GRE Quantitative Comparison & strategy, Introduce with the properties of Real Numbers,Know the rule of thumb about Mental Math, Addition, and Subtraction

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GRE Quantitative Comparison ( QC)

You will find a minimum of 40% Quantitative Comparison math in the real GRE exam, but most of the candidates do not know exactly what this QC wants? In short what QC test?

Let,s talk about the DNA of Quantitative Comparison,

(A) means that Quantity A is always greater, in every way, in all cases. So we can say "A" is arrogant, who does not care about others. Therefore, "A" is America the superpower.  

(B) means that Quantity B is always greater; it couldn't possibly be anything other than greater than Quantity A. So we can say "B" is a competitor, who always loves to compare with "America". Therefore, "B" is Russia the competitor superpower. 

(C) means at all times, in all ways, for all possible values of a relevant variable, the two Quantities are always equal. So we can say "C" is a moderator, who always tries to establish a peach between "A" & "B". Therefore, "C" is India that believes in equality & cooperation. 

(D) Means that things don't always stay one way. In other words, different values of the variables or different numbers within the specified ranges allow different choices that produce different results. So we can say "D" is a destroyer, who always tries to influence others for disagreement. Therefore, "D" is China that believes in inequality. 

 

 How to approach Quantitative Comparison ( QC) questions? 

1.     Do not do more calculations: Always try to simplify options A & B, you should not bother about detailed calculations. For example, see the below math,

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You can cancel "C" from both sides, after that, you have two variables; namely a  & b. Now you can easily say that the two quantity will not be equal, that means, if positive then b is greater, if negative then, a is greater. So, this is a double game that means C. On the other hand, you can simply choose some numbers & put them in the prompt, which will be time-consuming.  

Smart calculation: Closely inspect units of the math, what unit they have used, about, minimum, maximum, approximate. Say math wants an approximate value, then take a round number, for example, you have found a number 10.7, in that case rounding number will be 11. 

Know the matching operations: 

• You can add or subtract the same digit to both quantities i.e. A & B  
• You can multiply or divide by the same positive digit to both quantity i.e. A & B
•  Simplify QC problems before performing any calculations. 

Know the formula: Say, you have encountered circle math & want circumference, if you do not know the formula, how will you solve the math. You may be able to do that in some cases, but knowing the formula will save valuable time.

       

      Memorizing short-cut:

•   If denominators are the same then the greater numerator value of the fraction will be greater.
•   If numerators are the same then the smaller denominators' value of the fraction will be greater.
•   If both quantities show numerical value (fractions, decimals, %) then D will not be an answer.       Answer must be A, B, C

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Key points: 

1. GRE does not care about long calculations. For QC, you have to make quick comparisons.
2. Apply estimation technique, use part-wise comparisons & comparison to round integers.
3.  Look at the QC problems & think how too easily a comparison can make.  

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          Introduce with the properties of Real Numbers:

1.       0 is an even positive number. In GRE, the number means real numbers

2.       A real number can be any number of number line i.e. 0, negative/positive integer, decimals, fractions, and π.

3.       An integer is like a whole number but it can be either greater than 0, called positive, or less than 0, called negative. Zero is neither positive nor negative. Zero is called the origin of the number line, and it's neither negative nor positive. 0 is an integer.

4.       Counting Numbers are whole numbers, but without the zero because we can't "count 0.

5.        Natural Numbers can mean either counting numbers or "whole numbers". But remember, when the test says “number” this number could be any number on the number line. It could be positive, negative, or zero. It could be a whole number or a fraction or a decimal. 

     6. Whole numbers are 0 to positive numbers. It can’t be a negative number, fractions, or decimals.

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What are zero (0) Properties: 

• Multiple of all numbers
• All numbers are factors of 0
• It's a Neutral number
•       It  is a digit

Why is 0 an even number?

If we divide any number by 2 & get an integer or whole number without remainder then that number will be even.

For example, 4÷2=2, here, divide result is without the remainder, so, 2 is a whole number & integer

0÷2=0, here we do not get any of the remainders, so, 0 is also an integer  & an even number.

Case-1: Is 36 a factor of 0 or not?

2×4=8, so, 2 is a factor of 8, also 4 is a factor of 8

8×1=8, so, 8 is a factor of 8, also 1 is a factor of 8

36×0=0, so, 36 is a factor of 0, also 0 is a factor of 0

Know the answer to some questions:

What is non-negative number?

0 & positive integer or 0 with the right side number of the number line

What is non-positive number?

0 & negative integer or 0 with the left side number of the number line

Why 0 is a digit?

There are 10 digits, such as 0-9. Zero (0) falls into these rang, therefore 0 is a digit

What is the difference between digits & numbers?

Digits are base numbers such as 0-9; other hand numbers can be anything that we can put on the number line.  

How many numbers are there among -5 to 5, inclusive?

Count every number on number line with 5

How many numbers are there among -5 to 5, exclusive?

Count every number on number line without 5

If GRE ask

Integer that will be {....-4,-3,-2,-1, 0, 1,2,3,4....} [Fraction, decimal, π will not be integer] 

 Positive integer that will be {1, 2, 3, 4....} & Negative integer= {....-4,-3,-2,-1}

1.      Result of "+"  that will be "Sum"

2.       Result of "-" that will be "Difference"

3.       Result of "×" that will be "Product"

4.        Result of "÷" that will be  "Quotient"  

Commutative Property: The ability to switch the order that means a+b=b+a, here order does not matter.  Addition & Multiplication are commutative but division & subtraction are not generally.  Such as, 7-4≠ 4-7 but 7-4=7+ (-4) = (-4) +7 is commutative.

Associative Property: The ability to group numbers in different groupings that means a+(b+c)=(a+b)+c. Addition & Multiplication is Associative but division & subtraction are not

 Distributive property: a×(b+c)=a×b+a×c  or a(b-c)=a×b-a×c

Trick-1: Multiply or divide by 1 gives the same result. Such as a×1=a    & a/1=a

Trick-2: Multiply by 0 or  Anything times 0= always gives 0. Such as a×0=0

The zero product property: If the product of two numbers is 0, then one of the factors must be 0. Such as If a*b=0, then a=0 OR b=0 Here, OR is a part of math.  

Trick: When any non-zero number is divided by itself, the quotient has to equal 1. If a ≠ 0, then a/a=1 

Know the positive & negative rules: 

Rule-1: (Negative) - (Positive) /-2-(+2) will be the sum with a negative sign or (Negative) + (Negative)/-2+ (-2)  also will be negative sum[You can memorize this by remembering  "NP" or "Double N"]

Rule-2:( Small positive) - (Big positive)/2-(+10) will be minus with positive sign or (Small positive) + (Big negative)/ 2+ (-10) will be minus with negative sign[You can memorize this by remembering SB Positive or SB Negative]

Rule-3: We can always factor out the negative sign.

For example, (-46)-37=-(46+37) =-83

(-22)+ (-61) =-(22-61) =83

Rule-4: Factoring out a negative sign reverses the order of subtraction, such as 23-64=-(64-23)=-41

26-63=-(63-26) =-37

Know the rule of thumb about Mental Math, Addition, and Subtraction

• Every two-digit number can be written as the sum of a multiple of 10 & a single-digit number. Such as 37=30 (multiple of 10) +7(a single-digit number)
• If each digit of the first number (the tens digit & the one's digit) is bigger than the corresponding digit in the second number, then we can separate by digits to simplify the subtraction. Such as 59-31= (50+9) - (30+1)

  = (50 -30) + (9-1) =28

•       If the smaller of the two numbers, the one you subtract has a bigger one's digit, then apply a-b=(a+k) - (b+k)   such as  56-19= 57- 20=37   &  71-26=75-30=45

Know the additional rule of Positive and Negative Numbers 

Multiplication Rules:

(Positive)×(positive) =Positive

(Negative)×(negative) =positive

(Positive)×(negative) =negative

 Same sign=Positive product

 Different signs=Negative product

Divide Rules:

(Positive)÷ (positive) =positive

(Negative)÷ (negative) =Positive

(Negative)÷ (Positive) =negative

Same sign=Positive quotient

Different signs=Negative quotient

The combined sign Rules:

1.     If we multiply or divide numbers with the same sign, the result is positive.

2.      If we multiply or divide numbers with the different signs, the result is negative.

Absolute value: It makes everything positive except zero because zero is neither positive nor negative.

Absolute value=distance from 0

• The absolute value of a number gives the distance of the number from the origin.

E.g. |x|=the distance of x from the origin.

Distance of x from +5= |x-5|

Distance of x from -3= |x-(-3)|

                                    = |x+3|

Distance of x from +2= |x-2|

Distance of x from -2= |x+2|

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