Strategic Analysis of GRE Arithmetic

Arithmetic is one of the most important sections of GRE quantitative. You can master the section with logical understanding and developing mathematical skills by practicing more and more.

Terminology for Operations

There are four principal operations of arithmetic

• Addition

It gives the result known as the Sum or total that denotes adding number together, e.g. The sum of 8 and 4 is 12.

• Subtraction

The result given by this operation known as Difference which means deducting one number from another, e.g. The difference between 8 and 4 is 4.

• Multiplication

The result given by this operation known as Product where numbers are being multiplied together, e.g. The product of 8 and 4 is 32. Anything multiplied with 0 gives a product of 0. In question ‘of’ represent the multiplication, e.g. 5 of 6 means 5x6=30.

• Division 

Division gives a result known as Quotient which is obtained by dividing one number by another, e.g. The quotient is 2 when 8 is divided by 4. Anything divided by 0 gives a quotient as 0 but if 0 is divided by anything (x / 0) gives a quotient which is mathematically undefined.

• Order of Operation

PEMDAS is common to remember as the order of arithmetic operation which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction consecutively. It needs to maintain to get the correct answer in the test.

Notions for Numbers

A number in GRE always means a real number. A real number is a number on the number line, that includes whole numbers, fractions, and decimals: all the number shown on line and the numbers aren’t shown existing between every two numbers. They can be positive, negative and zero.

• Integer  
• Integers include all positive and negative whole numbers. The set of integer denotes the positives or the negatives. 
• The fractions or decimal numbers are not integers or whole number.
• Zero is the only integer number that is neither positive nor negative.
• Odd and Even

The numbers are evenly divisible by 2 without any remainder is known as the even number. On the other hand, odd numbers are not evenly divisible by 2 where we get remainder after the arithmetic operation of division.

Important remarks on even and odd numbers are presented for conceptual perception in GRE test:

Even + Even = Even; (4+6= 10);     Even + Odd = Odd; (4+5= 9);

Odd + Even = Odd; (1+4= 5);    Odd + Odd= Even;  (5+7= 12);

Even x Even = Even; (4x6= 24);     Even x Odd = Even; (4x5= 20);

Odd x Even = Even; (1x4= 4);       Odd x Odd= Odd;  (5x7= 35).

• Prime Numbers

The numbers which are evenly divisible only by 1 and itself are known as prime numbers. 2,3,5,7,13 etc. are prime numbers while 0 and 1 are not set to be considered as prime numbers. One should know the way to determine whether a number is a prime number or not:

• Firstly, one should approximately suppose a square root of that number.
• Then, all prime numbers less than the approximate convenient square root is utilized to try to divide the stipulated number.
• If the number is not divisible of any of the prime numbers tried then it is a prime number itself. If the number is divisible any of the primes then it is not a prime number.
• For example, we can consider 97 where we can approximate it to 100 to suppose a square root 10. Now all primes less than 10 are 7,5,3,2. Here, 97 is not divided by any of the four and so 97 is a prime number.
• Factor and Multiple
• Factors of any number are the numbers for which the stipulated number is divisible, e.g. The factors of 21 is 3 and 7; factors of 12 are 2,3,4,6. Reversely, 21 are the multiple of 3 and 7 while 12 is the multiple of 2,3,4 and 6. It should be remembered that if a number X is divisible by a number Q, then P is also divisible by the all factors of Q; e.g. 60 is divisible by 12 and it is also divisible by the factors of 12 i.e. 2,3,4 and 6.

Approximation

Approximations are vital time-saving ways to deal with GRE test with enormous efficacy.  

• Method of Rounding Numbers
• If the number in the tenth's place is 0 to 4, then it rounds down. If the number in the tenth's place is 5 through 9, then it rounds up, directly, e.g. 5.4 gets rounded down to 5. When the number is in the tenths place is 5 or greater, we round up—5.5 gets rounded up to 6.
• It is important in GRE test to remember that any digits to the right of the tenths place do not make any difference. The numbers 5.4, 5.445, 5.494, and 5.499 all have a 4 in the tenths place, so all of them get rounded down to 5.
• Double rounding is so an error if one looks at something like 6.57 and rounds to the nearest integer twice. For example, if anyone looks at 6.57, first rounds the 7's up to 6, and then rounds that up to 7 finally. That is 100% incorrect and invalid. It is a very common mistake in GRE test.
• In order to round to a place, look only at the digit in the next place to the right, the very next smallest place. The one needs to follow the usual rules of rounding off. For example, if one needs to round 85,347 off to a place towards nearer thousand, then he/she considers the number in the immediate lower next position (of thousandth place) i.e hundredth place.  That number is a 3, less than 5 and 3 is between 0 to 4; so it can be rounded down which gives 85,000, as the nearest thousand. If the number is 85,547 then the number or digit in the hundredths place is 5 and so the number is rounded off as 86,000 as the nearest thousand.

Properties of Arithmetic

The fundamental arithmetic properties are common to all real numbers.

• Commutative property
• It denotes the switch of the order. In case of addition, e.: (a+b) =(b+a). Both of the sides will give the same answers. In case of multiplication, it follows and supports the same as, ab=ba.
• So, addition and multiplication are commutative but division and subtraction, in general, are not commutative.
• Associative Property
• The associative Property is the ability to group things. In general the property is represented as: a+(b+c) = (a+b)+c .
• Addition and multiplication are associative. Subtraction and division, in general, are not associative.
• Distribution Property:
• The Distributive Property is an algebra property is used to multiply a single term and two or more terms inside a set of parentheses, e.g. ax(b+c) = ab + ac or, ax(b-c) = ab – ac. All multiplication distributes over addition and subtraction and so it is called the ‘distributing the multiplier’.

Simplification of Operations

Every three digit number can be written as the sum of multiple of 100, multiple of 10 and a single digit number & every two digit number can be written as the sum of a multiple of ten and a single digit number, e.g. 531=5x100+3x10+1 or 53=5x10+3. We can use this idea to simplify and multiply numbers, e.g. 531x60= (5x100+3x10+1)x60=(500+30+1)x60=30000+1800+60=31860 and save time mentally.

Absolute Value

• The definition of absolute value is that it makes the positive and negative numbers ‘positive’. The absolute value of five is 5. The absolute value of negative 5 is also 5.
• The absolute value of zero is zero. Zero's not a positive number.
• In a mathematical sense, the absolute value of a number gives the distance of the number from the origin. The absolute value of five is 5 denotes a distance of five from zero whether in right or left direction on the basis of +5 or -5.
• The absolute value of X is the distance of X from the origin. The absolute value of X-5 is the distance of X from +5. The absolute value of X+5 is the distance of X from -5.
• So, an equation like this, |X-5| > 3 i.e. the absolute value of X minus 5 is greater than 3, provides information as the distance between X and 5 is greater than positive 3. 

The art of arithmetic strategy adapted from this analysis is to advantageous to score smartly in GRE test’s quant section and so one needs to apply these strategies with rules and regulations by regular practices.