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Backsolving

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Backsolving

by nkaur » Mon Apr 18, 2011 4:44 am
Hi Guys,

Can someone please shorty explain how "Backsolving" works? I read about it in certain books but still did not understand the concept completely.

When can this concept be used? Is this a particular type of questions? And is it true that when using backsolving, one should start favoring either Answer B or D?

Thanks
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by manpsingh87 » Mon Apr 18, 2011 5:57 am
nkaur wrote:Hi Guys,

Can someone please shorty explain how "Backsolving" works? I read about it in certain books but still did not understand the concept completely.

When can this concept be used? Is this a particular type of questions? And is it true that when using backsolving, one should start favoring either Answer B or D?

Thanks
Backsolving is a technique of trying out the answer choices in GMAT questions. Each choice is selected one at a time and is assumed to be the right choice. Analyze the value or conditions by injecting the choices into the question. Backsolving is incredibly time saving in algebraic equations and number properties problems.

Case 1: To find the value of an equation
Example: Which number gives the lowest value for the equation?
1/X + (1/X) ^2 + 5X

Approach 1: Start from the middle value in the answer choice.
For example, if the five answer choices are 14,5,19,8,21

Plug 14 into the equation and check the value. If the value is too high, find out an answer choice with a much smaller value, like 5.

Approach 2: If you inspect the equation, the first two terms will result in small numbers, smaller than 1, according to the given answer choices. However, the 3rd term of the equation is the leading one. A small value in 5X will make the equation's net value smaller, and a big value in 5X will make the equation's net value bigger. So, the answer is almost immediately x = 5.


Case 2: To satisfy a condition
Example: If y # 3 and 2x/y is a prime integer greater than 2, which of the following must be true?

I. x = y
II. y = 1
III. x and y are prime integers.

(A) None

(B) I only

(C) II only

(D) III only

(E) I and II

For 1: Let us plug-in x=y=1
2x/y = 2, which is not greater than 2

For 2: y=1
Therefore 2x/y = 2x
Plug-in value for x:
x = 1, 2x = 2 which is equal to 2 and not greater than 2

For 3: x and y are prime integers

x = 13 y= 2
2x/y = 13, which is a prime integer greater than 2
x=2 y=13
2x/y = 0.3, which is less than 2 and not a prime integer

Note: If a certain value satisfies the condition, always try with other values before confirming the answer.

Correct Answer : A

For 3: x=13, y=2 satisfied the condition (Where x is a large prime number and y, a small one), but when we plugged in opposite numbers, x=2 and y =13 (Where x is a small prime number and y, a large prime number) the condition failed. Always test with opposite numbers to validate the condition.


PS: i found the above article at F1GMAT site..!!!
O Excellence... my search for you is on... you can be far.. but not beyond my reach!
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by Brian@VeritasPrep » Mon Apr 18, 2011 9:04 am
Hey guys,

Great explanation, manpsingh! Just a few things to add:

1) You asked about whether you should start with B or D. The benefit to doing so is that the answer choices will almost always be in ascending or descending order, so especially if you have a hunch that you need a smaller or larger number you can take fewer steps by choosing B or D. For example, if a question asks about something divided in two and wants the smaller portion, it's quite likely that choice D or E will be a "trap" answer giving you the larger number, so you'll likely want a number closer to A. If you choose B:

-If B is too big, the only smaller number is A, so it must be A. Or, if B is correct, you're done there, too. So you have a 40% chance (or slightly higher, actually, since you may have predicted that you'd need a larger number) of being done in one step.

-If B is too small, then the answer is either C, D, or E, and if you then choose D to split the remaining answer choices, then if D is too big it's C and if D is too small it's E. So you're just about guaranteed to be done in two steps. (Of course, if the equation is a little more nuanced you may not know whether you need a bigger or smaller number...say it's a quadratic or has a variable in multiple spots, it can be trickier...)



2) Perhaps more importantly than just backsolving as a concept is the notion that ANSWER CHOICES ARE ASSETS!! You can use them to backsolve; you can use them to see if a number property will suffice to answer the question (say, does the correct answer have to have a units digit of 5?); you can use them to get an idea of what type of calculation is going to be necessary. One of my favorite quick classroom examples I use - say your calculations got you to 124/93. It's not all that intuitive, I don't think, to immediately recognize that both numerator and denominator are divisible by 31. But you don't have to, because there will be answer choices:

A) 6/5
B) 5/4
C) 4/3
D) 3/2
E) 2/3

If you look at those choices, you'll recognize that A is impossible (you can't factor out the denominator to get a 5, as 93 is not divisible by 5). So is B (124 is not divisible by 5 so you can't get a numerator of 5). Only C is a plausible choice, so you don't actually have to factor in this case.

My point - keep an eye on the answer choices because they tend to give you clues about what you need to do, or they can give you a much more efficient way to work a problem without having to invent all the parameters yourself.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep

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by Tani » Mon Apr 18, 2011 4:48 pm
Backsolving can be used whenever the answers are numbers, not variables. It works on two principles:
1) one of the five answers is right, and
2) the answers are arranged from lowest to highest.

Initially the probability of any given answer being correct is 20%.

If you check B, you have a 20% chance that it is correct. However, if B is too big, the answer must be A, the only smaller number choice. That means you have a 40% chance of getting the right answer on the first try.

Then check D. If D is correct, you are done. If it is too small, the answer is E, and if it is too big the answer is C. Therefore you never need to check more than two answers.

You can start with either D or B and the system will work. Before you start, take a quick look to see whether the answer is likely to be one of the larger or one of the smaller choices. If it is clear that B is too small, start with D (and vice versa).
Tani Wolff
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