Equations - GMAT Quantitative

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Question

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x, x^{2}, x^{3}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right |>1$

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Answer

Assume Statement 1 alone. If is a negative number, then of the three given powers of in the set, only $x^{2}$ (the only even power) is positive. This makes $x^{2}$ the greatest of the three, thereby giving an answer to the question.

We show that Statement 2 alone is inconclusive by examining two values of whose absolute values are greater than 1 - namely, $\left \{-2, 2 \right \}$ .

Case 1: . Then $x^{2}= 4$ and $x^{3} = 8$ , making $x^{3}$ the greatest number of the three.

Case 2: . Then $x^{2}= 4$ and $x^{3} = -8$ , making $x^{2}$ the greatest number of the three.

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Question

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x, x^{2}, x^{3}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right |>1$

Tap to reveal answer

Answer

Assume Statement 1 alone. If is a negative number, then of the three given powers of in the set, only $x^{2}$ (the only even power) is positive. This makes $x^{2}$ the greatest of the three, thereby giving an answer to the question.

We show that Statement 2 alone is inconclusive by examining two values of whose absolute values are greater than 1 - namely, $\left \{-2, 2 \right \}$ .

Case 1: . Then $x^{2}= 4$ and $x^{3} = 8$ , making $x^{3}$ the greatest number of the three.

Case 2: . Then $x^{2}= 4$ and $x^{3} = -8$ , making $x^{2}$ the greatest number of the three.

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Question

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x, x^{2}, x^{3}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right |>1$

Tap to reveal answer

Answer

Assume Statement 1 alone. If is a negative number, then of the three given powers of in the set, only $x^{2}$ (the only even power) is positive. This makes $x^{2}$ the greatest of the three, thereby giving an answer to the question.

We show that Statement 2 alone is inconclusive by examining two values of whose absolute values are greater than 1 - namely, $\left \{-2, 2 \right \}$ .

Case 1: . Then $x^{2}= 4$ and $x^{3} = 8$ , making $x^{3}$ the greatest number of the three.

Case 2: . Then $x^{2}= 4$ and $x^{3} = -8$ , making $x^{2}$ the greatest number of the three.

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Question

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x, x^{2}, x^{3}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right |>1$

Tap to reveal answer

Answer

Assume Statement 1 alone. If is a negative number, then of the three given powers of in the set, only $x^{2}$ (the only even power) is positive. This makes $x^{2}$ the greatest of the three, thereby giving an answer to the question.

We show that Statement 2 alone is inconclusive by examining two values of whose absolute values are greater than 1 - namely, $\left \{-2, 2 \right \}$ .

Case 1: . Then $x^{2}= 4$ and $x^{3} = 8$ , making $x^{3}$ the greatest number of the three.

Case 2: . Then $x^{2}= 4$ and $x^{3} = -8$ , making $x^{2}$ the greatest number of the three.

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Question

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x, x^{2}, x^{3}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right |>1$

Tap to reveal answer

Answer

Assume Statement 1 alone. If is a negative number, then of the three given powers of in the set, only $x^{2}$ (the only even power) is positive. This makes $x^{2}$ the greatest of the three, thereby giving an answer to the question.

We show that Statement 2 alone is inconclusive by examining two values of whose absolute values are greater than 1 - namely, $\left \{-2, 2 \right \}$ .

Case 1: . Then $x^{2}= 4$ and $x^{3} = 8$ , making $x^{3}$ the greatest number of the three.

Case 2: . Then $x^{2}= 4$ and $x^{3} = -8$ , making $x^{2}$ the greatest number of the three.

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Question

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x, x^{2}, x^{3}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right |>1$

Tap to reveal answer

Answer

Assume Statement 1 alone. If is a negative number, then of the three given powers of in the set, only $x^{2}$ (the only even power) is positive. This makes $x^{2}$ the greatest of the three, thereby giving an answer to the question.

We show that Statement 2 alone is inconclusive by examining two values of whose absolute values are greater than 1 - namely, $\left \{-2, 2 \right \}$ .

Case 1: . Then $x^{2}= 4$ and $x^{3} = 8$ , making $x^{3}$ the greatest number of the three.

Case 2: . Then $x^{2}= 4$ and $x^{3} = -8$ , making $x^{2}$ the greatest number of the three.

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Question

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x, x^{2}, x^{3}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right |>1$

Tap to reveal answer

Answer

Assume Statement 1 alone. If is a negative number, then of the three given powers of in the set, only $x^{2}$ (the only even power) is positive. This makes $x^{2}$ the greatest of the three, thereby giving an answer to the question.

We show that Statement 2 alone is inconclusive by examining two values of whose absolute values are greater than 1 - namely, $\left \{-2, 2 \right \}$ .

Case 1: . Then $x^{2}= 4$ and $x^{3} = 8$ , making $x^{3}$ the greatest number of the three.

Case 2: . Then $x^{2}= 4$ and $x^{3} = -8$ , making $x^{2}$ the greatest number of the three.

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Question

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x, x^{2}, x^{3}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right |>1$

Tap to reveal answer

Answer

Assume Statement 1 alone. If is a negative number, then of the three given powers of in the set, only $x^{2}$ (the only even power) is positive. This makes $x^{2}$ the greatest of the three, thereby giving an answer to the question.

We show that Statement 2 alone is inconclusive by examining two values of whose absolute values are greater than 1 - namely, $\left \{-2, 2 \right \}$ .

Case 1: . Then $x^{2}= 4$ and $x^{3} = 8$ , making $x^{3}$ the greatest number of the three.

Case 2: . Then $x^{2}= 4$ and $x^{3} = -8$ , making $x^{2}$ the greatest number of the three.

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Question

Solve for .

Statement 1:

Statement 2:

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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 $x,y,\ and\ z$ .

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Question

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

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

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

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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

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Question

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

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

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

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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.

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Question

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

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

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

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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 = $\dpi{100} \small 2\times 4=8$

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Question

Data Sufficiency Question- do not actually solve the problem

Solve for .

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

1. $\small x=7$

2. $\small y=2$

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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.

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Question

Solve for $x,y,and\ z.$

Statement 1:

Statement 2:

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Answer

To solve for three unknowns, we need three equations. Here we have three equations if we use both statements 1 and 2. We don't need to solve any further. Because this is a data sufficiency question, it doesn't matter what the actual values of x, y, and z are. The important fact is the we could find them if we wanted to.

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Question

The volume of a fixed mass of gas varies inversely with the atmospheric pressure, in millibars, acting upon it, given that all other conditions remain constant.

At 12:00, a balloon was filled with exactly 100 cubic yards of helium. What its current volume?

Statement 1: The atmospheric pressure at 12:00 was 109 millibars.

Statement 2: The atmospheric pressure is now 105 millibars.

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Answer

You can use the following variation equation to deduce the current volume:

$V_{1} P _{1} = V_{2} P _{2}$

or, equivalently,

$V_{2} = \frac{V_{1} P _{1} }{P _{2}}$

To find the current volume $V_{2}$ , you therefore need three things - the initial volume $V_{1}$ , which is given in the body of the question; the initial pressure $P _{1}$ , which you know if you are given Statement 1; and the current pressure, $P _{2}$ , which you know if you are given Statement 2. Just substitute, and solve.

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Question

The volume of a fixed mass of gas varies inversely with the atmospheric pressure, in millibars, acting upon it, given that all other conditions remain constant.

At 12:00, a balloon was filled with exactly 100 cubic yards of helium. What is its current volume?

Statement 1: It is now 2:00.

Statement 2: The atmospheric pressure is now 105 millibars.

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Answer

The first statement is unhelpful; the time of day is irrelevant to the question.

You can use the following variation equation to deduce the current volume:

$V_{1} P _{1} = V_{2} P _{2}$

or, equivalently,

$V_{2} = \frac{V_{1} P _{1} }{P _{2}}$

To find the current volume $V_{2}$ , you therefore need three things - the initial volume $V_{1}$ , which is given in the body of the question; the current pressure $P_{2}$ , which you know if you use Statement 2, and the initial pressure $P_{1}$ , which is not given anywhere.

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Question

The electrical current through an object in amperes varies inversely as the object's resistance in ohms, given that all other conditions are equal.

Four batteries hooked up together run an electrical current of 3.2 amperes through John's flashlight. How much current would the same batteries run through John's radio?

Statement 1: The radio has resistance 20 ohms.

Statement 2: The flashlight has resistance 15 ohms.

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Answer

Let $I_{1},R_{1}$ be the current through and the resistance of the flashlight; let $I_{2},R_{2}$ be the current through and the resistance of the radio.

The variation equation here would be:

$\frac{I_{1}}{R_{1}} = \frac{I_{2}}{R_{2}}$

or equivalently:

$\frac{I_{1}}{R_{1}} \cdot R_{2}= \frac{I_{2}}{R_{2}} \cdot R_{2}$

$I_{2} = \frac{I_{1}R_{2}}{R_{1}}$

So in order to find the current in the radio, you need to know three things - the current $I_{1}$ in the flashlight, which you know from the body of the problem; the resistance from the flashlight $R_{1}$ , which you know if you are given Statement 2; and the resistance from the radio $R_{2}$ , which you know if you are given Statement 1. Just substitute and solve.

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Question

Evaluate:

$\log_{x} 100y - \log_{x} y$

Statement 1:

Statement 2:

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Answer

The difference of two logarithms with the same base is the same-base logarithm of the quotient of the numbers. Therefore, we can simplify this expression as

$\log_{x} 100y - \log_{x} y =\log_{x} \frac{100y}{y} = \log_{x} 100$

We need only know the value of , given to us in Statement 1, in order to evaluate this expression.

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Question

is a positive integer

True or false?

Statement 1:

Statement 2: is even.

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Answer

By the zero product principle, we can solve by setting each linear binomial to zero and solving. This yields three solutions:

$y - 2 = 0 \Rightarrow y = 2$

$y - 4 = 0 \Rightarrow y =4$

$y - 6= 0 \Rightarrow y = 6$

Neither statement alone narrows to one of these three solutions, but the two together do.

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Question

Solve for :

$\left | 4x-3\right |-17=y^{2}-z^{2}$

(1)

(2)

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Answer

Solution

To solve for x, we need the value of $y^{2}-z^{2}$

$y^{2}-z^{2}=(y-z)(y+z)$

Therefore, we need both statements in order to solve for x.

$\left | 4x-3\right |-17= 4\times12$

$\left | 4x-3\right |-17=48$

$\left | 4x-3\right |=65$

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Or

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