Algebra - GMAT Quantitative Reasoning

Card 0 of 20

Question

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

Answer

Simplify the expression:

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

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

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

$= y^{2}$

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

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

Compare your answer with the correct one above

Question

Evaluate the expression $(x-2y)^{2}+8xy$ .

Statement 1:

Statement 2:

Answer

Simplify the expression:

$(x-2y)^{2}+8xy = x^{2}-4xy+4y^{2} +8xy = x^{2}+4xy+4y^{2}$

This is equal to $(x+2y)^{2}$

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.

Compare your answer with the correct one above

Question

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

Statement 1:

Statement 2:

Answer

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

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

Compare your answer with the correct one above

Question

What is the value of ?

(1)

(2)

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.

Compare your answer with the correct one above

Question

Is ?

$\frac{d}{6} = e - 2$

Answer

(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.

Compare your answer with the correct one above

Question

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

Statement 1:

Statement 2:

Answer

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

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

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

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

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

Compare your answer with the correct one above

Question

Evaluate the expression for positive :

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

Statement 1:

Statement 2:

Answer

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

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

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

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

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

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

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

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

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

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

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

Compare your answer with the correct one above

Question

True or false?

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

Statement 1:

Statement 2:

Answer

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 is true. Since we need to know the values of both variables to answer this, both statements are necessary and sufficient.

Compare your answer with the correct one above

Question

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:

Statement 2:

Answer

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

Compare your answer with the correct one above

Question

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.

Answer

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.

Compare your answer with the correct one above

Question

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.

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

Compare your answer with the correct one above

Question

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.

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 $x^{7}+ x^{2}$ 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 $x^{2}+ x^{7}$ .

Compare your answer with the correct one above

Question

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.

Answer

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.

Compare your answer with the correct one above

Question

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.

Answer

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.

Compare your answer with the correct one above

Question

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.

Answer

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.

Compare your answer with the correct one above

Question

Solve for .

Statement 1:

Statement 2:

Answer

To solve for three unknowns, we need three equations. Therefore no combination of statements 1 and 2 will provide enough information to solve for $x,y,\ and\ z$ .

Compare your answer with the correct one above

Question

If $\dpi{100} \small z+x=y$ , what is the value of $\dpi{100} \small x$ ?

(1) $\dpi{100} \small z=7$

(2) $\dpi{100} \small y+7=z$

Answer

$\dpi{100} \small z+x=y$

Therefore, $\dpi{100} \small x=y-z$

(1) If $\dpi{100} \small z=7$ , then $\dpi{100} \small y-z=y-7$ , and the value of $\dpi{100} \small y-7$ can vary. $\dpi{100} \small \rightarrow$ NOT sufficient

(2) Subtracting both $\dpi{100} \small z$ and 7 from each side of $\dpi{100} \small y+7=z$ gives $\dpi{100} \small y-z=-7$ .

The value of $\dpi{100} \small y-z$ can be determined. $\dpi{100} \small \rightarrow$ SUFFICIENT

Compare your answer with the correct one above

Question

If $\dpi{100} \small x> 0$ , what is the value of x?

Statement 1: $\dpi{100} \small x> 1$

Statement 2: $\dpi{100} \small x^{2}-4=0$

Answer

We are looking for one value of x since the quesiton specifies we only want a positive solution.

Statement 1 isn't sufficient because there are an infinite number of integers greater than 1.

Statement 2 tells us that x = 2 or x = –2, and we know that we only want the positive answer. Then Statement 2 is sufficient.

Compare your answer with the correct one above

Question

What is the value of $\dpi{100} \small 2x+2y$ ?

Statement 1: $\dpi{100} \small x-3y=4$

Statement 2: $\dpi{100} \small x+y=4$

Answer

We know that we need 2 equations to solve for 2 variables, so it is tempting to say that both statements are needed. This is actually wrong! We aren't being asked for the individual values of x and y, instead we are being asked for the value of an expression.

$\dpi{100} \small 2x+2y$ is just $\dpi{100} \small 2\left ( x+y \right )$ , and statement 2 gives us the value of $\dpi{100} \small x+y$ . For data sufficiency questions, we don't actually have to solve the question, but if we wanted to, we would simply multiply statement 2 by 2.

$\dpi{100} \small 2x+2y=2\left ( x+y \right )=2$ * Statement 2 = $\dpi{100} \small 2\times 4=8$

Compare your answer with the correct one above

Question

Data Sufficiency Question- do not actually solve the problem

Solve for .

$\small 8vxy+9xz+10vyz=14$

1. $\small x=7$

2. $\small y=2$

Answer

In order to solve an equation with 4 variables, you need to know either 3 of the variables or have a system of 4 equations to solve.

Compare your answer with the correct one above

Tap the card to reveal the answer