Practicing Data Sufficiency Flashcards
Is √x an integer?
(1) x/6 is an odd integer.
(2) x2 is a multiple of 16.
Correct.
Stat.(1): if x/6 is an integer, then 6 is a factor of x. x can be represented as x=6*(some odd integer). Try plugging in x, which can be 6, 18, 30, 42. It doesn’t look like x can be a perfect square. Under test conditions, this would be a good point to stop if you’re pushed for time and decide that √x can never be an integer and Stat.(1)->No->S->AD.
If you could spare a moment, you could try and crack the question algebraically. x can also be represented as x=2·3·(some odd number). You know that the odd number cannot have 2 as a prime factor (it’s odd), so x only has one 2 as a prime factor, no more. If x only has one 2 as a prime factor, then x cannot be a perfect square - the prime factors of a square of an integer must come in pairs. √x can never be an integer.
Stat.(2): plug in x²=16, so x=4 and √x is an integer, 2. Plug in x²=32, so x=√32 and √x is not an integer, √(√32). Hence, whether √x is an integer cannot be determined. Stat.(2)->IS->A.
If a, b and c are integers, and a+b=c, is a=b?
(1) c is a multiple of a.
(2) c is a multiple of b.
Stat.(1)->Maybe->IS->BCE.
According to Stat. (2),
it can be that a=10, b=5, c=15, or that a=5, b=5, c=10.
Stat.(2)->Maybe->IS->CE.
According to Stat. (1+2),
c is a multiple of both a and b. Now that is tricky. There is no problem to show that a=b, for example a=5, b=5, c=10, but don’t fall into the trap - think DOZEN F!
You need to find a special case where c is the both the sum of a and b and a multiple of both of them. How is that possible? What if c=0, and a and b are positive and negative numbers which cancel each other out? for example:
–> a=-1, b=1, c=0
Thus you have found a case where a and b are not equal and the answer is “No”.
Stat.(1+2)->Maybe->IS->E.
Data Sufficiency: Plugging into Yes/No Data Sufficiency
The basic steps for Yes/No Data Sufficiency Plugging In:
Figure out the issue before you dive deeper into the statements. (The issue in Yes/No Data Sufficiency means which number(s) yield a Yes or a No). The issue is also an important step in solving Must Be questions.
Regard the statements as facts that cannot be broken. Only then should you plug in the numbers you used in the question stem.
Ask yourself every time you get a Yes or No: “Is it always true, for any number?” This is completely equivalent to Must Be questions.
Remember: only a definite Yes/No is Sufficient.
Remember the DOZEN F:
Different (e.g. even vs. odds, prime vs. multiple, fractions vs. integers, positive vs. negative, etc.)
One
Zero
Equal numbers for different variables
Negatives
Fractions
DS: Yes/No Basic Technique
Answering a definite “Yes” or a definite “No” means Sufficient.
If the answer is sometimes “Yes” and sometimes “No”, it means Maybe, which means Insufficient. The only way a statement will be insufficient is if it allows both a “Yes” and a “No” answer.
After that, follow the Data Sufficiency FlowChart to get the final answer.
Data Sufficiency: The Question Stem - What is the Type?
When solving a DS question you must first extract all possible data from the question stem:
Type: Determine whether it is a value question or a yes/no question.
Issue: Figure out the GMAT knowledge field that is tested (e.g averages, quadrilateral area formulas etc.)
Missing piece: While focusing on the question stem, determine what is the piece of information that’s needed to answer the question asked.
Always try to figure out the issue of the stem before you approach the statements. This way, you’ll be more focused on what you are looking for, thereby determining whether the statement is Sufficient or Insufficient more efficiently.
Data Sufficiency: Basic Work Order
Always begin by reading the question stem and try to figure out the issue of the problem (i.e which data is required to answer the question). Read statement (1) alone and decide sufficient/insufficient. POE like so: If statement (1) is sufficient POE BCE, hence statement (1)-\>S-\>AD If statement (1) is insufficient POE AD, hence statement (1)-\>IS-\>BCE Read statement (2) alone and decide sufficient/insufficient. Don't forget to POE as you go along. Make sure you don't use words such as "yes" or "no" for they may add to the confusion that is already there.
If Polygon X has fewer than 9 sides, how many sides does Polygon X have?
(1) The sum of the interior angles of Polygon X is divisible by 16.
(2) The sum of the interior angles of Polygon X is divisible by 15.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
The relationship between the number of sides in a polygon and the sum of the interior angles is given by 180(n – 2) = (sum of interior angles), where n is the number of sides. Thus, if we know the sum of the interior angles, we can determine the number of sides. We can rephrase the question as follows: “What is the sum of the interior angles of Polygon X?”
(1) SUFFICIENT: Using the relationship 180(n – 2) = (sum of interior angles), we could calculate the sum of the interior angles for all the polygons that have fewer than 9 sides. Just the first two are shown below; it would take too long to calculate all of the possibilities:
polygon with 3 sides: 180(3 – 2) or 180 × 1 = 180
polygon with 4 sides: 180(4 – 2) or 180 × 2 = 360
Notice that each interior angle sum is a multiple of 180. Statement 1 tells us that the sum of the interior angles is divisible by 16. We can see from the above that each possible sum will consist of 180 multiplied by some integer.
The prime factorization of 180 is (2 × 2 × 3 × 3 × 5). The prime factorization of 16 is (2 × 2 × 2 × 2). Therefore, two of the 2’s that make up 16 can come from the 180, but the other two 2’s will have to come from the integer that is multiplied by 180. Therefore, the difference (n – 2) must be a multiple of 2 × 2, or 4. Our possibilities for (n – 2) are:
3 sides: 180(3 – 2) or 180 × 1
4 sides: 180(4 – 2) or 180 × 2
5 sides: 180(5 – 2) or 180 × 3
6 sides: 180(6 – 2) or 180 × 4
7 sides: 180(7 – 2) or 180 × 5
8 sides: 180(8 – 2) or 180 × 6
Only the polygon with 6 sides has a difference (n – 2) that is a multiple of 4.
(2) INSUFFICIENT: Statement (2) tells us that the sum of the interior angles of Polygon X is divisible by 15. Therefore, the prime factorization of the sum of the interior angles will include 3 × 5. Following the same procedure as above, we realize that both 3 and 5 are included in the prime factorization of 180. As a result, every one of the possibilities can be divided by 15 regardless of the number of sides.
The correct answer is A.
In 2003 Acme Computer’s price for each of its computers was five times the price for each of its printers. What was the ratio of its gross revenue from computers to its gross revenue from printers in 2003?
(1) In the first half of 2003, Acme sold computers and printers in a ratio of 3:2; in the second half of 2003, Acme sold computers and printers in the ratio of 2:1.
(2) Acme’s 2003 price for each of its computers was $1,000.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
The question asks us to find the ratio of gross revenue from computers to that from printers, given that the price of a computer is five times the price of a printer.
(1) INSUFFICIENT: Statement (1) says that the ratio of computers to printers sold in the first half of 2003 was 3 to 2; one set of values satisfying this statement is 3 computers and 2 printers. Using example prices of $5 per computer and $1 per printer, we arrive at a gross revenue of $15 from computers and $2 from printers.
During the second half of 2003, the ratio of computers to printers sold was 2 to 1; one set of values satisfying this statement is 2 computers and 1 printer, grossing $10 and $1 respectively if we use the sample prices chosen above. With these numbers, then, the full-year gross revenues are $25 from computers and $3 from printers.
Alternatively, for the second half of 2003 Acme may have sold 4 computers and 2 printers, still in the required ratio of 2 to 1. In this case, Acme would have grossed $20 and $2 from computers and printers, respectively; with these numbers, then, the full-year gross revenues are $35 from computers and $4 from printers, yielding a different overall ratio. Therefore, statement (1) is insufficient to give us a definitive answer.
(2) INSUFFICIENT: Statement (2) tells us that one of Acme’s computers cost $1,000 in 2003, but it tells us nothing about the ratio or numbers of computers and printers sold.
(1) AND (2) INSUFFICIENT: Statement (2) fixes the price of a computer at $1,000, but, if we inflate the sample prices from $5 and $1 to $1,000 and $200, both of the solutions given in the explanation of statement (1) will still be possible. (The arithmetic will be identical; the prices will be 200 times as great.) Therefore, statements (1) and (2) together are still insufficient.
The correct answer is E.
The contents of one full cylindrical silo are to be transferred to another, larger cylindrical silo. The contents of the smaller silo will fill what portion of the larger silo?
(1) The larger silo has twice the base radius, and twice the height, of the smaller one.
(2) The smaller silo has a circular base with a radius of 10 feet.
The volume for a cylinder can be calculated by multiplying the area of the base times the height. The base is a circle with an area of r2, where r is the radius of the circle. Thus the volume of a cylinder is r2 × h, where h represents the height of the figure.
(1) SUFFICIENT: If we call the radius of the smaller cylinder r, and the height of the smaller cylinder h, the volume of the smaller cylinder would be r2h. If the radius of the larger cylinder is twice that of the smaller one, as is the height, the volume of the larger cylinder would be (2r)2(2h) = 8r2h. The volume of the larger cylinder is eight times larger than that of the smaller one. If the contents of the smaller silo, which is full, are poured into the larger one, the larger one will be 1/8 full.
(2) INSUFFICENT: This question is about proportions, and this statement tells us nothing about volume of the smaller silo relative to the larger one.
The correct answer is A.
If m, n, and p are integers, is m + n odd?
(1) m = p2 + 4p + 4
(2) n = p2 + 2m + 1
(1) INSUFFICIENT: Given that m = p2 + 4p + 4,
If p is even:
m = (even)2 + 4(even) + 4
m = even + even + even
m = even
If p is odd:
m = (odd)2 + 4(odd) + 4
m = odd + even + even
m = odd
Thus we don’t know whether m is even or odd. Additionally, we know nothing about n.
(2) INSUFFICIENT: Given that n = p2 + 2m + 1
If p is even:
n = (even)2 + 2(even or odd) + 1
n = even + even + odd
n = odd
If p is odd:
n = (odd)2 + 2(even or odd) + 1
n = odd + even + odd
n = even
Thus we don’t know whether n is even or odd. Additionally, we know nothing about m.
(1) AND (2) SUFFICIENT: If p is even, then m will be even and n will be odd. If p is odd, then m will be odd and n will be even. In either scenario, m + n will be odd.
The correct answer is C.
Is x% of x% of y equal to x% less than y ?
(1) x(x + 100) = 10,000
(2) y(y + 1) = 1
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What is the distance between Harry’s home and his office?
(1) Harry’s average speed on his commute to work this Monday was 30 miles per hour.
(2) If Harry’s average speed on his commute to work this Monday had been twice as fast, his trip would have been 15 minutes shorter.
Distance = Rate × Time, or D = RT.
(1) INSUFFICIENT: This statement tells us Harry’s rate, 30 mph. This is not enough to calculate the distance from his home to his office, since we don’t know anything about the time required for his commute.
D = RT = (30 mph) (T)
D cannot be calculated because T is unknown.
(2) INSUFFICIENT: If Harry had traveled twice as fast, he would have gotten to work in half the time, which according to this statement would have saved him 15 minutes. Therefore, his actual commute took 30 minutes. So we learn his commute time from this statement, but don’t know anything about his actual speed.
D = RT = (R) (1/2 hour)
D cannot be calculated because R is unknown.
(1) AND (2) SUFFICIENT: From statement (1) we learned that Harry’s rate was 30 mph. From Statement (2) we learned that Harry’s commute time was 30 minutes. Therefore, we can use the rate formula to determine the distance Harry traveled.
D = RT = (30 mph) (1/2 hour) = 15 miles
The correct answer is C.
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