Data-Sufficiency Questions - GMAT Quantitative

To solve for three variables, you must have three equations. Statements 1 and 2 together only give two equations, so the statements together are not sufficient.

If we only look at statement 1, we might think the equation is not linear because of the term. But statement 2 tells us the is a constant. Then the equation is linear. We need both statements to answer this question.

When solving an equation with 2 variables, a second equation or the solution of 1 variable is necessary to solve.

Using statement 1 we can set up a series of equations and solve for both and .

Additionally, the information in statement 2 indicates that there is only one possible solution that satisfies the requirement that both are positive integers.

To solve with 2 unknowns, we must create a system of equations with at least 2 equations. Using statement 2 as a second equation we can easily get our answer. Solve statement 2 for x or y, and plug in for the corresponding variable in the equation given by the problem.

So, solving statement 2 for x, we get x=12+y. Replacing x in the equation from the problem, we get 4(12+y) + 7y=169. We can distribute the 4, and combine terms to find 48+11y=169. Subtract 48 from both sides, we get 11y=121. So y=11. Reusing either equation and plugging in our y value gives our x value. So x - 11=12, or x=23. This shows that x > y, and statement 1 is true. But even though it's true, it is completely unneccessary information. Therefore the answer is that we only need the information from statement 2, and statement 1 is not needed.

We will let and stand for the radii of the bases of Cylinders 1 and 2, respectively, and and stand for their heights.

The surface area of Cylinder1 can be calculated from radius and height using the formula:

;

similarly, the surface area of Cylinder 2 is

Therefore, we are seeking to determine which, if either, is greater, or .

Statement 1 alone tells us that , but without knowing anything about the heights, we cannot compare to . Similarly, Statement 2 tells us that

, or, equivalently, , but without any information about the radii, again, we cannot determine which of and is the greater.

Now assume both statements to be true. Substituting for and for , Cylinder 1 has surface area:

.

Cylinder 2 has surface area

, so , and Cylinder 1 has the greater surface area.

If the slopes of the lines are not equal, then the lines intersect at one solution; if they are equal, then they do not intersect, or the lines are the same line. Write each equation in slope-intercept form, :

The slopes of the lines are .

We need to know both and in order to determine their equality or inequality, and only if they are unequal can we answer the question.

Set and .

The slopes are unequal, so the lines intersect at one point; the system has exactly one solution.

If the slopes of the lines are not equal, then the lines intersect at one solution. If the slopes are equal, then there are two possibilties: either they do not intersect or they are the same line. Write each equation in slope-intercept form:

The slopes of these lines are .

If Statement 1 is true, then we can rewrite the first slope as , meaning that the lines have unequal slopes, and that there is only one solution. Statement 2 tells us the value of , which is irrelevant.

The equivalent question is "does have a real root?"

If you know only that is negative, you need to know whether is even or odd; negative numbers have real odd-numbered roots, but not real even-numbered roots.

If you know only that is even, you need to know whether is negative or nonnegative; negative numbers do not have real even-numbered roots, but nonnegative numbers do.

If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.

Therefore, the answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

In order to solve an equation set, one requires a number of equations equal to the number of variables. Therefore, either of the statements allow the problem to be solved.

To know the perimeter of a rectangle, we need to know length and width.

Statement (1) provides Area, which does not provide the necessary information. NOT SUFFICIENT

Statement(2) provides one side, but we still need the other to determine perimeter. NOT SUFFICIENT.

Both together are sufficent however since if we know one side and the area, we can find the remaining side using division, and then use the two side lengths to find the perimeter.

In order to solve an equation set, one requires a number of equations equal to the number of variables. Therefore, three equations are needed and both statements are required to solve the problem.

Let be the sidelength of the square. Then its perimeter is .

From Statement 1 alone, the length of the rectangle is , and, if is the width, the perimeter of the rectangle is . Therefore, we can prove that the rectangle has the greater perimeter.

From Statement 2 alone, the width of the rectangle is , and its perimeter is at least - but unless we know the length, we do not know whether the total perimeter is greater than, equal to, or less than .

Assume both statements are true. Statement 1 establishes that has two congruent legs, making it a 45-45-90 triangle. Statement 2 establishes that has a hypotenuse that has length times that of a leg, making it also a 45-45-90 triangle. The triangles have the same angle measures, so they are similar by the Angle-Angle Postulate. However, we are not given any actual lengths or any relationship between the lengths of the corresponding sides of different triangles, so we cannot determine whether the triangles are congruent or not.

The multiples of 3 between 29 and 50 are 30, 33, 36, 39, 42, 45, and 48 - therefore, has seven elements total.

The number of subsets in a set can be calculated by raising 2 to the power of the number of elements. Therefore, the answer to our question is .

To find perimeter we need length and width. We are given the length.

I) Is irrelevant.

II) We are given a way of relating area and width. Since we know that area is length times width, we can use II to set up an equation where we substitute in the known length along with the given statement to solve for our width

Solve the second one for w and you're good to go!

.

Therefore the perimeter would be:

Solve for in each statement.

Statement 1:

Statement 2:

From either statement alone, it can be deduced that .

This is a question that may initially seem to require more information to solve than it actually does, requiring you to leverage assets in order to “move up” the data sufficiency ladder. For this problem, be wary of "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient". Multiples tend to follow set patterns, so if you can find one of those patterns, you should be able to use small sets of smaller numbers to prove a rule, meaning that you need less information than you think you do.

From the information given, you know that T contains only multiples of 14. Based on your knowledge of factors and multiples, you should recognize that you can break up your numbers into factors to make them more manageable. If you are looking for multiples of 21, you should recognize that each multiple of 21 must also be a multiple of 3 and 7, since those are the prime factors of 21. Similarly, every multiple of 14 will also be a multiple of 2 and 7. Leverage what you know here: since every multiple of 14 is already a multiple of 7, that means that you are looking for how many multiples of 14 are also multiples of 3 so that you can satisfy the factors of 21 (3 and 7).

Statement (1) may at first seem insufficient since so little information is given. If there are 30 integers in the set, you should ask yourself: does the number of multiples of 3 depend more on where the set starts or how many items are in the set? You can come to a conclusion by using your printing press. Multiples of 14 start with 0 and continue:

0, 14, 28, 42, 56, 70, 84, 98… etc.

Notice that every third number (0, 42, 84, etc.) is a multiple of 3. Based on this, you should recognize that, as long as the number of terms in the set is divisible by 3, it doesn’t matter where you start. If you need to prove this to yourself, you can take 3 sets:

Set 1: 0, 14, 28 Set 2: 14, 28, 42 Set 3: 28, 42, 56

You can see that, although you start at different points in the pattern, because each set has three consecutive terms, you are guaranteed to have a single multiple of 3 in each set (remember that 0 is a multiple of all numbers). If you extrapolate from here, you should see that a set of 30 numbers would have ten times as many multiples of 3, or 10 multiples of 3 (and therefore of 21).

Statement (1) is therefore sufficient; eliminate "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked", "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient", and "Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed". Statement 2 gives only a starting point for the Set and no end point. Thus, the Set could have 1 multiple of 21 or an infinite number of multiples of 21. There is no way of telling. Therefore (2) is insufficient, eliminating answer choice "EACH statement ALONE is sufficient to answer the question asked".

Notice that if you hadn’t done the work to leverage statement (1) you may have concluded that you needed to know the starting point of the set in order to come to a conclusion and might have chosen "Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient". Remember that, especially for harder questions like this one, to be leery of choosing any “easy” answer – generally the correct answer is going to require you to put in some work to make your answer choices work.

Choice "Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked" is correct.

This problem heavily rewards those who "play the game" of Data Sufficiency effectively. While the two statements should look just about identical, those who Play Devil's Advocate and/or ask "Why Are You Here?" can spot the ever-important difference and avoid the trap answer.

Your inclination on both statements should be to use algebraic mirroring to factor coefficients and arrive at the expression j + k on the left hand side. For statement 1 that's:

m(j + k) = 2m

And for statement 2 that's:

5(j + k) = 10

Note that in statement 2, you can simply divide both sides by 5 and arrive at j + k = 2, making statement 2 sufficient.

Most people try to do the same thing on statement 1, dividing both sides by m. But you cannot do that! Why? Because m could equal 0, and you cannot divide by 0. You can demonstrate that by setting m equal to 0, in which case statement 1 would be:

0j + 0k = 0(2), in which case j and k could be absolutely anything.

So statement 1 is insufficient and the correct answer is "Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked". And the lesson: you can avoid that trap (note that most examinees choose "EACH statement ALONE is sufficient to answer the question asked.") by:

Playing Devil's Advocate - when a statement seems a little too easy, ask yourself whether negative numbers, fractions, zero, or any other "edge cases" might give a different answer.

Asking "Why Are You Here?" - when one statement is extremely easy (as statement 2 is here), that's a signal that the more-nuanced statement likely has some difficulty to it, and that the easy statement might provide a clue. The difference between the statements here is statement 1 uses a variable where statement 2 uses the coefficient 5. Why would that distinction matter? Because if you can't rule out 0 as a value of a variable, you can't divide by it.