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Data Sufficiency Procedure/Approach
Minimal Implementation
Einstein Approach requires not getting bogged down in Details, but Determining Conclusiveness of
reaching Definitive Answer for each Statement.
1. Read Question Stem. Understand and Write down the rephrased question posed. Take inventory of
any information that was implied or provided as part of the question stem.
2. Identify Question as 1) “What is or How much is or How old is” something? Or 2) “Is
something true?”
“What is” questions must be dealt with typically by predetermining your “need to know”
information, and by seeking out the pieces of information in the two statements presented to you.
“ Is something true?” questions must be dealt with on the basis of creating scenarios that will attempt
to create two Conflicting scenarios, one in which the ‘something being tested’ is true, and the other in
which it is not.
As long as the answer to the question is “Always true” or “Always false; ie: NEVER true” then you
will find data sufficiency.
If you can conjure up the two scenarios (typical strategy: pick different numbers) then there is no
sufficiency. Again, if either statement, on their own, will determine that the statement is always true or
always false, then the answer is D)
If the statements, taken together, support a finding that sometimes the stem is true, and sometimes
false, then the answer is E)
3. Work from bottom up. Ask yourself: What do I know about this concept area? For example, if
the problem defines X as a digit, ask yourself:
What do I know about digits? I know that a digit is an integer having a value in the range between 0
and 9. You need to use this knowledge along with any additional information that is provided in the
question stem or in the statements.
4. Predetermine and Write Down your Need To Know information? so you can look for and
recognize relevant information when presented to you in Each Statement. In a majority of cases, you
should be able to do this. You need to do this so that you can ????.
5. Read Statement 1. Write down Statement 1 and Take inventory of any information that was implied
or provided as part of statements 1. Play with scenarios, Don’t ONLY consider Integers UNLESS
Explicitly Stated. For example, if a statement reads that X is an integer, be sure to test scenarios
involving x = 0, x =1, and x = 2. In some situations, you may need to test scenarios involving negative
values and fractional values.
6. Seek CONFLICT across scenarios. A conflict is a sure sign that a definite answer is not possible.
Remember: CONFLICT leads to a “may be” answer, not a definite yes or no answer. Remember to
associate answer choices 1 and 4 with statement 1. Follow the procedure described for eliminating and
keeping choices after examining the statements one by one.
7. Purge Statement 1 by Covering Statement 1 with Hand. Read Statement 2. Write down Statement 2
and Take inventory of any information that was implied or provided ONLY as part of statement 2,
PLUS the information contained in the question stem.. Play with scenarios, Don’t ONLY consider
Integers UNLESS Explicitly Stated. For example, if a statement reads that X is an integer, be sure to
test scenarios involving x = 0, x =1, and x = 2. In some situations, you may need to test scenarios
involving negative values and fractional values.
8. Seek CONFLICT across scenarios. A conflict is a sure sign that a definite answer is not possible.
Remember: CONFLICT leads to a “may be” answer, not a definite yes or no answer. Remember to
associate answer choices 2, 3, and 5 with statement 2. Follow the procedure described for eliminating
and keeping choices after examining the statements one by one.
9. When there appears to be a conflict, look for commonality.
If asked to find the value of X, you may be given two quadratic equations. Typically, quadratic
equations yield two answers.
If statement 1, when solved, indicates that X = + or – 5, and
Statement 2, when solved, indicates that X has two roots, namely 11 and –5
Then, even though it appears that there are three or four different answers to the value of X, there
nonetheless is data sufficiency, because the two statements, when considered together, yield a unique
value for X, namely X = -5. Hence, the answer is C)
10. Scratch paper work
Here is how you should organize your scratch paper for data sufficiency problems
Stem facts
and Question which has to be answered
Statement # 1
Its separate Facts
| Statement # 2
| Its separate facts
This way, when you analyze statement 1, in its own right, it is easy to look vertically only at the facts
that apply. When you then examine statement 2, in its own right, it is easy to disassociate yourself
from the facts of statement # 1, by covering them up. Lastly, if neither 1 nor 2 alone are sufficient, you
need (are allowed to) to look at ALL the information. So, for the first time, you look at your ENTIRE
scratch paper
1)
2)
3)
4)
5)
6)
Problem Solving Procedure/Approach
Full Implementation
Recognize Word problem by its 5 Multiple Choice Answer Choices – usually the answer
choices are discreet numbers, like 8, 16, 24, 25, 30 – and, of course, the fact, that the problem is
rendered in prose (hence, WORD problem) rather than being given in simple equation form.
Record on scratch paper all Numbers and Quantifiable words (twice, half)
Write Question, so you identify precise value & units seeking
Identify how many variables: 1, 2, or 3
- Objective: Number of Variables should = Number of Equations to be formed
- Anchor Variables at single specific point in time (due to moving targets i.e. age changes
every year) or specific meaning of variable Q before working with altered equation(s)
Impute: Additional Necessary Facts/Formulae
Now formulate the appropriate number of necessary equations
7) System of solving 2 Simultaneous equations
- Strategy: 1) Man in Middle, 2) Addititive, 3) Substitution
Man in the Middle
I.
C + D = 195
II.
C + F = 240
III.
F = 2D
Deduction: C + F = 240 = C + 2D
Man in the Middle Equation: C + 2D = 240
A better example might deal with interior angles in a triangle. Suppose in one triangle, the angles
are called, x, 2x and y. In another triangle, the angles are called x, y and 2y.
We know that x + 2x + y =180
We also know that x + y + 2y = 180
The 180 becomes the man in the middle, so we write this as:
x + 2x + y =180= x + y + 2y
x + 2x + y = x + y + 2y
3x + y
= x + 3y
x
= y
Therefore, each of x and y = 45 degrees
Additive
3X – Y = 23
Y + X = 21
Priority: See if you can Add or Subtract to eliminate a variable
3X – Y = 23
X + Y = 21
4X
= 44
Substitution
3X + Y = 14
5X + 4Y = 28
Priority: Choose Equation with the “Easiest” numbers to isolate 1 Variable
i.e. Z = X + - Y…
3X + Y = 14 Identify Easiest equation to Isolate
5X + 4Y = 28
Y = 14 – 3X Isolated Variable Deduction
5X + 4Y = 28 Unmanipulated Equation
5X + 4[14 – 3X] = 28 Substitute Isolated Variable Deduction in other equation and Solve
Once you have the value of X determined, substitute the value of X back into the first equation and
compute Y.
Strategy: Occasionally, you will see problems where you have 3 variables but only two equations.
While this will prevent you from SOLVING for every variable in absolute terms, you probably will be
asked to express ‘t in terms on n’ – and that is possible.
If the equations involve n, m and t, then you should strategically choose to eliminate the variable m.
This simply means that you want to derive a manipulated equation in the form of m= …n…t
You then substitute this value of m into the other equation, and no more m’s will appear. Therefore,
your resulting equation involves only t’s and n’s. Simply pull all of the elements containing the
variable t over to the left, and all other terms on the right, simplify on both sides, and ultimately divide
on the left (and right) by the same factor so that the co-efficient for t becomes 1. Whichever is left on
the right (containing the variable n, and maybe some constant terms) is the solution for ‘express t in
terms on n’
Back Solve Approach
1) Start by Testing Answer Choice B. In other words, ASSUME that B is correct.
- At Most 2 Calculations
- 40% Chance of getting Correct Answer on First Trial
2) You have 20% chance that answer IN FACT is B. If the answer is too high, and it intuitively
looks as though that by LOWERING the value of the variable the calculation would result in a smaller
number, then the answer is A)
2) If First Trial is Inconclusive, and you are looking for an variable determination that would
result in a higher answer, then Test Answer Choice D. If right-on, then the correct answer id
D) – if not, then the answer must be E)
3) During Ramp-up, Also Plug in/Test Final Answer choice in Hypothetical Equation for Sanity
Check
Average/Aggregate Approach
1) Aggregate ( ∑ ) = Average * # of Values
2) Aggregate = ( ∑ ) Field Set Characteristics
3) Average * # of Values = ( ∑ ) Field Set
Boys Club Example
12 , 14 , 16 , 18 are agestof existing four members
Additional Fifth Member – Now the average is 16 Years
Question – How Old is # 5
To Compute, Need to deal in Aggregates, and Dynamic Change
Average
# of People
Aggreagate
Avg (1) = 15
4
60
Avg (2) = 16
5
80
Dynamic Change
20
Dynamic Event: 5th Member entered the club – that was the ONLY change, therefore he is SOLELY
responsible for the dynamic change – ergo his age is is 20
3 Puppy Example
2 Labradors (Identical Twins & 1 German Shepard
German Shepard increases weight 15% - the other one gain nothing – at the beginning, they had
identical weight
Average
Avg (Day 1) = 12 lbs
Avg (Day 30) = 13 lbs
# of People
3
3
Aggreagate
36
39
Dynamic Change
3
3 = 15
X
100
Period
Before
After
Labrador #1
8
8
Labrador #2
8
8
German Shepard
20
23
Symmetrical Integer Series
A typical problem might read: What is the sum of the 21 smallest positive integers
Or : Let set A be defined as all the consecutive even integers between 100 and 200, plus the 19 largest
negative integers. What is the average value of all the elements in set A?
To attempt to compute the sum of these sequence of integers would be suicidal. You don’t have the
time to do it; chances are very good that you’d make a computational error; and this is not what the
GMAT is all about. Nobody wants to test you skill at adding up large columns of numbers.
You need to employ a systemic, or sampling approach, and you need to apply the basic tenet that
whenever you deal with numbers that are equally (symmetrically) spaced out along the number line,
the middle number is also the average number.
Benignly, most sequences would involve an ODD number of integers (the principle does not only
apply to integers, but almost universally, these types of problems deal in integers). For instance, the
middle number in the sequence 1, 2, 3, 4, 5 is 3 – there are two numbers that are smaller on its left, and
two numbers that are larger, on its right.
Remembering that the [Formulaeic approach] Aggregate (of any set) is the AVERAGE value of the
elements, times the NUMBER of elements (and also, empirically, is the simple sum of all the
elements), we therefore know that in order to sum up a sequence of evenly spaced numbers (such as
consecutive integers) all we need to determine, is its median element (which in the case of an odd set
of numbers is also the average) and multiply this median value by the number of elements in the
sequence.
In the foregoing example, the median/average is 3, and there are five elements, so the aggregate = 15.
Ergo, the sum of 1 + 2 + 3 + 4 + 5 = 15. Simple, so why the formula?
Well let’s go back to the type of problems cited at the beginning of this section.
Suppose the sequence is all the even integers between 97 and 103. What is their sum?
Visually, we see that we can ‘bracket’ them, and in fact combine 99 with its symmetrical counterpart,
101, and easily compute that they add up to 200. Very conveniently, 98 and 102 ALSO add up to 200.
Thus, by bracketing, we can compute the easy calculation that we got a couple of 200’s, together with
the solitary 100 which forms the anchor around which the rest of the integers symmetrically add up.
Simple to compute the average: The aggregate is 500, and divided by 5 yields the answer: The average
of the sequence is 100, and their sum is 500.
98
99
100
101
102
200
200
∑=
500
Avg = 100
100
Avg = 100
∑ =1100
88
∑=792
Avg = 88
88
792=9*Avg
Avg =88
Once you get used to the notion of bracketing symmetrical number sequences, it is easy to transition to
the formula that Σ (a1…..a 51) = a26 x 51
Just make sure to check the symmetry of the sequence – make sure, as you designate the 26th element
as the middle or median element, that you can visualize (compute) that there is an equal amount of
numbers to the right of element # 26, and to the left of element # 26. In this case, there are 25 initial
elements, then follows element # 26 (the anchor, the median) and then there are 25 elements to the
right of the anchor. Altogether, 51 elements, an odd number, which makes the computation real easy.
Deli Line (Inclusive)
3…………376
Process:
1) Subtract Smaller number from Larger number
2) Add 1 to the result to reach total inclusive set
376
- 3
373
+ 1
374
Movie Line (Between)
8…………15
Process:
15
1) Subtract Smaller number from Larger number
-8
2) Subtract 1 from the result to reach total set in between 7
-1
6
Observations
Importance of Computing Value of 1 Unit
Smallest - Largest always moves from Left to Right. Think negative numbers
Largest Negative Number is -1.