GMAT Math : DSQ: Simplifying algebraic expressions

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Example Question #1 : Simplifying Algebraic Expressions

Evaluate the expression $(x+y)^{2}-x(x+3y)+xy$ $\displaystyle (x+y)^{2}-x(x+3y)+xy$

1) x = 3 $\displaystyle x = 3$

2) y = 2 $\displaystyle y = 2$

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.

BOTH statements TOGETHER are insufficient 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.

Correct answer:

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

Explanation:

Simplify the expression:

$(x+y)^{2}-x(x+3y)+xy$ $\displaystyle (x+y)^{2}-x(x+3y)+xy$

$= x^{2}+2xy+y^{2}-x^{2}-3xy+xy$ $\displaystyle = x^{2}+2xy+y^{2}-x^{2}-3xy+xy$

$= x^{2}-x^{2} +xy+2xy-3xy+y^{2}$ $\displaystyle = x^{2}-x^{2} +xy+2xy-3xy+y^{2}$

$= y^{2}$ $\displaystyle = y^{2}$

Therefore, we only need to know $\displaystyle y$ - If we know y = 2 $\displaystyle y = 2$, we calculate that $(x+y)^{2}-x(x+3y)+xy = y^{2} = 2^{2} = 4$ $\displaystyle (x+y)^{2}-x(x+3y)+xy = y^{2} = 2^{2} = 4$

The answer is that Statement 2 alone is sufficient to answer the question, but Statement 1 is not.

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Example Question #2 : Simplifying Algebraic Expressions

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

Statement 1: x = 5 $\displaystyle x = 5$

Statement 2: y=3 $\displaystyle y=3$

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.

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.

Correct answer:

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

Explanation:

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

Therefore, you only need to know the value of $\displaystyle y$ to evaluate this; knowing the value of $\displaystyle x$ is neither necessary nor helpful.

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Example Question #1 : Simplifying Algebraic Expressions

Evaluate the expression for positive x,y $\displaystyle x,y$:

$\frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{(2x^{2}y^{2})^{2}}$ $\displaystyle \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{(2x^{2}y^{2})^{2}}$

Statement 1: x = 3 $\displaystyle x = 3$

Statement 2: y = 4 $\displaystyle y = 4$

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.

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.

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:

$\frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{(2x^{2}y^{2})^{2}}$ $\displaystyle \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{(2x^{2}y^{2})^{2}}$

$= \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{2^{2} (x^{2})^{2}(y^{2})^{2}}$ $\displaystyle = \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{2^{2} (x^{2})^{2}(y^{2})^{2}}$

$= \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{4 x^{2\cdot 2}y^{2\cdot 2}}$ $\displaystyle = \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{4 x^{2\cdot 2}y^{2\cdot 2}}$

$= \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{4 x^{4}y^{4}}$ $\displaystyle = \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{4 x^{4}y^{4}}$

$= \frac{4x^{2}\cdot 3x^{2}y^{2}\cdot 5}{y^{-4}\cdot 8\cdot 4 x^{4}y^{4}}$ $\displaystyle = \frac{4x^{2}\cdot 3x^{2}y^{2}\cdot 5}{y^{-4}\cdot 8\cdot 4 x^{4}y^{4}}$

$= \frac{4 \cdot 3\cdot 5 \cdot x^{2}\cdot x^{2}\cdot y^{2}}{8\cdot 4 \cdot x^{4} \cdot y^{4}\cdot y^{-4}}$ $\displaystyle = \frac{4 \cdot 3\cdot 5 \cdot x^{2}\cdot x^{2}\cdot y^{2}}{8\cdot 4 \cdot x^{4} \cdot y^{4}\cdot y^{-4}}$

$= \frac{60 \cdot x^{2+2}\cdot y^{2}}{32 \cdot x^{4} \cdot y^{4-4}}$ $\displaystyle = \frac{60 \cdot x^{2+2}\cdot y^{2}}{32 \cdot x^{4} \cdot y^{4-4}}$

$= \frac{60 \cdot x^{4}\cdot y^{2}}{32 \cdot x^{4}}$ $\displaystyle = \frac{60 \cdot x^{4}\cdot y^{2}}{32 \cdot x^{4}}$

Cancel the $x^{4}$ $\displaystyle x^{4}$ from both halves:

$= \frac{60 y^{2}}{32 } = \frac{15 y^{2}}{8 }$ $\displaystyle = \frac{60 y^{2}}{32 } = \frac{15 y^{2}}{8 }$

As can be seen by the simplification, it turns out that only the value of $\displaystyle y$, which is given only in Statement 2, affects the value of the expression.

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Example Question #1 : Simplifying Algebraic Expressions

What is the value of x+y $\displaystyle x+y$?

(1) 7x=7-7y $\displaystyle 7x=7-7y$

(2) $\displaystyle 13x+13y=13$

Possible Answers:

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient.

BOTH statements TOGETHER are not sufficient to answer the question.

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

Correct answer:

EACH statement ALONE is sufficient to answer the question.

Explanation:

(1) Add $\displaystyle 7y$ to both sides to make 7x+7y=7 $\displaystyle 7x+7y=7$. Then divide through by 7 to get

x+y=1 $\displaystyle x+y=1$. This statement is sufficient.

(2) Divide both sides by 13. The equation becomes x+y=1 $\displaystyle x+y=1$. This statement is sufficient.

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Example Question #1 : Simplifying Algebraic Expressions

Is d > e $\displaystyle d > e$?

(1.) $\displaystyle (1.)$ $\displaystyle d = e + 5$

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

Possible Answers:

Statement 1 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.

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

EACH statement ALONE is sufficient to answer the question.

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 $\displaystyle e$ to make it equal to $\displaystyle d$, it follows that d>e $\displaystyle d>e$. This statement is sufficient.

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

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

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

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

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Example Question #6 : Simplifying Algebraic Expressions

True or false?

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

Statement 1: x = 10 $\displaystyle x = 10$

Statement 2: y = 7 $\displaystyle y = 7$

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.

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 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.

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

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

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

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

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

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

The inequality, therefore, is equivalent to

$4x \geq 2$ $\displaystyle 4x \geq 2$,

the truth or falsity of which depends only on the value of $\displaystyle x$

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Example Question #7 : Simplifying Algebraic Expressions

True or false?

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

Statement 1: x = 7 $\displaystyle x = 7$

Statement 2: y = 4 $\displaystyle y = 4$

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.

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.

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.

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

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

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

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

Using similar algebra, you can simplify the other expression:

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

The question, assuming the variables have nonzero values, is equivalent to asking whether 16x<12y $\displaystyle 16x< 12y$ 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 #8 : Simplifying Algebraic Expressions

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

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

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:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true.

$x^{2} + x^{3} + x^{5 }$ $\displaystyle x^{2} + x^{3} + x^{5 }$ and $x^{2} + x^{4} + x^{4 }$ $\displaystyle x^{2} + x^{4} + x^{4 }$ 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 #9 : Simplifying Algebraic Expressions

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

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

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:

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.

EITHER 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:

Both statements together are insufficient.

$2x^{2}+ 3x^{2}$ $\displaystyle 2x^{2}+ 3x^{2}$ and $2x^{2}+ 2x^{3}$ $\displaystyle 2x^{2}+ 2x^{3}$ 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 #1 : 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.

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

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.

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.

EITHER 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 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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GMAT Math : DSQ: Simplifying algebraic expressions

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