided by the last quantity is our probablility.
But the two given statements together only tell us that eighteen marbles are not yellow. This is not enough information. For example, if there are two yellow marbles, the probability of drawing a yellow marble is
Therefore, the answer is that both statements together are insufficient to answer the question.
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Several decks of playing cards are shuffled together. One card is drawn, shown, and put aside. Another card is dealt. What is the probability that the dealt card is red, assuming the first card is known?
-
The card removed before the deal was red.
-
The cards were shuffled again between the draw and the deal.
Several decks of playing cards are shuffled together. One card is drawn, shown, and put aside. Another card is dealt. What is the probability that the dealt card is red, assuming the first card is known?
-
The card removed before the deal was red.
-
The cards were shuffled again between the draw and the deal.
Tap to reveal answer
To answer this question you need to know two things: the number of red cards left and the number of total cards left. The second statement is irrelevant, as a reshuffle does not change the composition of the deck. The first statement tells you that there is one fewer red card than black cards, but it does not tell you how many of each there are, as you do not know how many decks of cards there were.
And that information, which is not given, affects the answer. For example, if there were four decks, there were 103 red cards out of 207; if there were six decks, there were 155 red cards out of 311. The probabilities would be, respectively,
and
,
a small difference, but nonetheless, a difference.
The correct answer is that both statements together are insufficient to answer the question.
To answer this question you need to know two things: the number of red cards left and the number of total cards left. The second statement is irrelevant, as a reshuffle does not change the composition of the deck. The first statement tells you that there is one fewer red card than black cards, but it does not tell you how many of each there are, as you do not know how many decks of cards there were.
And that information, which is not given, affects the answer. For example, if there were four decks, there were 103 red cards out of 207; if there were six decks, there were 155 red cards out of 311. The probabilities would be, respectively,
and
a small difference, but nonetheless, a difference.
The correct answer is that both statements together are insufficient to answer the question.
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