DSQ: Understanding real numbers - GMAT Quantitative
Card 0 of 248
Is z a negative number?
(1) 13z>14z
(2) 4+z is positive
Is z a negative number?
(1) 13z>14z
(2) 4+z is positive
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(1) Subtracting 13z from both sides of 13z>14z shows that 0>z, so z must be negative rightarrow SUFFICIENT
(2) Subtracting 4 from both sides of 4+z>0 gives 
. Therefore, for this statement, z can be negative but is not necessarily negative. rightarrow NOT SUFFICIENT.
(1) Subtracting 13z from both sides of 13z>14z shows that 0>z, so z must be negative rightarrow SUFFICIENT
(2) Subtracting 4 from both sides of 4+z>0 gives
If 
, 
, and 
are points on the number line, what is the distance between 
and 
?
(1) The distance between 
and 
is 7.
(2) The distance between 
and 
is 12.
If
(1) The distance between
(2) The distance between
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Looking at statements (1) and (2) separately, there is no information about the distance between 
and 
. Looking at statements (1) and (2) together, there are two possibilities: 1) 
and 
are on different sides of 
, and the distance between them is 19; 2) 
and 
are on the same side of 
, and the distance between them is 5. Therefore, we cannot get the only possible answer to this question based on the two statements.
Looking at statements (1) and (2) separately, there is no information about the distance between
Data sufficiency question- do not actually solve the question
How many children are attending a party?
1. If 6 more children attend the party, there will be 24 children there.
2. If 10 children leave the party, there will be less than 12 children in attendance.
Data sufficiency question- do not actually solve the question
How many children are attending a party?
1. If 6 more children attend the party, there will be 24 children there.
2. If 10 children leave the party, there will be less than 12 children in attendance.
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Statement 1 precisely tells how many children are attending the party while Statement 2 only provides a range.
Statement 1 precisely tells how many children are attending the party while Statement 2 only provides a range.
A sandwich shop offers a discount in which customers pay $2.99 for each additional sandwich after purchasing their first sandwich. Emily bought her first sandwich for $4.95. If she purchased 5 additional sandwiches to take home, then the total amount she paid is equivalent to which of the following?
A sandwich shop offers a discount in which customers pay $2.99 for each additional sandwich after purchasing their first sandwich. Emily bought her first sandwich for $4.95. If she purchased 5 additional sandwiches to take home, then the total amount she paid is equivalent to which of the following?
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The price of 6 sandwiches can be expressed as 
. Regrouping is needed since this is not in the answer choices. If $3.00 is used instead of $2.99, each of the five additional sandwiches is .01 too high and the total is .05 too high. Therefore, this amount needs to be subtracted from the $4.95.
The price of 6 sandwiches can be expressed as
Given that 
and 
, is 
positive, negative, or zero?
-
-
Given that
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If it is known that 
, it can be determined that 
.
Then 
Since 
, 
, and the question is answered.
But if we only know that 
, it is not sufficient.
Case 1:
yields 
Case 2:
yields 
The answer is that Statement 1 is sufficient, but not Statement 2.
If it is known that
Then
Since
But if we only know that
Case 1:
Case 2:
The answer is that Statement 1 is sufficient, but not Statement 2.
Which of the following is the decimal representation of an irrational number? (Assume any observed patterns continue)
Which of the following is the decimal representation of an irrational number? (Assume any observed patterns continue)
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A number is rational if and only if its decimal representation is terminating or repeating.
and 
are terminating decimals, which are rational (note: the latter number is not equal to 
). Both can be eliminated.
has a repeating digit and 
has a repeating group of five digits, and are therefore repeating decimals, which are also rational. Both can be eliminated.
has an infinite pattern - the number of zeroes that precede each one is increasing by one with each group - but since the pattern is not repeating, it is neither a terminating decimal nor a repeating decimal. It is irrational.
A number is rational if and only if its decimal representation is terminating or repeating.
Jenkins wanted to find all the real solutions of the following expression. Find them for him.
I) 
has a vertex at the point 
.
II) 
is never undefined.
Jenkins wanted to find all the real solutions of the following expression. Find them for him.
I)
II)
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To find the real solutions, we need to know what c is. An easy way to do that would be if we had a point on the function.
I) gives us such a point. Therefore, we are able to substitute in our values and solve for c using algebraic operations.
Thus our equation becomes:
.
From here we can try to factor to find the real solutions or you can use the quadratic formula.
Plugging in our values we get the solution 
.
II) Is not really relevant, we are dealing with a parabola, so it shouldn't be undefined anyway.
To find the real solutions, we need to know what c is. An easy way to do that would be if we had a point on the function.
I) gives us such a point. Therefore, we are able to substitute in our values and solve for c using algebraic operations.
Thus our equation becomes:
From here we can try to factor to find the real solutions or you can use the quadratic formula.
Plugging in our values we get the solution
II) Is not really relevant, we are dealing with a parabola, so it shouldn't be undefined anyway.
Order 
from least to greatest.
Order
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We can find each in terms of 
.
In ascending order, the four values are:
The correct choice is 
.
We can find each in terms of
In ascending order, the four values are:
The correct choice is
Evaluate 
.
Statement 1: 
Statement 2: 
Evaluate
Statement 1:
Statement 2:
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From Statement 1 alone, since 1 raised to the power of any real number is equal to 1, we know that
.
However, we cannot determine the value of 
knowing only that 
. If 
is a nonzero number, then 
. However, if 
, then
,
which is an undefined expression.
From Statement 1 alone, since 1 raised to the power of any real number is equal to 1, we know that
However, we cannot determine the value of
which is an undefined expression.
True or false: 
is an integer.
Statement 1: The multiplicative inverse of 
is not an integer.
Statement 2: The additive inverse of 
is an integer.
True or false:
Statement 1: The multiplicative inverse of
Statement 2: The additive inverse of
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Assume Statement 1 alone. The multiplicative inverse of a number is the number which, when multiplied by that number, yields a product of 1. If the multiplicative inverse of 
is not an integer, it is possible for 
to be an integer or to not be one, as is shown in these examples:
If 
, which is an integer, then, since 
, the multiplicative inverse of 
is 
, which is not an integer, If 
, which is not an integer, then, since 
, the multiplicative inverse of 
is 
, which is not an integer.
Assume Statement 2 alone. The additive inverse of a number is the number which, when added to that number, yields a sum of 0. If 
is the additive inverse of the number 
, then
, or
by Statement 2, 
is an integer; .
, the product of integers, is itself an integer.
Assume Statement 1 alone. The multiplicative inverse of a number is the number which, when multiplied by that number, yields a product of 1. If the multiplicative inverse of
If
Assume Statement 2 alone. The additive inverse of a number is the number which, when added to that number, yields a sum of 0. If
by Statement 2,
Evaluate 
.
Statement 1: 
Statement 2: 
Evaluate
Statement 1:
Statement 2:
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Assume Statement 1 alone.
, since 0 multiplied by any number yields a product of 0.
Assume Statement 2 alone.
, since 0 added to any number yields the sum 0. However, without knowing 
and 
, or their product, it is impossible to determine the result.
Assume Statement 1 alone.
Assume Statement 2 alone.
is a real number. True or false: 
is a rational number.
Statement 1: A square whose side has length 
has area 
.
Statement 2: 
is a rational number.
Statement 1: A square whose side has length
Statement 2:
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From Statement 1 alone, since a side of a square with area 
has as its sidelength
,
.
This is the quotient of two integers; by definition, this is rational.
From Statement 2 alone, if we let 
, then 
. 
is rational, and so is 6, so their product, 
, is also rational.
From Statement 1 alone, since a side of a square with area
This is the quotient of two integers; by definition, this is rational.
From Statement 2 alone, if we let
is a real number. True or false: is a rational number.
Statement 1: 
is a rational number.
Statement 2: 
is a rational number.
Statement 1:
Statement 2:
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Statement 1 provides insufficient information to determine whether 
is rational or not.
If 
, which, being an integer, is rational, then 
, which, being an integer, is rational.
If 
. which is irrational, then 
, which, being an integer, is rational.
Statement 2 is, however, sufficient. If 
for some rational number 
, then 
. 1, being an integer, is rational, and as the difference of rational numbers, 
is rational.
Statement 1 provides insufficient information to determine whether
If
If
Statement 2 is, however, sufficient. If
is a real number. True or false: is a rational number.
Statement 1: 
is an irrational number.
Statement 2: 
is an irrational number.
Statement 1:
Statement 2:
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Assume Statement 1 alone. 
, the quotient of integers, is rational. The sum of any two rational numbers is rational, so if 
is rational, then 
is a rational number. However, 
is irrational, so, contrapositively, 
is irrational.
Assume Statement 2 alone. 
, the quotient of integers, is rational. The product of any two rational numbers is rational, so if 
is rational, then 
is a rational number. However, 
is irrational, so, contrapositively, 
is irrational.
Assume Statement 1 alone.
Assume Statement 2 alone.
is a real number. True or false: is a rational number.
Statement 1: 
Statement 2: 
Statement 1:
Statement 2:
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Assume Statement 1 alone. The polynomial expression in
can be factored as follows:
Either 
, in which case 
, which, as an integer is also rational, or
, in which case:
or either 
or 
, both irrational.
Therefore, Statement 1 provides insufficient information.
Assume Statement 2 alone. If 
, then 
, making 
a fourth root of 4. A rational root of an integer must itself be an integer, and there is no integer which, when taken to the fourth power, is equal to 4. Therefore, 
must be irrational.
Assume Statement 1 alone. The polynomial expression in
can be factored as follows:
Either
or either
Therefore, Statement 1 provides insufficient information.
Assume Statement 2 alone. If
is a real number, with 
a positive integer. True or false: is a rational number.
Statement 1: is a multiple of 9.
Statement 2: 
is a multiple of 12.
Statement 1:
Statement 2:
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The two statements together provide insufficient information to answer the question. For example, 36 and 72 are both multiples of both 9 and 12. However, the positive square root of 36 is 6, making 
an integer and, consequently, rational. However, the positive square root of 72 is not an integer, since 72 falls between consecutive perfect squares 64 and 81 (the squares of 8 and 9, respectively), and any square root of an integer that is not itself an integer must be irrational.
The two statements together provide insufficient information to answer the question. For example, 36 and 72 are both multiples of both 9 and 12. However, the positive square root of 36 is 6, making
is a real number. True or false: is an integer.
Statement 1: 
Statement 2: 
Statement 1:
Statement 2:
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Assume Statement 1 alone. The polynomial expression in
can be factored as follows:
Either 
, in which case 
, or
- which is impossible for any real value of 
, since
, and
Therefore, 
is an integer.
Assume Statement 2 alone. Again, we factor:
Similarly to the polynomial in Statement 1, 
, 
, or 
. All three possible values of 
are integers.
Assume Statement 1 alone. The polynomial expression in
can be factored as follows:
Either
Therefore,
Assume Statement 2 alone. Again, we factor:
Similarly to the polynomial in Statement 1,
Evaluate 
.
Statement 1: 
Statement 2: 
Evaluate
Statement 1:
Statement 2:
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Assume Statement 1 alone. By the commutative property of addition,
.
By substitution,
.
Assume Statement 2 alone. By the commutative property of multiplication,
and 
By two substitutions,
Assume Statement 1 alone. By the commutative property of addition,
By substitution,
Assume Statement 2 alone. By the commutative property of multiplication,
By two substitutions,
Evaluate 
.
Statement 1: 
Statement 2: 
Evaluate
Statement 1:
Statement 2:
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By the distributive property of multiplication over addition,
Assume Statement 1 alone. 
, so, by substitution,
.
But, without knowing 
, 
, or their product, it is impossible to determine the value of the expression. Statement 2 provides insufficient information, for similar reasons.
Now assume both statements to be true. Then
Since 
and 
, by substitution,
.
By the distributive property of multiplication over addition,
Assume Statement 1 alone.
But, without knowing
Now assume both statements to be true. Then
Since
Evaluate 
.
Statement 1: 
Statement 2: 
Evaluate
Statement 1:
Statement 2:
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Assume Statmemt 1 alone. By the commutative property of multiplication:
By the addition property of equality,
By substitution,
Statement 2 alone, however, does not answer the question. If 
, then by substitution,
;
however, without further information, we cannot evaluate this expression.
Assume Statmemt 1 alone. By the commutative property of multiplication:
By the addition property of equality,
By substitution,
Statement 2 alone, however, does not answer the question. If
however, without further information, we cannot evaluate this expression.