# Basic Operations and PEMDAS

The following are some simple rules to keep in mind regarding whole numbers, fractions, and decimals:

• Ordering is the process of arranging numbers from smallest to greatest or from greatest to smallest. The symbol > is used to represent “greater than,” and the symbol < is used to represent “less than.” To represent “greater than or equal to,” use the symbol ≥; to represent “less than or equal to,” use the symbol ≤.

• The Commutative Property of Multiplication is expressed as a ×  b = b  × a, or ab = ba.

• The Distributive Property of Multiplication is expressed as a(b + c) = ab + ac.

• The order of operations for whole numbers can be remembered by using the acronym PEMDAS

PEMDAS:

• P - First do the operations within the parentheses, if any.

• E - Next, do the exponents.

• MD - Next, do the multiplication and division from left to right.

• AS - Finally, do the addition and subtraction from left to right.

• When a number is expressed as the product of two or more numbers, it is in factored form. Factors are all of the numbers that will divide evenly into one number.

• A number is called a multiple of another number if it can be expressed as the product of that number and a second number. For example, the multiples of 4 are 4, 8, 12, 16, etc., because 4 × 1 = 4, 4 × 2 = 8, 4 × 3 = 12, 4 × 4 = 16, etc.

• The Greatest Common Factor (GCF) is the largest integer that will divide evenly into any two or more integers.

• For example, the GCF of 24 and 36 is 12, because 12 is the largest integer that will divide evenly into both 24 and 36.

• The Least Common Multiple (LCM) is the smallest integer into which any two or more integers will divide evenly.

• The LCM of 24 and 36 is 72, because 72 is the smallest integer into which both 24 and 36 will divide evenly.

• Multiplication

• Multiplying and dividing both the numerator and the denominator of a fraction by the same nonzero number will result in an equivalent fraction.

• When multiplying fractions, multiply the numerators to get the numerator of the product, and multiply the denominators to get the denominator of the product. For example, 35×78=2140.35×78=2140.

• Divison

• To divide fractions, multiply the first fraction by the reciprocal of the second fraction. For example, 13÷14=13×41,13÷14=13×41, which equals 43.43.

• When adding and subtracting like fractions, add or subtract the numerators and write the sum or difference over the denominator. So, 18+28=38,18+28=38, and 47−27=2747-27=27

• When adding or subtracting unlike fractions, first find the Least Common Denominator. The Least Common Denominator is the smallest integer into which all of the denominators will divide evenly.

• For example, to add 3434 and 56,56, find the smallest integer into which both 4 and 6 will divide evenly. That integer is 12, so the Least Common Denominator is 12.

• Multiply 3434 by 3333 to get 912,912, and multiply 5656 by 2222 to get 1012.1012. Now add the fractions: 912+1012=1912,912+1012=1912, which can be simplified to 1712.1712.

• Place value refers to the value of a digit in a number relative to its position. Moving left from the decimal point, the values of the digits are 1s, 10s, 100s, etc. Moving right from the decimal point, the values of the digits are 10ths, 100ths, 1000ths, etc.

• When converting a fraction to a decimal, divide the numerator by the denominator.

Here’s an example question: