DSQ: Simplifying algebraic expressions - GMAT Quantitative
Card 1 of 120
Carleton's teacher challenged him to fill in the circle and the square below to form a polynomial with degree 
.
Did Carleton succeed?
Statement 1: The sum of the numbers Carleton filled in the two shapes was 
.
Statement 2: Carleton wrote a 
in the circle.
Carleton's teacher challenged him to fill in the circle and the square below to form a polynomial with degree
Did Carleton succeed?
Statement 1: The sum of the numbers Carleton filled in the two shapes was
Statement 2: Carleton wrote a
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The degree of a polynomial is the highest of the degrees of any of its terms; the degree of a term with one variable is the exponent of the variable.
Statement 1 alone provides insufficient information, since, for example, the degree 
polynomial 
and the degree 6 polynomial 
both satisfy the statement. Statement 2 alone is also insufficient, since it says nothing about the number in the square.
The two statements together are sufficient; if the number in the circle is 
, and the numbers add up to 
, the number in the square is 
, making the polynomial the degree 
polynomial 
.
The degree of a polynomial is the highest of the degrees of any of its terms; the degree of a term with one variable is the exponent of the variable.
Statement 1 alone provides insufficient information, since, for example, the degree
The two statements together are sufficient; if the number in the circle is
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Evaluate the expression 
-
-
Evaluate the expression
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Simplify the expression:
Therefore, we only need to know 
- If we know 
, we calculate that 
The answer is that Statement 2 alone is sufficient to answer the question, but Statement 1 is not.
Simplify the expression:
Therefore, we only need to know
The answer is that Statement 2 alone is sufficient to answer the question, but Statement 1 is not.
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Evaluate the expression 
.
Statement 1: 
Statement 2: 
Evaluate the expression
Statement 1:
Statement 2:
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Simplify the expression:
This is equal to 
Knowing the sum or the difference of 
and 
is not enough to determine the value of this expression. But knowing both allows you to solve a system of equations:
The expression can now be evaluated by substitution.
The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.
Simplify the expression:
This is equal to
Knowing the sum or the difference of
The expression can now be evaluated by substitution.
The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.
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Evaluate: 
Statement 1: 
Statement 2: 
Evaluate:
Statement 1:
Statement 2:
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Therefore, you only need to know the value of 
to evaluate this; knowing the value of 
is neither necessary nor helpful.
Therefore, you only need to know the value of
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What is the value of 
?
(1) 
(2) 
What is the value of
(1)
(2)
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(1) Add 
to both sides to make 
. Then divide through by 7 to get
. This statement is sufficient.
(2) Divide both sides by 13. The equation becomes 
. This statement is sufficient.
(1) Add
(2) Divide both sides by 13. The equation becomes
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Is 
?
Is
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(1) Since 5 must be added to 
to make it equal to 
, it follows that 
. This statement is sufficient.
(2) Multiply both sides by 6 to obtain 
. Thus, whether 
or
depends on the value selected for 
and 
. For instance, 
implies
, (such that 
) but 
implies 
,
(such that 
). Therefore, this statement is insufficient.
(1) Since 5 must be added to
(2) Multiply both sides by 6 to obtain
(such that
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Evaluate the expression 
Statement 1: 
Statement 2: 
Evaluate the expression
Statement 1:
Statement 2:
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We simplify to make sure of whether we need to know one or both of 
and 
:
Indeed, the values of both 
and 
must be known to evaluate this expresson.
We simplify to make sure of whether we need to know one or both of
Indeed, the values of both
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Evaluate the expression for positive 
:
Statement 1: 
Statement 2: 
Evaluate the expression for positive
Statement 1:
Statement 2:
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Cancel the 
from both halves:
As can be seen by the simplification, it turns out that only the value of 
, which is given only in Statement 2, affects the value of the expression.
Cancel the
As can be seen by the simplification, it turns out that only the value of
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True or false?
Statement 1: 
Statement 2: 
True or false?
Statement 1:
Statement 2:
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Simplify both expressions algebraically.
Using similar algebra, you can simplify the other expression:
The question, assuming the variables have nonzero values, is equivalent to asking whether 
is true. Since we need to know the values of both variables to answer this, both statements are necessary and sufficient.
Simplify both expressions algebraically.
Using similar algebra, you can simplify the other expression:
The question, assuming the variables have nonzero values, is equivalent to asking whether
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True or false?
Statement 1: 
Statement 2: 
True or false?
Statement 1:
Statement 2:
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Simplify each expression.
The inequality, therefore, is equivalent to
,
the truth or falsity of which depends only on the value of 
Simplify each expression.
The inequality, therefore, is equivalent to
the truth or falsity of which depends only on the value of
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Stephanie was challenged by her teacher to create a monomial of degree 5 by filling in the square and the circle in the figure below.
Did Stephanie succeed?
Statement 1: Stephanie wrote a 5 in the square.
Statement 2: Stephanie wrote a 7 in the circle.
Stephanie was challenged by her teacher to create a monomial of degree 5 by filling in the square and the circle in the figure below.
Did Stephanie succeed?
Statement 1: Stephanie wrote a 5 in the square.
Statement 2: Stephanie wrote a 7 in the circle.
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The degree of a monomial with one variable is the exponent of that variable. Therefore, only the number Stephanie wrote in the circle is relevant. Statement 1 is unhelpful; Statement 2 alone proves that Stephanie was unsuccessful.
The degree of a monomial with one variable is the exponent of that variable. Therefore, only the number Stephanie wrote in the circle is relevant. Statement 1 is unhelpful; Statement 2 alone proves that Stephanie was unsuccessful.
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Michelle was challenged by her teacher to create a monomial of degree 
by filling in the square and the circle in the figure below.
Did Michelle succeed?
Statement 1: Michelle wrote the same positive integer in both the circle and the square.
Statement 2: Michelle wrote a 
in the circle.
Michelle was challenged by her teacher to create a monomial of degree
Did Michelle succeed?
Statement 1: Michelle wrote the same positive integer in both the circle and the square.
Statement 2: Michelle wrote a
Tap to reveal answer
The degree of a monomial with more than one variable is the sum of the exponents of the two variables.
If Statement 1 alone is assumed, then, since the same integer is in both shapes, the sum must be twice this integer. The monomial must therefore have even degree—and, specifically, its degree cannot be 
.
Statement 2 alone proves nothing; if Michelle wrote a 
in the circle, she could have formed, for example, 
, which has degree 5, or 
, which has degree 
.
The degree of a monomial with more than one variable is the sum of the exponents of the two variables.
If Statement 1 alone is assumed, then, since the same integer is in both shapes, the sum must be twice this integer. The monomial must therefore have even degree—and, specifically, its degree cannot be
Statement 2 alone proves nothing; if Michelle wrote a
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Gary was challenged by his teacher to write some whole numbers in the shapes in the diagram below in order to form a simplified expression.
Did Gary succeed?
Statement 1: Gary wrote the same whole number in the square and in the diamond.
Statement 2: Gary wrote different whole numbers in the circle and the triangle.
Gary was challenged by his teacher to write some whole numbers in the shapes in the diagram below in order to form a simplified expression.
Did Gary succeed?
Statement 1: Gary wrote the same whole number in the square and in the diamond.
Statement 2: Gary wrote different whole numbers in the circle and the triangle.
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A polynomial in one variable is simplified if and only if no two of its terms are like—that is, if and only if no two terms have the same exponent. Statement 1 alone only addresses the coefficients, which are not relevant to whether the expression can be simplified further. By Statement 2 alone, Gary wrote different exponents, so he succeeded in forming a simplified polynomial.
A polynomial in one variable is simplified if and only if no two of its terms are like—that is, if and only if no two terms have the same exponent. Statement 1 alone only addresses the coefficients, which are not relevant to whether the expression can be simplified further. By Statement 2 alone, Gary wrote different exponents, so he succeeded in forming a simplified polynomial.
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The figure below shows a binomial with its coefficients and exponents replaced by shapes:
Each shape replaces a whole number.
Is this a simplified expression?
Statement 1: The square and the circle are replacing the same integer.
Statement 2: The diamond and the triangle are replacing different integers.
The figure below shows a binomial with its coefficients and exponents replaced by shapes:
Each shape replaces a whole number.
Is this a simplified expression?
Statement 1: The square and the circle are replacing the same integer.
Statement 2: The diamond and the triangle are replacing different integers.
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Both statements together are insufficient.
and 
each match the conditions of both statements. However, the former is the sum of like terms - the exponents are the same - and can be simplified; the latter is the sum of unlike terms - the exponents are different - and cannot.
Both statements together are insufficient.
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The figure below shows a trinomial with its exponents replaced by shapes:
Each shape replaces a whole number.
Is this a simplified expression?
Statement 1: The sum of the three exponents is 10.
Statement 2: The circle and the triangle are replacing different numbers.
The figure below shows a trinomial with its exponents replaced by shapes:
Each shape replaces a whole number.
Is this a simplified expression?
Statement 1: The sum of the three exponents is 10.
Statement 2: The circle and the triangle are replacing different numbers.
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Assume both statements are true.
and 
each fits the conditions of both statements. However, the first polynomial, having no like terms - three different exponents - is a simplified expression; the second, having like terms - both with exponent 4 - is not.
Assume both statements are true.
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Carleton's teacher challenged him to fill in the circle and the square below to form a polynomial with degree .
Did Carleton succeed?
Statement 1: The sum of the numbers Carleton filled in the two shapes was .
Statement 2: Carleton wrote a in the circle.
Carleton's teacher challenged him to fill in the circle and the square below to form a polynomial with degree
Did Carleton succeed?
Statement 1: The sum of the numbers Carleton filled in the two shapes was
Statement 2: Carleton wrote a
Tap to reveal answer
The degree of a polynomial is the highest of the degrees of any of its terms; the degree of a term with one variable is the exponent of the variable.
Statement 1 alone provides insufficient information, since, for example, the degree polynomial and the degree 6 polynomial both satisfy the statement. Statement 2 alone is also insufficient, since it says nothing about the number in the square.
The two statements together are sufficient; if the number in the circle is , and the numbers add up to , the number in the square is , making the polynomial the degree polynomial .
The degree of a polynomial is the highest of the degrees of any of its terms; the degree of a term with one variable is the exponent of the variable.
Statement 1 alone provides insufficient information, since, for example, the degree
The two statements together are sufficient; if the number in the circle is
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Evaluate the expression
-
-
Evaluate the expression
Tap to reveal answer
Simplify the expression:
Therefore, we only need to know - If we know , we calculate that
The answer is that Statement 2 alone is sufficient to answer the question, but Statement 1 is not.
Simplify the expression:
Therefore, we only need to know
The answer is that Statement 2 alone is sufficient to answer the question, but Statement 1 is not.
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Evaluate the expression .
Statement 1:
Statement 2:
Evaluate the expression
Statement 1:
Statement 2:
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Simplify the expression:
This is equal to
Knowing the sum or the difference of and is not enough to determine the value of this expression. But knowing both allows you to solve a system of equations:
The expression can now be evaluated by substitution.
The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.
Simplify the expression:
This is equal to
Knowing the sum or the difference of
The expression can now be evaluated by substitution.
The answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.
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Evaluate:
Statement 1:
Statement 2:
Evaluate:
Statement 1:
Statement 2:
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Therefore, you only need to know the value of to evaluate this; knowing the value of is neither necessary nor helpful.
Therefore, you only need to know the value of
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What is the value of ?
(1)
(2)
What is the value of
(1)
(2)
Tap to reveal answer
(1) Add to both sides to make . Then divide through by 7 to get
. This statement is sufficient.
(2) Divide both sides by 13. The equation becomes . This statement is sufficient.
(1) Add
(2) Divide both sides by 13. The equation becomes
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