GMAT Math : Algebra

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Example Questions

Example Question #894 : Data Sufficiency Questions

Is d > e ?

(1.) d = e + 5

(2.) $\frac{d}{6} = e - 2$

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

EACH statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

BOTH statements TOGETHER are not sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

Explanation:

(1) Since 5 must be added to to make it equal to , it follows that d>e . This statement is sufficient.

(2) Multiply both sides by 6 to obtain d = 6e - 12 . Thus, whether d > e or

d < e depends on the value selected for and . For instance, e = 0 implies

d = -12 , (such that e > d ) but e = 20 implies d = 108 ,

(such that e < d ). Therefore, this statement is insufficient.

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Example Question #1 : Dsq: Simplifying Algebraic Expressions

True or false?

$\frac{(x + y)^{2} - (x - y)^{2} }{y} \geq \frac{(x + y)^{2} + (x - y)^{2} }{x^{2}+y^{2}}$

Statement 1: x = 10

Statement 2: y = 7

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

Simplify each expression.

$\frac{(x + y)^{2} - (x - y)^{2} }{y}$

$= \frac{(x^{2}+2xy + y^{2}) - (x^{2}-2xy + y^{2}) }{y}$

$= \frac{4xy }{y} = 4x$

$\frac{(x + y)^{2} + (x - y)^{2} }{x^{2}+y^{2}}$

$= \frac{(x^{2}+2xy + y^{2}) + (x^{2}-2xy + y^{2}) }{x^{2}+y^{2}}$

$= \frac{2x^{2} + 2y^{2}}{x^{2}+y^{2}} = \frac{2 \left ( x^{2} + y^{2} \right ) }{x^{2}+y^{2}} = 2$

The inequality, therefore, is equivalent to

$4x \geq 2$ ,

the truth or falsity of which depends only on the value of

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Example Question #2 : Dsq: Simplifying Algebraic Expressions

True or false?

$\frac{(x + 4y)^{2} - (x - 4y)^{2} }{y} < \frac{(x + 3y)^{2} - (x - 3y)^{2} }{x}$

Statement 1: x = 7

Statement 2: y = 4

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Simplify both expressions algebraically.

$\frac{(x + 4y)^{2} - (x - 4y)^{2} }{y}$

$=\frac{(x^{2} + 2\cdot x\cdot 4y+ (4y)^{2} ) - (x^{2} - 2\cdot x\cdot 4y+ (4y)^{2} ) }{y}$

$=\frac{(x^{2} + 8 xy+ 16y^{2} ) - (x^{2} -8 xy+ 16y^{2} ) }{y}$

$=\frac{16xy }{y} =16x$

Using similar algebra, you can simplify the other expression:

$\frac{(x + 3y)^{2} - (x - 3y)^{2} }{x} = 12y$

The question, assuming the variables have nonzero values, is equivalent to asking whether 16x<12y is true. Since we need to know the values of both variables to answer this, both statements are necessary and sufficient.

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Example Question #1 : Simplifying Algebraic Expressions

The figure below shows a trinomial with its exponents replaced by shapes:

$x^{\bigcirc } + x^{\square} + x^{\bigtriangleup }$

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.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true.

$x^{2} + x^{3} + x^{5 }$ and $x^{2} + x^{4} + x^{4 }$ 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.

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Example Question #3 : Dsq: Simplifying Algebraic Expressions

The figure below shows a binomial with its coefficients and exponents replaced by shapes:

$\square x^{\bigcirc } + \lozenge x ^{ \bigtriangleup }$

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.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Both statements together are insufficient.

$2x^{2}+ 3x^{2}$ and $2x^{2}+ 2x^{3}$ 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.

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Example Question #4 : Dsq: Simplifying Algebraic Expressions

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.

$\square x^{\bigcirc }$

Did Stephanie succeed?

Statement 1: Stephanie wrote a 5 in the square.

Statement 2: Stephanie wrote a 7 in the circle.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

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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Example Question #11 : Dsq: Simplifying Algebraic Expressions

Evaluate the expression $(x + y)^{2} - (x-y)^{2}$

Statement 1: x = 9

Statement 2: y = 5

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

We simplify to make sure of whether we need to know one or both of and :

$(x + y)^{2} - (x-y)^{2}$

$= \left (x ^{2}+ 2xy + y^{2} \right ) -\left ( x ^{2}-2xy + y^{2} \right )$

$= x ^{2}+ 2xy + y^{2} - x ^{2}+2xy - y^{2} \right )$

= 4xy

Indeed, the values of both and must be known to evaluate this expresson.

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Example Question #106 : Algebra

Carleton's teacher challenged him to fill in the circle and the square below to form a polynomial with degree .

$x^{\bigcirc } + x^{\square }$

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.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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 $x^{7}+ x^{2}$ and the degree 6 polynomial $x^{6}+ x^{3}$ 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 $x^{2}+ x^{7}$ .

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Example Question #12 : Dsq: Simplifying Algebraic Expressions

Michelle was challenged by her teacher to create a monomial of degree by filling in the square and the circle in the figure below.

$x^{\square }y^{\bigcirc}$

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.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

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, $x^{2}y^{3}$ , which has degree 5, or $x^{3}y^{3}$ , which has degree .

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Example Question #904 : Data Sufficiency Questions

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.

$\square x^{\bigcirc } + \lozenge x ^{ \bigtriangleup }$

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.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

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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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #894 : Data Sufficiency Questions

Example Question #1 : Dsq: Simplifying Algebraic Expressions

Example Question #2 : Dsq: Simplifying Algebraic Expressions

Example Question #1 : Simplifying Algebraic Expressions

Example Question #3 : Dsq: Simplifying Algebraic Expressions

Example Question #4 : Dsq: Simplifying Algebraic Expressions

Example Question #11 : Dsq: Simplifying Algebraic Expressions

Example Question #106 : Algebra

Example Question #12 : Dsq: Simplifying Algebraic Expressions

Example Question #904 : Data Sufficiency Questions

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